STUDY OF THE STRESS STATE UNDER RADIAL COMPRESSION OF A CYLINDER

Yu. S. BOGOMOLOV, L. M. SEDOKOV, L. G. TSEKHANOVA

(Recommended for publication by the scientific seminar of the Department of Strength of Materials)

Analysis of the stress state of cylinders under radial compression between parallel press plates in the plastic region is of great interest, since it serves as the basis for studying the processes of transverse forging and rolling.

The study was carried out on specimens of strongly work-hardening steel Kh18N9T by the hardness-measurement method. The principal characteristics of this steel are as follows: yield strength \(26\ \text{kg}/\text{mm}^2\), ultimate strength \(62\ \text{kg}/\text{mm}^2\), conventional relative elongation \(\delta = 46.7\%\), and conventional relative reduction \(\psi = 55\%\). The choice of steel for setting up the experiment is explained by the very considerable change in its hardness during plastic deformation.

Hardness was measured on a hardness tester on the \(D\) scale under a load of \(100\ \text{kg}\). This scale proved to be the most convenient for the selected material, with an initial hardness \(H_{\mathrm{B}} = 150\). To preserve similarity of the impressions, a diamond conical indenter was used.

The relationship between the hardness of the steel and the degree of its deformation (the intensity of the stressed state) was determined under linear compression.

In order to obtain significant deformations under axial compression and to avoid any substantial influence of friction at the ends, the specimens were deformed successively in seven stages. The initial dimensions of the specimens—height \(h_0 = 60\ \text{mm}\) and diameter \(d_0 = 24\ \text{mm}\)—were selected with the ratio

\[ \frac{h_0}{d_0} = 2.5. \]

This ratio was adopted on the basis of an analysis of compression curves obtained by M. A. Bolshanina [1]. The use of specimens with an even larger height-to-diameter ratio would lead to loss of stability of the specimen during its plastic deformation.

In the first stage, loading is carried out in steps of 10 tons. After each increase in load, the dimensions of the deformed specimen and its hardness are measured. When the ratio

\[ \frac{h_i}{d_{\max}} \approx 1, \]

which corresponds to a load of 100 tons, the deformed specimens were remachined. Here and subsequently, \(h_i\) denotes the height of the specimen changing during the deformation process, and \(d_{\max}\) its maximum diameter at the same moment. During remachining, the height of the specimens remained ...

remains unchanged, while the diameter is reduced to the value \(d_i\) so as to obtain the ratio \(\frac{h_i}{d_i}\) equal to 1.5. As the experiments showed, specimens with diameter \(d_i < 20\) mm at a ratio \(\frac{h_i}{d_i} > 1.5\) under loads of several tens of tons become skewed. Therefore, in what follows, at the remaining loading stages the ratio \(\frac{h_i}{d_i}\) is taken to be no more than 1.5.

After machining at the second stage, the specimens were again subjected to compression until the ratio \(\frac{h_i}{d_i} \simeq 1\) was reduced, and both the hardness and the changing dimensions of the specimens were measured. This was followed by a second machining with a reduction in the specimen diameter, in order to obtain the ratio \(\frac{h_i}{d_i}\) equal to 1.5; then came the next stage of deformation, etc.

At the seventh stage of deformation, after the sixth machining, the dimensions of the specimens were \(h_i = 7.5\) mm and \(d_i = 5\) mm.

The stress intensity \(\sigma_i\) under axial compression is equal to the true compressive stress

\[ \sigma_i = S_{\text{comp}} = \frac{P}{F_0}(1 - \varepsilon_{\text{comp}}). \tag{1} \]

Here \(P\) is the load in kg,
\(F_0\) is the area of the initial section in \(\text{mm}^2\),
\(\varepsilon_{\text{comp}}\) is the relative deformation in compression.

The strain intensity under axial compression is determined from the relation:

\[ e_i = \ln \frac{h_0}{h_1} + \frac{\sigma_i}{3G} \simeq \ln \frac{h_0}{h_1}, \tag{2} \]

where \(h_0\) is the initial height of the specimen before deformation, in mm; \(h_1\) is the height of the specimen after application of the load, in mm; \(G\) is the elastic shear modulus, kg/mm\(^2\).

Fig. 1. Dependence between stress (strain) intensity and hardness.

On the basis of the experimental results, curves have been plotted for the dependence of the stress intensity and strain intensity on hardness (Fig. 1). The calibration graph obtained should be regarded as approximate, since, despite the measures taken, the deformation of the specimens was nonuniform, and the hardness was measured at the end faces, where, as is known [1], the deformation is somewhat less than that determined by formula (2).

From the theory of small elastic-plastic deformations it is known [2] that the stress intensity is a quite definite function of the strain intensity, independent of the character of the stress...

state. As a first approximation, it may be assumed that the dependence of stress on hardness (Fig. 1) under axial compression can be extended to the case of radial compression, which represents a plane stress state.

Thus, the graphs in Fig. 1 can be used so that, from the magnitude of the changing hardness of the plastically deformed cylinder, one may judge the intensity of deformation (stress).

