Investigation of Radial Compression of a Cylinder by Hardness Measurement
Abstract Generated abstract
Radial compression of a cylinder between parallel plates is examined for the plane strain conditions typical of forging and rolling, where stresses and strains are highly nonuniform. The study uses hardness measurements on compressed steel specimens, calibrated by torsion tests, and interprets the resulting stress intensity field through a slip-line based solution of the equilibrium equations. The method yields slip-line networks and distributions of principal stresses and strains for several reductions, with calculated forming forces agreeing reasonably with measured loads at smaller reductions but showing larger error at the highest reduction. The results confirm earlier qualitative findings and indicate that the central stress state changes from mixed tension and compression at small reductions to stresses of the same compressive sign at larger reductions.
Full Text
INVESTIGATION OF RADIAL COMPRESSION OF A CYLINDER BY HARDNESS MEASUREMENT
Yu. S. Bogomolov, G. D. Del, L. M. Sedokov, L. G. Tsekhanova
(Presented by the scientific seminar of the Department of Strength of Materials)
In the process of radial compression of a cylinder between parallel plates, which is one of the common cases in metal forming by pressure, a very nonuniform field of stresses and strains is produced.
The theoretical distribution of stresses along the vertical diameter of an ideally plastic cylinder was obtained by A. D. Tomlenov [1].
Papers [2, 3] present the results of an experimental investigation of the stressed and strained states by the photoelastic method [2] and by hardness measurement [3]. For the numerical determination of the magnitudes of the principal stresses from the results of hardness measurements, paper [3] used their ratios obtained by the photoelastic method, when the degree of plastic deformation was simulated by removing a flat and an elastic state close to the limiting one was considered [2]. The ratio of the length to the diameter of the cylinder was small, and the stressed state was close to plane stress.
In the present work, radial compression under conditions of plane strain, which are more characteristic of forging and rolling processes, is investigated by hardness measurement. The results of hardness measurement were decoded by the method described in [4].
The essence of the decoding method consists in solving, by the slip-line method, the system of differential equilibrium equations when the intensity of shear stresses \(K\) is known at various points.
Relations on the slip lines:
\[ \frac{\sigma}{2k} - \Theta = -A_{\alpha}, \qquad \frac{\sigma}{2k} + \Theta = -A_{\beta}, \tag{1} \]
where \(\sigma\) is the hydrostatic pressure, and \(\Theta\) is the angle of inclination of the slip line to the \(x\)-axis. The quantities \(A_{\alpha}\) and \(A_{\beta}\) are determined by approximate integration. Thus, for example, at point 3 in Fig. 3, \(A_{\alpha}\) and \(A_{\beta}\) are found from the known \(A_{\alpha}\) and \(A_{\beta}\) at points 1 and 2 from the relations:
\[ \left. \begin{aligned} (A_{\alpha})_{3} &= (A_{\alpha})_{2} -\frac{1}{2k_{3}}\cdot \frac{k_{1}-k_{3}}{l_{1-3}}\cdot l_{2-3};\\[6pt] (A_{\beta})_{3} &= (A_{\beta})_{1} -\frac{1}{2k_{3}}\cdot \frac{k_{2}-k_{3}}{l_{2-3}}\, l_{1-3}, \end{aligned} \right\} \tag{2} \]
where \(l_{1-3}\) and \(l_{2-3}\) are the distances between points 1 and 3, and 2 and 3.
From the known \(A_\alpha\) and \(A_\beta\), \(\Theta\) and \(\sigma\) were determined:
\[ \left. \begin{aligned} \Theta &= \frac{1}{2}(A_\alpha - A_\beta),\\ \sigma &= -k(A_\alpha + A_\beta). \end{aligned} \right\} \tag{3} \]
Stress components:
\[ \left. \begin{aligned} \sigma_x &= \sigma - k\sin 2\Theta,\\ \sigma_y &= \sigma + k\sin 2\Theta,\\ \sigma_{xy} &= k\cos 2\Theta. \end{aligned} \right\} \tag{4} \]
The study of the stress–strain state under radial compression of cylindrical bodies was carried out on specimens made of self-hardening steel Kh18N9T \((\sigma_{\mathrm{t}} = 26\ \text{kg}/\text{mm}^2;\ \sigma_{\mathrm{uts}} = 62\ \text{kg}/\text{mm}^2;\ \delta = 46.7\%;\ \psi = 70\%)\).
