Study of the Stress-Strain State During Axisymmetric Upsetting of a Cylinder
Abstract Generated abstract
This study examines the stress and strain state produced during axisymmetric upsetting of cylindrical specimens, with particular attention to the validity of the Haar and Kármán assumption that the circumferential stress equals one of the principal stresses in meridional planes. The authors use hardness measurements, calibrated against compression tests for a strain hardening steel, to determine strain intensity and stress intensity throughout compressed cylinders, and then calculate radial, circumferential, axial, and principal components. The results show a markedly nonuniform strain distribution, especially along the cylinder axis, and reveal that circumferential and radial strains, and the corresponding stresses, are not equal. The findings indicate that common estimates of mean strain intensity from diameter change may overstate deformation and that the Haar and Kármán assumption is not confirmed for unlubricated axisymmetric upsetting.
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INVESTIGATION OF THE STRESS–STRAIN STATE IN AXISYMMETRIC UPSETTING OF A CYLINDER
V. A. Ogorodnikov, G. D. Del
(Presented by the scientific seminar of the Department of Strength of Materials)
Many technological processes of cold metal forming by pressure are accompanied by axisymmetric deformation of the metal being worked. Examples of such processes are: upsetting of a cylindrical billet, piercing, extrusion (pressing), drawing, forging of turbine disks, flanges, etc.
Analytical methods for determining the stress–strain state under axisymmetric plastic deformation have not yet been sufficiently developed. In most cases, solutions to such problems have been obtained for an ideally plastic material under the Haar–Kármán assumption that the circumferential stress is equal to one of the principal stresses in the meridional planes [1].
Under the same assumption, it is possible to determine the stressed state of a strain-hardening material from the hardness distribution [2]. Determination of the stressed state guided by the Haar–Kármán hypothesis is justified in certain cases [3], while in other cases it leads to errors of unknown magnitude [4].
The purpose of the present work is to determine the stress–strain state during axisymmetric upsetting of a cylinder and to verify, under these conditions, the Haar–Kármán assumption.
The stress–strain state was determined by the hardness-measurement method. The strains were calculated by the method described in article [5].
A calibrated plot “stress intensity $\sigma_i$—hardness, $H_B$—strain intensity $e_i$” was constructed from the results of testing cylindrical specimens of the material under study in axial compression and subsequent measurement of the hardness of specimens compressed to various degrees of deformation. Compression of the calibrated specimens was carried out with lubrication of the end faces using a paste of colloidal graphite and glycerin. Hardness was measured in meridional sections.
The investigation of the stress–strain state was performed on three cylindrical specimens of high-work-hardening steel 45G17Yu3Kh, 54 mm in height and 36 mm in diameter. This steel was selected in order to achieve the greatest sensitivity of hardness to increments of plastic deformation.
The test specimens were subjected to axial compression without lubrication of the end faces in order to obtain a stress–strain state,
taking place in the forging of cylindrical billets. The amount of upsetting of the cylinders, \(\ln \dfrac{H_0}{H}\), was 0.18, 0.13, and 0.48.
After compression, the end faces of the specimens were polished; hardness was determined by the Brinell method with a ball \(D=10\) mm under a load of 1000 kg. The indicated load was chosen in order to obtain the largest number of hardness measurements. The hardness of the calibration specimens was determined in the same way; therefore, the reduced hardness could not affect the test results. To obtain the hardness distribution over the volume, layers of material were ground off the specimens in planes perpendicular to the axis.
The high surface finish and the use of a tool microscope made it possible to measure the diameters of the impressions with an accuracy of up to 0.01 mm, which corresponded to an accuracy in determining hardness of up to 3 Brinell units.
From the measured hardness, the strain intensity \(e_i\) was determined from the calibration graph.
In studying axisymmetric deformation, we have three equations [5]:
\[ \left. \begin{aligned} \frac{\sqrt{2}}{3} \sqrt{(e_r-e_\varphi)^2+(e_r-e_z)^2+(e_\varphi-e_z)^2+\frac{3}{2}\gamma_{rz}^{\,2}} &= e_i,\\ e_r+e_z+e_\varphi&=0, \end{aligned} \right\} \tag{1} \]
\[ \frac{de_\varphi}{dr}=\frac{1}{r}(e_r-e_\varphi). \tag{2} \]
Along the axis of symmetry, from the condition of symmetry \(e_r=e_\varphi\), \(\gamma_{rz}=0\), therefore
\[ e_r=e_\varphi=-\frac{1}{2}e_z=\frac{1}{2}e_i. \tag{3} \]
The radial \(e_r\) and circumferential strains \(e_\varphi\) were also calculated in the horizontal plane of symmetry perpendicular to the \(z\)-axis. At point 1 of this plane, coinciding with the vertical axis of symmetry, we have \(e_r=e_\varphi=\pm \dfrac{1}{2}e_i\). At the next point 2, located at a distance \(\Delta r=r_2-r_1=2\) mm, the strains were calculated by the formulas
\[ e_{r_2}= \frac{-B(2C+1)+1.73\sqrt{e_i^{\,2}(1+C+C^2)-B^2}} {2(1+C+C^2)}, \tag{4} \]
\[ e_{\varphi_2}=B+Ce_r, \tag{5} \]
where
\[ \left. \begin{aligned} B&=\frac{r_2\left[2r_1e_{\varphi_1}+\Delta r\,(e_{r_1}-e_{\varphi_1})\right]} {r_1(2r_2+\Delta r)},\\ C&=B+Ce_r \end{aligned} \right\}. \tag{6} \]
The circumferential and radial strains at the remaining points of the horizontal plane of symmetry were calculated in the same way. The calculation performed was checked by comparing the resulting value of the circumferential strain on the contour with the results of calculation by the formula
\[ e_\varphi=\ln\frac{D_b}{D_0}, \tag{7} \]
where \(D_0\) and \(D_b\) are the diameters before and after deformation in the average (over height) section of the specimens.
