DEFORMED STATE DURING UPSETTING OF HOLLOW CYLINDERS

V. A. Ogorodnikov, V. D. Del

(Presented by the scientific seminar of the Laboratory of Plastic Deformations)

Analysis of the deformed state during upsetting of hollow cylinders is of theoretical and practical interest. Knowledge of deformations is necessary for the rational design of this technological operation. L. A. Shofman [1] studied the position of the conventional surface separating flow as a function of the geometry of the specimen and the degree of deformation. Another solution of the problem under consideration was obtained in the work of I. Ya. Tarnovskii and coauthors [2]. A. V. Altykis investigated the closing of the internal cavity during upsetting of hollow cylindrical forgings [3]. However, the literature known to us contains no information on the distribution of deformations during upsetting of hollow cylinders as a function of a number of technological factors.

The aim of the present work is to investigate the deformed state during upsetting of hollow cylinders.

To investigate the deformed state, the hardness-measurement method was used. This method, which has found broad application in the study of a number of technological problems [4, 5, 6, 7], makes it possible to determine the deformed state during upsetting without violating the continuity of the specimens under investigation in the course of deformation.

To investigate the deformed state by hardness measurement, it is necessary to have a calibration graph of “strain intensity—hardness.” The calibration graph was constructed from the results of testing cylindrical specimens of the material under study (15Kh steel) in axial compression and subsequent hardness measurement of specimens compressed to various degrees of deformation. Compression of the calibration specimens was carried out with lubrication of the end faces by a paste of colloidal graphite and glycerin, using lead foil. Hardness was measured by the Brinell method at 4–5 points of the meridional section of the upset specimens with a ball 5 mm in diameter under a load of 500 kg.

For the test, 9 cylinders of 15Kh steel were made, the dimensions of which are given in Table 1. The restraint coefficient indicated in the table is determined by the ratio of the ring width to its height,

\[ m=\frac{D_0-d_0}{2h_0}, \tag{1} \]

where \(h_0\), \(d_0\), and \(D_0\) are, respectively, the height and the internal and external diameters of the specimen before deformation.

Deformation of the specimens was carried out on a 500-ton hydraulic press. Upsetting was performed between dry hardened plates.

Table 1

Specimens having the same braking coefficient were upset by 10, 20, and 30%. They were then cut along the middle section in height. The resulting surface was ground, and hardness was measured at 18–20 points under the same conditions as in testing the calibration specimens. From the hardness, using the calibration graph “hardness—strain intensity,” the value of the strain intensity, $e_i$, was determined.

The possibility of determining $e_i$ from the hardness of a deformed body is shown in [5].

When determining the strains along the horizontal axis of symmetry of upset cylinders, the following system of equations is solved [4]:

\[ \left. \begin{aligned} e_r^2 + e_\varphi^2 + e_r e_\varphi &= \frac{3}{4} e_i^2,\\ \frac{d e_\varphi}{d r} &= \frac{e_\varphi - e_r}{r}. \end{aligned} \right\} \tag{2} \]

The interpretation was carried out according to the method set forth in [4]. As the initial condition, the strains on the inner surface were used, determined from the relation

\[ e_\varphi = \ln \frac{R}{R_0}. \]

Here $R_0$ and $R$ are the radii of the inner surface of the specimen before and after deformation. The radial strain on the inner surface is found by solving the first of equations (2).

The method of numerical integration of the indicated system was programmed on a Minsk-1 digital computer. The simplicity of the experiment and the automation of the interpretation of its results made it possible to study the deformed state as a function of the amount of upsetting and the dimensions of the cylinders investigated. The strain diagrams obtained are presented in Figs. 1, 2, and 3 (the curve numbers indicate the specimen numbers).

In all cases, a considerable nonuniformity of the radial strain $e_r$ is observed. The radial-strain diagram has maxi-

...maximum near the inner surface and decreases toward the periphery. As the degree of deformation increases, the nonuniformity of the radial strain increases.

It should be noted that, as in the upsetting of solid cylinders, the circumferential strains are not equal to the radial strains [7]. The greatest discrepancy

Fig. 1

is observed on the inner surface and reaches 90%. The disagreement between the radial and circumferential strains means that Kármán’s hypothesis that the circumferential stress is equal to one of the principal stresses in the meridional planes is, under these conditions, in contradiction with experiment. The circumferential strains are distributed more uniformly.

The magnitude of the circumferential strain on the inner surface is close to zero and, as the degree of deformation increases, remains almost unchanged. Only for specimen No. 6, at 30% upsetting, was closure of the inner cavity observed (\(e_\varphi < 0\)). The data obtained on closure of the inner cavity agree with the data of L. A. Shofman [1]. In the direction toward the outer surface, the circumferential strain increases monotonically. As the...

Fig. 2

Fig. 3

with an increase in the friction coefficient \(m\), the deformation diagrams are distributed more nonuniformly. For equal \(m\), changing \(\dfrac{D_0}{d_0}\) in the direction of increase leads to a decrease in the nonuniformity of deformation.

Verification of the results of the deformation calculation was carried out by comparing the calculated circumferential strain on the outer lateral surface with the actual value

\[ e_\varphi=\ln \frac{D}{D_0}, \]

where \(D_0, D\) are the outside diameters of the specimen before and after deformation. In most cases the error did not exceed 20%.

REFERENCES

1. L. A. Shofman. Fundamentals of the Calculation of Stamping and Pressing Processes. Mashgiz, 1961.

2. I. Ya. Tarnovskii, A. A. Pozdeev, O. A. Ganago. Deformations and Forces in Metal Forming. Mashgiz, 1959.

3. A. V. Altykis. Formation of laps during free upsetting of cylindrical forgings with a central hole. Forging and Stamping Production, No. 6, 10, 1960.

4. G. D. Del’. Determination of the deformed state in the plastic region from the distribution of hardness. Izv. TPI, vol. 157, Tomsk, 1969.

5. G. D. Del’. Hardness of deformed metal. Izv. AN SSSR, “Metally”, No. 4, 1967.

6. G. D. Del’, V. A. Ogorodnikov. Determination of the stress–strain state in the plastic region from the fibrous macrostructure and the distribution of hardness. Izv. vuzov, “Ferrous Metallurgy”, No. 6, 1969.

7. E. S. Vaksman, G. D. Del’, V. A. Ogorodnikov. The deformed state in plane and axisymmetric upsetting. Izv. TPI, vol. 183, Tomsk, 1968.