Hardness was measured on a hardness tester with a diamond cone under a load of \(100\ \text{kg}\), with readings on the D scale. The initial hardness of the steel was \(HRD = 24\), which corresponds to \(HB = 150\).
A calibration curve relating hardness to the intensities of stresses and strains was constructed from the results of torsion tests of the steel under investigation. A cylindrical specimen of radius \(R = 18\ \text{mm}\) was twisted to a specified value of the specific twist angle \(\varphi\). From it the shear in the outer layer of the specimen was determined, \(\gamma = R\varphi\), and from the latter—the logarithmic shear [5]
\[ g = \ln \left[ 1 + \frac{\gamma}{2}\left(\gamma + \sqrt{4+\gamma^2}\right) \right]. \tag{5} \]
The strain intensity is
\[ e_i = \frac{1}{\sqrt{3}}\,g. \]
The shear stress was determined from the torque by the relation [6]
\[ \tau = \frac{3}{2\pi R^3}\left(M + \frac{1}{3}\gamma \frac{dM}{d\gamma}\right). \tag{6} \]
The stress intensity in plane strain is \(\sigma_i = \sqrt{3}\,\tau\). Hardness was measured at ten points along the outside diameter. The hardness, averaged from these ten measurements, was compared with the stress and strain intensities found by the method described above.
The aged calibration curve is presented in Fig. 1.
The relation between the length \(l\) and the diameter \(d\) at which plane strain occurs in the middle part of the specimen was determined beforehand. For this purpose, on three specimens \(30\ \text{mm}\) in diameter and \(30\), \(60\), and \(90\ \text{mm}\) in length, points were marked on small flats made along the length of the specimens. The specimens were subjected to radial compression in two stages of deformation, to a reduction
\[ \delta = \frac{d-h}{d}\,100\%, \]
respectively, \(25.7\) and \(40\%\).
Measurement of the distances between the marked points after upsetting made it possible to determine the axial strain \(e_z\). The distribution of \(e_z\) along the length of the specimens at a reduction of \(40\%\) is shown in Fig. 2.
It is evident from this graph that at a distance of one diameter from the end face of the specimen the axial strain \(e_z\) is practically absent.
Thus, taking for the main series specimen dimensions \(d = 30\ \text{mm}\) and \(l = 60\ \text{mm}\), one can obtain plane deformation in the middle cross section along the length.
The magnitude of the load \(P\), the width of the contact area \(a\), the corresponding central angles \(\alpha\), the section heights \(h\), and the nominal degree of
\[ \sigma_i=\frac{3\sqrt{3}}{2\pi R^3}\left(M+\frac{1}{3}l\frac{dM}{dl}\right) \]
\[ e_i=\frac{1}{\sqrt{3}}\ln\left[1+\frac{\gamma}{2}\left(\gamma+\sqrt{4+\gamma^2}\right)\right] \]
Fig. 1. Calibration graph
Fig. 2. Dependence of axial deformation on specimen length
deformation for the tested specimens are given in the table.
