Determination of shear force in cutting at microspeeds
Abstract Generated abstract
This paper addresses the determination of shear force during metal cutting at microspeeds by relating chip deformation to stress and strain intensities in the plastic deformation zone. It develops a calculation scheme using chip shrinkage, the shear-plane angle, relative shear, and the flow curve of the work material, and examines whether the flow curve obtained in cutting can be replaced by that from compression or tension tests. Experiments on steel 15 and brass L62 used grid distortion and hardness measurements in chip roots, together with calibration from mechanical tests, to compare stress and strain states. The results support use of a single flow curve for cutting and compression or tension under comparable temperature and deformation-rate conditions, enabling shear force to be calculated from standard test data.
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PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
OF THE ORDER OF THE OCTOBER REVOLUTION
AND THE ORDER OF THE RED BANNER OF LABOR
named after S. M. Kirov
Vol. 224 1976
DETERMINATION OF THE SHEAR FORCE IN CUTTING AT MICROSPEEDS
G. L. KUFAREV, M. G. GOLDSHMIDT, V. A. GOVORUKHIN, F. K. PLESHAKOV
(Submitted by the scientific seminar of the departments of
machine tools and metal cutting and mechanical-engineering technology)
The shear force in cutting is understood to mean the force \(P_{\tau}\) acting along the upper boundary of the plastic-deformation zone and determining the degree of plastic deformation of the chip metal. With a sufficiently high degree of accuracy, the shear force is determined by the equation
\[ P_{\tau}=\tau \cdot \frac{a \cdot b}{\sin \Phi}, \tag{1} \]
where
\(\tau\) is the tangential stress at the upper boundary of the plastic-deformation zone;
\(a, b\) are the thickness and width of the layer being cut;
\(\Phi\) is the angle determining the direction of the upper boundary of the plastic-deformation zone.
If the chip shrinkage \(\xi\) is assumed to be known, then the angle \(\Phi\) is determined by the relation
\[ \tan \Phi=\frac{\cos \gamma}{\xi-\sin \gamma}, \tag{2} \]
where \(\gamma\) is the rake angle of the cutter.
In this formulation, the problem of determining the shear force reduces to determining the tangential stress \(\tau\) for a known degree of deformation of the chip metal. The degree of deformation of the metal in the cutting process should be estimated either by the value of the relative shear [1],
\[ \varepsilon=\frac{\xi^{2}-2\xi \cdot \sin \gamma+1}{\xi \cdot \cos \gamma}, \tag{3} \]
or by the value of the strain intensity \(e_i\).
Since deformation in the cutting process is carried out according to the scheme of simple shear, the relative shear is numerically equal to the true shear [2], and then the strain intensity in cutting should be determined from the relation [3].
\[ e_i=\frac{\varepsilon}{\sqrt{3}}. \tag{4} \]
It may be noted here that the process of simple shear is one type of complex loading, and therefore the quantity \(e_i\), determined in the general case by integrating the derivative of the strain intensity, whose value depends on the derivatives of the nine components of the strain tensor with respect to the parameter determining the change in the stress tensor, most fully reflects the physical content of the deformation process.
A generalized characteristic of the stressed state of a material in any deformation process is the stress intensity, \(\sigma_i\). For the case of plane deformation it is related [4] to the value of the shear-stress intensity \(K\) by the relation
\[ \sigma_i=\sqrt{3}\cdot K. \]
The upper boundary of the plastically deformed zone in cutting is a slip line, for which the equality \(K=\tau\) is valid. Then
\[ \sigma_i=\sqrt{3}\tau \quad \text{or} \quad \tau=\frac{\sigma_i}{\sqrt{3}}. \tag{5} \]
Thus, in order to determine the shear force at a known degree of deformation of the chip metal, it is necessary to know the dependence \(\sigma_i—e_i\) in the cutting process (for the case of simple shear).
The hypothesis of a single flow curve assumes that, for a strain-hardening material, the functional relationship between the stress and strain intensities \(\sigma_i=f(e_i)\) is, with practically sufficient accuracy, single-valued and independent of the type of stress–strain state. If the use of this hypothesis is justified, then the stress \(\tau\) in cutting can be determined from the results of static tensile or compressive tests of the metal being machined.
In the cutting laboratory of TTI, an experimental verification was carried out of the hypothesis of a single flow curve in the processes of cutting (plane deformation) and compression and tension (a volumetric stress–strain state), under practically identical temperature–velocity deformation conditions.
All studies in which the coincidence of the flow curve in cutting and compression has been checked up to the present time are methodologically insufficiently correct, since what was compared with the flow curve obtained in compression was not the flow curve in cutting, but only the final characteristics of deformation of the chip metal in cutting. Each of the experimental points for cutting, characterized by its coordinates \(\sigma_i\) and \(e_i\), was obtained either under changed machining regimes and cutting conditions, or with a different geometry of the cutting part of the tool.
