Investigation of Residual Stresses During Ring Expansion
Abstract Generated abstract
This paper presents a method for determining residual stresses in expanded rings when the stresses arise from plastic deformation. The approach uses hardness measurements to infer stress intensity across the ring radius, solves the axisymmetric plane-stress equilibrium and yield relations numerically, and combines the resulting plastic-deformation stresses with elastic unloading stresses. Experiments on steel and copper rings show that circumferential stresses during expansion increase toward the outer surface, residual circumferential stresses are significant, and residual radial stresses are small. The study also develops simplified analytical approximations based on linearized stress relations and nearly linear or constant stress-intensity distributions, finding reasonable agreement with numerical calculations and indicating dependence of residual stress magnitude on ring width and material strength.
Full Text
INVESTIGATION OF RESIDUAL STRESSES
IN RING EXPANSION
G. D. Del, G. Kh. Kim
(Presented by the scientific seminar of the Department of Strength of Materials)
If residual stresses are the result of plastic deformations, then their magnitude can be determined from the hardness distribution [1]. In a plane stress state with axial symmetry, such a determination can be carried out without destroying the part. In the present work, residual stresses in ring expansion are determined by this method.
Residual stresses are equal to the difference between the stresses during plastic deformation and the unloading stresses:
\[ \sigma_{\varphi 0}=\sigma_{\varphi}-\sigma_{\varphi p}, \qquad \sigma_{r0}=\sigma_{r}-\sigma_{rp}, \tag{1} \]
where \(\sigma_{\varphi 0}, \sigma_{r0}\) are the residual circumferential and radial stresses,
\(\sigma_{\varphi}, \sigma_{r}\) are the stresses during plastic deformation,
\(\sigma_{\varphi p}, \sigma_{rp}\) are the unloading stresses.
The stresses during plastic deformation are determined from the hardness distribution [2] from the system of equations:
\[ \left. \begin{aligned} \sigma_{r}^{2}+\sigma_{\varphi}^{2}-\sigma_{r}\sigma_{\varphi}&=\sigma_{i}^{2},\\ \frac{\partial \sigma_{r}}{\partial r}+\frac{\sigma_{r}-\sigma_{\varphi}}{r}&=0, \end{aligned} \right\} \tag{2} \]
where \(r\) is the radius of the point at which the stresses are determined.
The stress intensity \(\sigma_i\) is determined from the hardness. For this purpose, on the basis of tests of the metals under investigation in axial compression with lubrication and subsequent measurement of the hardness of the deformed specimens, a calibration graph was constructed relating stress intensity to hardness. The hardness of the specimens was measured in the meridional section with a diamond pyramid under a load of 30 kg in steel tests and 20 kg in copper tests. Ring expansion was carried out with lubrication on a specially designed device.
The device consisted of a conical mandrel made of hardened steel, onto which the rings under investigation were pressed. After expansion, the hardness was measured on the end surface of the ring. The measurements were made along arcs of different radius. From the measured hardness, using the calibration graph, the epure of stress intensity along the radius of the ring was constructed.
The system of equations (2) is solved by a numerical method. For this purpose, the radius over the width of the ring is divided into a series of segments of length \(в\) point-
points 1, 2, 3,... From the stress-intensity diagram the values of \(\sigma_i\) at these points are determined.
