Stress State Under Radial Compression of a Cylinder
Abstract Generated abstract
This paper investigates the stress state along the vertical axis of symmetry in cylindrical steel specimens subjected to radial compression between parallel press plates, a problem relevant to transverse forging, rolling, and strength testing of brittle materials. Using measured strains, hardness based stress calibration, and flow curves from axial compression tests, the authors determine principal stresses for work hardening St. 3 steel disks of different sizes and degrees of upsetting. The results show that compressive vertical stresses increase with upsetting and reach a maximum near the contact area, while horizontal stresses may change from compression near contact to tension toward the center. The study also indicates that the coefficient used to estimate strength in radial compression depends strongly on material plasticity, less on work hardening, and only weakly on specimen dimensions.
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PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV
Volume 173 1970
STRESS STATE UNDER RADIAL COMPRESSION OF A CYLINDER
Yu. S. BOGOMOLOV, G. D. DEL
(Presented by the scientific seminar of the Department of Strength of Materials)
Knowledge of the stress state under radial compression of cylindrical specimens between parallel plates is necessary for carrying out such pressure-working processes of metals as transverse forging and rolling more rationally, and also for substantiating more reliable methods of mechanical testing for a number of brittle materials.
The literature gives results of experimental investigations of stresses along the horizontal axis of symmetry [1, 2, 3]. However, both in the theory of pressure working of metals and in connection with testing brittle materials under radial compression, investigations of stresses along the vertical axis of symmetry are of the greatest importance. This is connected with the fact that plastic deformation originates in the vicinity of the contact areas, encompassing an ever larger part of the vertical axis of symmetry, and only at large degrees of upsetting does it extend to the entire horizontal axis of symmetry. When testing brittle materials, a crack arises at various points of the vertical axis of symmetry.
The analytical solution of A. D. Tomlenov [4], as a result of which the stresses along the vertical axis of symmetry were calculated, was obtained for an ideally plastic material, which somewhat limits its range of application.
The main task of the present work is the experimental determination of stresses along the vertical axis of symmetry of a cylinder made of a work-hardening material, compressed radially between parallel plates of a press.
The investigation was performed according to the method described in article [5]. To calculate stresses by this method it is necessary to know the transverse strain \(e_z\) and the intensities of strains and stresses. Experiments were carried out on disks of steel St. 3 with diameters of 160 and 32 mm and thicknesses of 50 and 10 mm, which made it possible to determine the influence of specimen dimensions on the nature and magnitude of the stressed-strained state.
To determine the strain \(e_x\), parallel scratches were applied along the axes of symmetry of the specimens. Measuring the distances between the scratches with a toolmaker’s microscope at various points of the disk after its upsetting, we find from the relation:
\[ e_x = \ln \frac{a}{a_0}, \]
where \(a_0\) and \(a\) are the distances between the scratches before and after upsetting.
Measurement of the transverse strain \(e_z\) was carried out with an indicator along the axes of symmetry of the disk when it was placed on a horizontal plate. The calculation formula was
\[ e_z=\ln \frac{h}{h_0}, \]
where \(h_0\) and \(h\) are the thickness of the disk before and after upsetting.
The strain \(e_y\) was determined from the condition of constancy of volume:
\[ e_x+e_y+e_z=0. \]
The strain intensity is
\[ e_i=\frac{\sqrt{2}}{3}\sqrt{(e_x-e_y)^2+(e_y-e_z)^2+(e_z-e_x)^2}. \]
Knowing the strain intensity, one can determine the stress intensity \(\sigma_i\) from the flow curve obtained in linear compression, as follows. From the material under investigation, 8 cylindrical specimens were made; these were compressed with lubrication in the axial direction to different degrees of upsetting. From the known relations, the stress intensities \(\sigma_i\) and strain intensities \(e_i\) were determined for each of the specimens (Fig. 1). At several points of the meridional section of the specimens, the Rockwell hardness \(HRB\) was determined. From the experimental results a calibration graph \(\sigma_i\)—\(HRB\) was constructed. One of the disks was compressed along the radius by 30%. From the distortion of the grid of divisions applied to the surface of the disk before upsetting, the strain intensity was determined at various points. Measurement of hardness at these same points of length made it possible, by means of the calibration graph \(\sigma_i\)—\(HRB\), to determine the stress intensity. The values of \(\sigma_i\) and \(e_i\) thus obtained at various points of the disk were plotted on the flow curve obtained in linear compression and gave a scatter within the limits of the natural experimental error (Fig. 1). This made it possible to use the flow curve for linear compression in our study.
Fig. 1. Flow curve of St. 3: ○ — axial compression, × — radial compression
The principal stresses are determined from the relation [5]:
\[ \sigma_{1,2}=\frac{\sigma_i}{e_i}\left(-e_z' \pm \sqrt{\frac{e_i^{\prime 2}-e_z^2}{3}}\right). \]
Upsetting of the disks under test was carried out with lubrication consisting of a mixture of graphite and glycerin in order to reduce the possibility of the appearance of stresses along the \(Z\) axis, i.e., to obtain a plane stress state. A disk of diameter 160 mm was upset by 3, 5, 6, and 11.7%.