The greatest discrepancy did not exceed 6%. The high accuracy of determining strains from the hardness distribution is also indicated by the results of checking the diagrams of strain intensity \(e_i\), constructed along the vertical axis \(z\). The discrepancy of the mean strain value \(e_i\), found from these diagrams with \(\ln \frac{l_0}{r}\), did not exceed 1%. The results obtained indicate a considerable nonuniformity in the distribution of strains along the \(Z\) axis. Along the horizontal axis of symmetry the strains are distributed more uniformly.
The greatest strain value occurs at the center of the specimen; the greatest hardness was also recorded there. The circumferential strain is not equal to the radial strain—the greatest discrepancy occurred in the most highly deformed specimen and was approximately 50%.
Fig. 1. Strains during upsetting of a cylinder
The diagrams obtained (Fig. 1) agree qualitatively with strain diagrams obtained by other methods [6].
From the measured hardness, the stress intensity \(\sigma_i\) was also determined from a calibration graph. On the contour of the deformed specimens, from the relations of deformation theories of plasticity, the circumferential stresses \(\sigma_\theta\) and the principal stress \(\sigma_3\), directed tangentially to the contour, were determined.
\[ \sigma_z = \frac{2}{3}\,\frac{\sigma_i}{e_i}\,(e_3 - e_z), \tag{8} \]
\[ \sigma_\theta = \frac{2}{3}\,\frac{\sigma_i}{e_i}\,(e_3 - e_\theta), \tag{9} \]
where \(e_3, e_2\) are the principal strains determined from system (1).
\[ e_\varphi = -\ln \frac{l_b}{L_0}. \]
Thus the stresses \(\sigma_\theta\) and \(\sigma_3\) were calculated on the contour of the deformed specimens (Fig. 2) in sections
\[ \frac{Z}{H} = 0 \]
(the mean over the height...
section, curves 1),
\[ \frac{Z}{H}=\frac{1}{8}\quad \text{(curves 2)},\qquad \frac{Z}{H}=\frac{1}{4}\quad \text{(curves 3)}, \]
\[ \frac{Z}{H}=\frac{3}{8}\quad \text{(curves 4)},\qquad \frac{Z}{H}=\frac{1}{2}\quad \text{(curves 5)}. \]
From the obtained diagrams it follows that the stress intensity \(\sigma_i\) and the principal stress \(\sigma_3\) increase with increasing degree of upsetting.
On the contour of the section
\[ Z=\frac{H}{2} \]
the principal stress increases with increasing degree of upsetting and then decreases. The circumferential stress \(\sigma_\theta\) amounts to about \(40\%\) of the principal stress \(\sigma_3\). Proceeding from the Haar–Kármán assumption, it should be equal to zero.
The investigation carried out permits the following conclusions to be drawn:
- On the average (over the height) of the section, without lubrication of the compressed specimen, the circumferential strain exceeds the radial strain. For this reason, the assumption that the mean strain intensity in axial compression may be determined as twice the logarithm of the ratio of the specimen diameters after and before compression is apparently erroneous, since the mean strain intensity is thereby overestimated.
Fig. 2. Stresses during upsetting of a cylinder
- Because of the inequality of the radial and circumferential strains, the corresponding stresses are also not equal; thus, the Haar–Kármán assumption is not confirmed under conditions of axisymmetric upsetting.
REFERENCES
- A. Yu. Ishlinskii. Applied Mathematics and Mechanics. Vol. VII, issue 3, 1944.
- G. D. Del’. Investigation of plastic deformation by hardness measurement. Izv. TPI, vol. 138, 1965.
- L. M. Kachanov. Fundamentals of the Theory of Plasticity. Gostekhizdat, 1956.
- R. Hill. Mathematical Theory of Plasticity. IL, 1956.
- G. D. Del’. Determination of the deformed state in the plastic region from the distribution of hardness. Izv. TPI, vol. 157, 1968.
- P. O. Pashkov. Plasticity and Fracture of Metals. Sudpromgiz, 1950.