The main series of experiments was carried out on 6 disks 45 mm in diameter and 20 mm long. These disks were subjected to radial compression, i.e., they were compressed between two parallel plates in such a way that the axis of the specimens remained parallel to the press plates. The magnitude of the load determining the degree of compression was 11.5, 20, 30, 50, 70, and 95 tons; the conventional relative deformation varied from 2.7 to 39%.

The predeformed specimens were faced by several millimeters in order to remove metal flows caused by deformation along the axis of the disks, and were ground. A grid of scratches with a pitch of 1.5–2 mm was made on the ends of the specimens with a scribing tool. At the points of intersection of the scratches, hardness was measured on half the area of the end faces, owing to the symmetry of the disks. For some specimens (compressed under loads of 20 and 30 tons), in order to check and average the results, hardness was measured over the entire area of the disk end face.

The data obtained on the magnitude of hardness at various points on the end faces of the specimens made it possible to construct isoclines (lines of equal hardness). The patterns of the isoclines (Fig. 2) correspond rather closely to photographs of isochromatics (lines of principal shear stresses) presented in [3] for the case of forging on rectangular dies. This circumstance makes it possible to conclude that, despite a number of assumptions, E. N. Unksov and V. N. Zavartseva obtained, by the photoelasticity method, a quite realistic qualitative picture of the distribution of stresses over the cross section of a shaft.

The processing of the research results for determining the stresses at points of the transverse section of the specimens was performed in two calculation variants. In the first variant, the dependence of the magnitude of the maximum shear stresses under axial compression on hardness was taken as the basis. The maximum shear stress for linear compression is

\[ \tau_{\max} = \frac{S_{\text{ж}}}{2} = \frac{\sigma_i}{2}, \tag{3} \]

therefore its magnitude is easily determined from the graph (Fig. 1).

Extending the dependence between the maximum shear stresses and hardness under linear compression to the case of radial compression of a disk, one can determine, from the changed hardness value, the values of the maximum shear stresses at points of the transverse section of the specimens.

In the second calculation variant, from the changed hardness at points of the transverse section of the cylinder under radial compression, the stress intensity is determined. To pass from the stress intensity \(\sigma_i\) to the principal normal and maximum shear stresses, the stress-distribution epures and the ratios of the principal stresses given in [3] were used.

The difference between the maximum shear stresses determined by the two calculation variants is small and amounts to 5–7%. Consequently, the proposed method for determining the compo-

Fig. 2. Isoclines at various degrees of compression of the specimens.

Visible subfigure labels:

• \(\alpha = 21.5^\circ\)
• \(\alpha = 37.5^\circ\)
• \(\alpha = 59^\circ\)
• \(\alpha = 73^\circ\)
• \(\alpha = 88^\circ\)
• \(\alpha = 110^\circ\)

Stress scales and labels shown in the diagrams include \(\tau\), \(\sigma_1\), and \(\sigma_3\), in \(\mathrm{kg/mm^2}\).

...components of the stressed state by hardness testing may be regarded as satisfactory.

The investigation carried out makes it possible to record the initial foci of the onset of plastic deformation and to estimate the degree of nonuniformity in the transition of the metal to the plastic state. Thus, Fig. 3 shows the dependence of the stress intensity for several points of the specimen cross section on the line load. These graphs are shown for those characteristic points at which, at different stages of deformation, the stress intensity was maximal.

At the first stage of deformation, under a load of up to \(600\ \mathrm{kg/mm}\), the maximum stress intensity is observed at point \(A\), lying practically on the area over which the compressive force acts. Subsequently, with an increase in the line load from \(1000\) to \(1500\ \mathrm{kg/mm}\), the maximum stress intensity is noted at point \(B\). Thus, the wave of propagation of the greatest stress intensity, and consequently of the greatest deformations, passes from the surface into the depth of the cylinder.

Fig. 3. Dependence of stress intensity for points of the specimen cross section on the magnitude of the line load.

Finally, when the line load is increased to \(2500\ \mathrm{kg/mm}\), the maximum stress intensity turns out to be at the center of the disk, at point \(O\). After this, a further increase in the load (in the experiments it reached \(4700\ \mathrm{kg/mm}\)) does not lead to a change in the position of the zone of maximum stress intensity. It remains in the central part of the cylinder.

Thus, as the degree of deformation increases, the yielding of the surface layers is at first maximal, then is retarded and proceeds inward, spreading to the central part of the cylinder. Evidently, unworked metal will exist until the maximum intensity appears at the center of the cylinder. This degree of deformation is characterized by a line load of \(2500\ \mathrm{kg/mm}\) and a conventional relative deformation \(\delta = 20\%\).

LITERATURE

1. V. D. Kuznetsov. Physics of the Solid State. Vol. 2, Tomsk, TSU Publishing House, 1941.
2. G. A. Smirnov-Alyaev. Resistance of Materials to Plastic Deformation. Mashgiz, 1961.
3. E. P. Unksov, V. M. Zavartseva. Investigation of stresses in metal during forging of shafts and rods. Vestnik mashinostroeniya, No. 3, 1955.