Studies of deformation in cutting by the method of dividing grids, in combination with the method for determining stress intensity by hardness measurement [5], make it possible to verify the hypothesis of a single flow curve by superimposing on the curve \(\sigma_i—e_i\), obtained in compression or tensile tests, the points characterizing the successively stressed–deformed state of an elementary volume of metal as it moves in the plastic zone of chip formation from its lower to its upper boundary.
When carrying out compression tests according to the procedure set forth in [5], the stress intensity was determined by the formula
\[ \sigma_i=\frac{4P}{\pi \cdot D^2}\ \text{kg/mm}^2, \]
where \(P\) is the current value of the compressive force, in kg,
\(D\) is the current value of the specimen diameter, in mm,
and the strain intensity is given by the formula
\[ e_i=\ln \frac{h_0}{h}, \]
where \(h_0\) and \(h\) are the initial and current height of the specimen.
In the study of sheet steel 15, the dependence \(\sigma_i\)—\(e_i\) was obtained by tensile testing ten specimens, which in the course of testing were brought to different degrees of deformation. The length of the specimens in this case was 10 times greater than the width, which ensured a simple tension scheme. The elongation of the specimen during deformation, \(\Delta l_0\), was recorded by the distance between scratches made on its surface. The stress and strain intensities were determined, respectively, by the formulas
\[ \sigma_i=\frac{P}{F}, \qquad e_i=\ln \frac{l_0+\Delta l_0}{l_0}, \]
or, in the case of neck formation,
\[ e_i=\ln \frac{F_0}{F}. \]
Here, \(l_0\) is the distance between scratches before deformation; \(F_0\) and \(F\) are the initial and minimum current cross-sectional areas of the specimen.
Based on the results of compression testing (tension testing), flow curves \(\sigma_i\)—\(e_i\) and graphs \(\sigma\)-H (hardness) were constructed for brass L62 and steel 15.
When cutting flat specimens at microspeeds, with a preliminary marked dividing grid, prepared from these metals, chip roots were obtained.
On a PMT-3 instrument, the hardness of the plastically deformed zone of each of these chip roots was measured, and lines of constant hardness were constructed. Using the calibration graph \(\sigma_i\)-H, obtained in compression experiments, the value of \(\sigma_i\) at each point of the plastically deformed zone was determined from the measured hardness. The strain intensity of each point of this zone was calculated from the distortion of the dividing grid. Thus, for each point of the zone plastically deformed during cutting, there was both a characteristic of the stress state—\(\sigma_i\), and a characteristic of deformation—\(e_i\).
Since, in the process of continuous chip formation, each row of points parallel to the cutting-speed direction during deformation characterizes the entire deformation history of a point on a given horizon from the beginning of plastic deformation to the upper boundary of the plastic zone, the flow curve in cutting was constructed from the characteristics of individual points of such a row or of several rows.
In Fig. 1, the solid line represents the flow curve obtained in tensile testing of steel 15; the experimental points describe the flow curves in cutting for metal particles located on different horizons through the thickness of the layer being removed.
Fig. 1.
Fig. 2
Fig. 3
In Fig. 2, the characteristics of points from one row of the chip root (dark points) are superimposed on the flow curve for compression of brass L62 (solid line), as well as points (light points) characterizing the final degree of deformation of chips removed during cutting at microspeed with cutters of different geometry and with different lubricants. For the latter points, \(\sigma_i\) was calculated from the forces measured by means of a dynamometer designed by VNII, and \(e_i\) from chip shrinkage. Similar results were also obtained in a special series of experiments on compression and cutting of brass L62 at an elevated test temperature (Fig. 3). The experimental (light) points for cutting in Fig. 3 characterize
also give only the terminal relationship \(\sigma_i - e_i\) for the fully deformed material of the chip.
The experimental data described make it possible to regard the hypothesis of the unity of the flow curve in the coordinates \(\sigma_i - e_i\) in cutting and in compression (tension) as valid, provided that the temperature–velocity regime of deformation remains constant.
This conclusion makes it possible to calculate the shear force in cutting at microspeeds in the following way. From the chip shrinkage, the angle \(\Phi\) is calculated (formula 2), as are the relative shear (formula 3) and the strain intensity (formula 4). From the value of \(e_i\) found, \(\sigma_i\) is determined from the flow curve constructed in a compression (tension) test; this, using equality (5), makes it possible to determine the magnitude of the shear stress and, consequently, to calculate the shear force by formula (1).
REFERENCES
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A. M. Rosenberg, A. N. Eremin. Elements of the Theory of the Metal-Cutting Process. Mashgiz, Moscow, 1956.
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I. P. Renne. On the question of taking account of the loading method in analyzing the deformed state. Proceedings of the Tula Mechanical Institute, issue 8, Oborongiz, Moscow, 1958.
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V. G. Osipov. On the characteristics of finite deformations, in the collection Problems of Metallurgy, Academy of Sciences of the USSR, Moscow, 1953.
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L. M. Kachanov. Fundamentals of the Theory of Plasticity. Moscow, GITTL, 1956.
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G. D. Del, M. G. Goldshmidt. Determination of stresses in the cutting zone. Izvestiya TPI, vol. 162, 1967.