After deformation, the specimens were cut in the middle along the length plane. Hardness was measured in the resulting section. The results of the hardness measurements were statistically processed. From the averaged hardness, using the calibration graph, the magnitude of the intens-
of the tangential stresses \(k = \frac{1}{\sqrt{3}}\sigma_i\). The distribution \(\kappa(x,y)\) is shown in Fig. 3.
| No. | \(P\), t | \(a\), mm | \(\alpha\), deg. | \(h\), mm | \(\delta\), % |
|---|---|---|---|---|---|
| 1 | 61 | 12 | 48 | 26.5 | 11.7 |
| 2 | 105 | 17 | 70 | 24 | 20 |
| 3 | 175 | 24 | 101 | 21 | 30 |
The boundary conditions used in the interpretation were taken as follows: the lateral surface is free of load; friction on the contact areas was neglected (we note that this assumption could affect only the results in the region \(CAF\), Fig. 3); along the axes of symmetry the angle of inclination of the slip lines is \(45^\circ\).
The interpretation was carried out as follows. In the region \(ABL\) the Cauchy problem was solved; in the regions \(BLD\), \(DACE\), and \(DEO\) mixed problems were solved.
As a result of the solution, the networks of slip lines shown in Fig. 3 were obtained for one quarter of the specimen, owing to its symmetry.
In the same figure are shown the diagrams of the strains \(e_x=-e_y\) for the horizontal and vertical diameters. From the graph of the relationship
Fig. 3. Radial compression of a cylinder
between hardness and strain intensity (Fig. 2), one can determine \(e_i\) at various points of the horizontal and vertical diameters.
The values of the principal strains are determined from the relation [4]
\[ e_1=-e_3=\frac{\sqrt{3}}{2}e_i. \tag{7} \]
The diagrams of \(e_x=-e_y\) for the horizontal diameter have a parabolic character, with a maximum at the center of the specimen at different degrees of reduction. For the vertical diameter, the maximum of the diagrams
\(e_x = -e_y\), at reductions of 11.7 and 20%, is located at some distance from the center of the specimen and only at \(\delta = 30\%\) shifts into the central part of the cylinder.
Figure 4 shows the stress diagrams \(\sigma_x\) and \(\sigma_y\) along the horizontal diameter of the deformed specimens. The stresses \(\sigma_y\) are shown by thick lines, and \(\sigma_x\) by thinner lines. Circles denote the experimental points at a reduction of 11.7%, crosses at \(\delta = 20\%\), and triangles at \(\delta = 30\%\).
The \(\sigma_y\) diagrams are not monotonic. They have two minima, which approximately coincide with the projection of the boundaries of the contact area, for the given degree of reduction, onto the horizontal diameter. As the degree of reduction increases, the maximum values of \(\sigma_y\) at the center of the specimen increase considerably; the local maxima for the outer parts of the diagram increase somewhat, and the values of the minima rise. The positive (tensile) stresses \(\sigma_x\) decrease and then change sign in the central part of the specimen. The maximum values of \(\sigma_x\) correspond to the minima on the \(\sigma_y\) diagrams.
Fig. 4. Diagrams of principal stresses along the horizontal diameter:
○—reduction 11.7%; ×—reduction 20%; Δ—reduction 30%.
From the \(\sigma_y\) diagram, the deforming force was calculated and then compared with the actual one. At reductions of 11.7% and 20%, the calculated force proved to be less than the actual force by 8 and 4%, respectively. At the greatest reduction, the calculated force proved to be overestimated by 23%. The considerable magnitude of the error in the latter case is evidently explained by ...
with accumulation of error in the process of the approximate solution of the differential equilibrium equations.
Consequently, the proposed decoding method [4] in a number of cases makes it possible to obtain reliable results, but methods for refining the solutions must be developed.
In the work carried out, the main qualitative conclusions of studies [2, 3] are confirmed, and the change in the principal stresses with increasing reduction of the specimens is quantitatively evaluated. At significant degrees of reduction \((\delta = 30\%)\), the results of the present work approximately correspond to the results of study [3]. In this case, the difference in the diagrams of \(\sigma_r\) is also not so substantial.
It may be considered established that, at small degrees of reduction, a stress state with stresses of different signs occurs in the central part of the specimen. At significant reductions, however, \(\sigma_r\) changes sign to negative, and in the center of the specimen a stress state with stresses of the same sign arises.
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