The nature of the change in the deformed state, studied in sufficient detail, was confirmed in the present work and is not considered in detail.
Figure 2 presents the diagrams of the principal stresses along the axes of symmetry of the disk. The stresses \(\sigma_x\) at the contact area are compressive—
are compressive and amount to 18–28 kg/mm\(^2\), depending on the degree of upset. Then they change sign and become tensile in the central part of the disk. The maximum of the stress \(\sigma_x\) at an upset of 11.7% is equal to 25 kg/mm\(^2\) and is located at the center of the disk. At smaller upsets, the maximum of the stress \(\sigma_x\) lies approximately at a distance of one third of the disk height from the center. The maximum of the stress \(\sigma_y\), amounting to 34–60 kg/mm\(^2\), is located at the contact area; the minimum is at the center of the disk.
Fig. 2. Plots of the principal stresses along the axes of symmetry of cylinders compressed radially between parallel plates.
Cylinder \(d = 160\) mm, \(l = 50\) mm.
Along the vertical axis: \(\circ\) — principal stresses at \(\delta = 3\%\), \(\times\) — principal stresses at \(\delta = 5.6\%\), \(\triangle\) — principal stresses at \(\delta = 11.7\%\).
Along the horizontal axis: \(\odot\) — \(\sigma_x\) at \(\delta = 11.7\%\), \(\bullet\) — \(\sigma_y\) at \(\delta = 11.7\%\).
Cylinder \(d = 32\) mm, \(l = 10\) mm, \(\square\) — principal stresses at \(\delta = 11.7\%\).
To estimate the dependence of the stress state on the specimen diameter, the stresses were determined at an upset of 11.7% for a disk of diameter 32 mm. The calculation results, shown in Fig. 2, indicate a comparatively small change in the stress state as the cylinder dimensions change.
Previously, a method was proposed for determining the strength limit of brittle materials under radial compression of cylindrical specimens.
The strength is estimated by the formula:
\[ \sigma_{\mathrm{tch}} = A \frac{P}{d l}, \]
where \(P\) is the breaking force,
\(d\) and \(l\) are the diameter and length of the specimen.
For a number of brittle materials, the coefficient \(A\), according to the first theory of strength and in accordance with the well-known Hertz solution, is approximately 0.64. For materials possessing some plasticity, an increase in the indicated coefficient has been observed. The coefficient \(A\), apparently,
also depends on the strain hardening of the material and, to some extent, on the size of the specimens.
The investigation carried out makes it possible to estimate the influence of plasticity and strain hardening of the material on the magnitude of the coefficient \(A\).
Taking as the strength criterion the greatest tensile stress according to the first theory of strength, one can obtain a graph of the dependence of the coefficient \(A\) on the degree of upsetting (Fig. 3). It is evident from the graph that the coefficient \(A\) does not remain constant and depends substantially on the plasticity of the material.
Fig. 3. Graph of the dependence of the coefficient \(A\) on the upsetting of the cylinder
To estimate the influence of strain hardening, a calculation of stresses was performed for \(\sigma_i=\mathrm{Const}\), i.e., for an ideally plastic material. The stresses were determined from the deformation of the disk mentioned above, of diameter \(160\ \mathrm{mm}\), upset by \(11.7\%\). In doing so, in accordance with previously performed studies [6], it was assumed that the deformed state is practically independent of the properties of the material.
As a result of the calculation it was established that, for the same upsetting, in a non-strain-hardening material the coefficient \(A\) is \(25\%\) lower than in a brittle material of the same strength as St. 3.
Conclusions
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In radial compression of a disk, the stresses \(\sigma_y\) along the vertical axis are compressive and increase with increasing upsetting. They attain their maximum value at the contact area, and their minimum value at the center of the cylinder. The stresses \(\sigma_x\), at small upsettings, are negative at the contact area and positive at the center of the cylinder.
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The ratio of the strength limit to the average stress in the diametral section when testing brittle materials in radial compression
compression depends strongly on the plasticity of the material, to a somewhat lesser extent on work-hardening, and practically does not depend on the dimensions of the specimen.
REFERENCES
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E. P. Unksov, V. M. Zavartseva. Investigation of stresses in metal during the forging of shafts and bars. Vestnik mashinostroeniya, No. 3, 1955.
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Yu. S. Bogomolov, L. M. Sedokov, L. G. Tsekhanova. Investigation of the stress state during transverse forging and rolling by the hardness-measurement method. Kuznechno-shtampovochnoe proizvodstvo, No. 6, 1964.
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Yu. S. Bogomolov, G. D. Del, L. M. Sedokov, L. G. Tsekhanova. Investigation of radial compression of a cylinder by hardness measurement. Izv. TPI, vol. 133. TGU Press, 1965.
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A. D. Tomlenov. On the features of calculating the stress state arising during forging with flat dies. Vestnik mashinostroeniya, No. 3, 1959.
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G. D. Del. Investigation of plastic deformation by hardness measurement. Proceedings of the Tomsk Polytechnic Institute, vol. 138, 1966.
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Yu. S. Bogomolov. Influence of the mechanical properties of the material on the deformed state of cylindrical specimens under radial compression, (in press).