At point 1 of the outer surface of the ring
\[ (\sigma_r)_1=0,\quad (\sigma_\varphi)_1=(\sigma_i)_1,\quad \frac{d\sigma_r}{dr}=\frac{(\sigma_\varphi)_1}{r_1}. \]
At point 2
\[ (\sigma_r)_2=-\left(\frac{\partial \sigma_r}{\partial r}\right)_1\cdot b =\frac{(\sigma_\varphi)_1}{r_1}\cdot b, \]
\[ (\sigma_\varphi)_2= \frac{(\sigma_r)_2+\sqrt{4(\sigma_i)_2^2-3(\sigma_r)_2^2}}{2}. \tag{3} \]
To determine \(\sigma_r\) at point 3, we write the equations of system (2) in finite differences for point 2:
\[ \frac{(\sigma_r)_1-(\sigma_r)_3}{2b} -\frac{(\sigma_r)_2-(\sigma_\varphi)_2}{r_2}=0, \]
whence we find
\[ (\sigma_r)_3=(\sigma_r)_1+\frac{(\sigma_r)_2-(\sigma_\varphi)_2}{r_2}\cdot 2b. \tag{4} \]
The stress \((\sigma_\varphi)_3\) is determined by a formula analogous to (3). At points 4, 5, 6, etc., the stresses are determined by formulas analogous to (3) and (4).
The unloading stresses are determined from the formulas of the theory of elasticity [3]:
\[ \sigma_{rp}=-\frac{pa^2}{b^2-a^2}\left(1-\frac{b^2}{r^2}\right),\quad \sigma_{\varphi p}=\frac{pa^2}{b^2-a^2}\left(1+\frac{b^2}{r^2}\right), \tag{5} \]
where \(b, a\) are the outer and inner radii of the ring,
\(r\) is the radius of the point at which the stresses are determined,
\(P\) is the pressure on the inner surface, numerically equal to the radial stress \(\sigma_r\) during plastic deformation.
Then, by formulas (1), the residual stresses are determined.
Investigations carried out by this method showed that the circumferential stresses during plastic deformation increase from the inner surface to the outer surface, and their absolute values are greater the greater the degree of deformation. Figure 1 gives the stress diagrams in a ring for two degrees of expansion \(\delta\):
\[ \delta=\frac{a-a_0}{a_0}\cdot 100\%, \tag{6} \]
where \(a_0, a\) are the inner radius of the ring before and after expansion. The stress diagrams during plastic deformation shown in Fig. 1 qualitatively coincide with analogous diagrams during expansion of a tube made of an ideally plastic material [4]. It follows from Fig. 1 that significant circumferential residual stresses arise in the ring; the radial residual stresses are small. Figure 2 gives diagrams of residual stresses for the same expansion of different rings. The results obtained show that the magnitude of the residual stresses depends on the width of the ring and the strength of the material; for wider rings they are greater, and, for the same width, they are greater in a stronger material.
According to the equilibrium condition, the area enclosed by the diagram of circumferential residual stresses must be equal to zero. It follows from Fig. 2 that this condition is satisfied with sufficient accuracy.
Fig. 1. Stress diagrams during expansion of a steel ring: 1, 3—diagrams of circumferential stresses under plastic deformation \(\delta = 7.5\) and \(\delta = 16\%\); 2, 4—diagrams of unloading stresses \((\delta = 7.5\) and \(\delta = 16\%)\); 5, 6—diagrams of radial stresses under plastic deformation and unloading for \(\delta = 7.5\%\)
Fig. 2. Residual stresses during expansion of rings
| Curve number | Material | \(\delta,\ \%\) | \(h_0,\ \mathrm{mm}\) | \(b_0,\ \mathrm{mm}\) | \(a_0,\ \mathrm{mm}\) |
|---|---|---|---|---|---|
| 1 | Steel St. 3 | 11 | 10 | 45 | 25 |
| 2 | Steel St. 3 | 11 | 10 | 35 | 25 |
| 3 | Steel St. 3 | 11 | 10 | 40 | 25 |
| 4 | Steel St. 3 | 11 | 10 | 35 | 25 |
| 5 | Copper | 11 | 10 | 35 | 25 |
The determination of residual stresses is simplified if one takes into account that, for \(\sigma_r < 0.6\sigma_\varphi\), relation (3) can be replaced by a linear function of the form:
\[ \sigma_\varphi = \sigma_i + 0.66\sigma_r. \tag{7} \]
The distribution of \(\sigma_i\) along the radius is also close to linear:
\[ \sigma_i = A - Br,\qquad B = \frac{\sigma_{iv} - \sigma_{in}}{b - a},\qquad A = \sigma_{iv} - Br, \tag{8} \]
where \(\sigma_{iv}, \sigma_{in}\) are the stress intensities on the inner and outer surfaces of the ring. Taking (7) and (8) into account, we reduce the system of equations (2) to the form:
\[ \frac{d\sigma_r}{dr} + \frac{0.33}{r}\sigma_r - \frac{A}{r} + B = 0. \tag{9} \]
For \(r=b\), \(\sigma_r=0\); therefore, according to (9),
\[ \sigma_r=3A-0.75Br+\sqrt[3]{\frac{b}{r}}\,(0.75Bb-3A) \]
or
\[ \sigma_r=3A-0.75Br+c\sqrt[3]{\frac{b}{r}}, \tag{10} \]
where
\[ C=0.75Bb-3A. \tag{11} \]
The circumferential stress is determined from (7) and (8). The relations obtained make it possible to determine the residual stresses at any point of interest along the radius of the ring, knowing only the stress intensity on the inner and outer surfaces; its value can be determined from the flow curve, knowing the strain intensity \(e_i\).
On the outer surface,
\[ e_{i\mathrm{n}}=\ln \frac{b}{b_0}, \tag{12} \]
where \(b_0, b\) are the radii of the outer surface of the ring before and after expansion.
Fig. 3. Comparison of diagrams of residual stresses determined by different methods. 1, 3—results of numerical integration; 2, 4—stresses obtained by linearizing equations (2); 5—distribution of residual stresses under the assumption that \(\sigma_i\) is constant.
On the inner surface, the circumferential and axial strains are equal to
\[ e_\varphi=\ln \frac{a}{a_0}, \qquad e_z=\ln \frac{h}{h_0}, \]
where \(h_0, h\) are the thicknesses of the ring before and after expansion.
The deformation along the radius \(e_r\) is found from the incompressibility condition:
\[ e_r=-e_\varphi-e_z. \]
The strain intensity is
\[ \begin{aligned} e_i&=\frac{\sqrt{2}}{3}\sqrt{(e_\varphi-e_r)^2+(e_\varphi-e_z)^2+(e_r-e_z)^2} \\ &=\frac{2}{\sqrt{3}}\sqrt{e_\varphi^2+e_\varphi e_z+e_z^2}. \end{aligned} \tag{13} \]
The determination of residual stresses can be simplified if it is taken into account that, in narrow rings \(\left(\text{for } \dfrac{a}{b}>0.7\right)\), the variation of \(\sigma_i\) along the radius is insignificant. Then, taking it to be constant along the radius of the ring and equal to
\[ \sigma_i=\frac{\sigma_{i\mathrm{v}}+\sigma_{i\mathrm{n}}}{2}, \]
the radial stress can be determined from (11) with \(A=\sigma_i\) and \(B=0\). Otherwise the calculation coincides with the preceding one.
Figure 3 gives diagrams of residual stresses in two rings, constructed by different methods. Comparison of these diagrams shows that the stress values calculated by the different methods are approximately the same, which confirms the reliability of the proposed method for determining residual stresses.
REFERENCES
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G. D. Del. Hardness of a deformed body. Izv. AN SSSR, Metals, Moscow, No. 4, 1967.
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G. D. Del. Determination of stresses under plastic deformation from the hardness distribution. Proceedings of the institutes of the State Committee for Standards, Measures, and Measuring Instruments, Leningrad, 1967.
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N. I. Bezukhov. Fundamentals of the Theory of Elasticity, Plasticity, and Creep. Vysshaya Shkola Publishing House, Moscow, 1961.
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L. M. Kachanov. Fundamentals of the Theory of Plasticity. State Publishing House of Technical-Theoretical Literature, Moscow, 1956.