Verification of the Haar-Kármán Assumption under Axisymmetric Settlement Conditions
Abstract Generated abstract
This paper examines whether the Haar-Karman assumption can be used to determine stresses during axisymmetric upsetting of cylindrical specimens. The authors compare stress components derived from measured strains, strain rates, hardness distributions, and grid distortion in Kh18N9T steel and lead with stresses calculated using the Haar-Karman condition and slip-line methods. The comparisons indicate substantial discrepancies for radial and circumferential stress components, especially near the contact end faces, while axial stress is reproduced more accurately in some regions. The study concludes that the Haar-Karman assumption is not generally confirmed for axisymmetric settlement and may lead to large errors in stress-state determination.
Full Text
VERIFICATION OF THE HAAR–KARMAN ASSUMPTION UNDER CONDITIONS OF AXISYMMETRIC SETTLEMENT
V. A. Ogorodnikov, G. D. Del
(Presented by the scientific seminar of the Department of Strength of Materials)
Axisymmetric deformation is one of the most important, but insufficiently developed, problems of plasticity. As in other spatial problems, the governing equations for the case of axial symmetry are in general not integrable; therefore the number of analytical solutions in this field is very limited. The solution of some problems has been obtained by introducing the concept of “complete plasticity,” proposed by Haar and Karman [1]. According to their hypothesis, in the plastic state the circumferential stress is equal to one of the principal stresses in the meridional plane. Serious objections have been raised against this condition [2]; nevertheless it is widely used in the engineering theory of plasticity [3].
The purpose of the present work is to verify the possibility of determining the stress state under axisymmetric settlement on the basis of the Haar–Karman assumption.
The stress state was studied in steel Kh18N9T and in lead. Stresses were calculated from strains and strain rates. The investigation of the deformed state of the strain-hardening material was carried out from the hardness distribution; the deformed state in lead was studied from the distortion of a dividing grid previously applied to one of the halves of a composite specimen.
In studying the stress–strain state by hardness measurement, a calibration curve is first constructed (stress intensity $\sigma$, hardness $H$, strain intensity $e_i$). The calibration curve for the material under investigation (steel Kh18N9T) was constructed by testing eight cylindrical specimens in axial compression with lubrication. Hardness was measured by the Brinell method with a ball of diameter 5 mm under a load of 500 kg.
The stress–strain state was determined on three cylindrical specimens of diameter 36 and height 54 mm. The specimens were subjected to axial compression without lubrication of the end faces. The magnitude of the settlement of the cylinders, $\ln \dfrac{H_0}{H}$, was 0.095, 0.191, and 0.28. After measuring the hardness at the center of the end face of the deformed specimens, the latter were cut. The resulting surface was ground, and at various points along its axis of symmetry the hardness was measured. From the hardness, using the calibration curves, the corresponding value of the strain intensity was determined.
$e_i$. The strain components $e_r$ and $e_\varphi$ were calculated by the method set forth in [5]. In addition, the “strain rates” were calculated by the following method. From the diagrams, using the expression
\[ e_\varphi=\ln \frac{R}{R_0}, \tag{1} \]
where $R_0$, $R$ are the radii before and after deformation, the initial radii $R_0$ of points in the plane of symmetry perpendicular to the $z$ axis were determined, and graphs of their variation along the radii were constructed. By specifying the initial radius, graphs were constructed for the variation of $e_\varphi$, $e_r$, and $e_i$ as functions of the change in specimen height. From the tangents of the inclination angle of the tangents to these graphs, the strain rates were determined, referred to the rate of change of the specimen height. Having data on the strains and strain rates, and also knowing the stress intensity $\sigma_i$, the components of the stress deviator were calculated from the relations of the deformation theories
\[ s_r=\frac{2}{3}\frac{\sigma_i}{e_i}e_r,\quad s_\varphi=\frac{2}{3}\frac{\sigma_i}{e_i}e_\varphi, \tag{2} \]
and from the relations of the flow theories
\[ s_r=\frac{2}{3}\frac{\sigma_i}{\dot e_i}\dot e_r,\quad s_\varphi=\frac{2}{3}\frac{\sigma_i}{\dot e_i}\dot e_\varphi, \tag{3} \]
where $s_r$ and $s_\varphi$ are the radial and circumferential components of the stress deviator.
Fig. 1. Components of the stress deviator on the radius of the barrel during upsetting of a cylinder, calculated from the relations of the deformation theories—1, from the flow theories—2, and under the Haar—Karman assumption—3.
In Fig. 1 the components of the stress deviator calculated in this way are compared with their determination from hardness under the Haar—Karman assumption [4]. The results obtained show that \(s_r\) and \(s_\varphi\) are determined under the Haar—Karman assumption with considerable error. The component \(s_z\), however, is determined with sufficient accuracy under the indicated assumption.
The greatest discrepancy of the stress-deviator components \(s_r\) and \(s_\varphi\) should apparently be expected at the contact of the specimens being upset. To determine \(s_r\) and \(s_\varphi\) on the end face, it is necessary to know the strain components \(e_r\) and \(e_\varphi\). The circumferential strain \(e_\varphi\) was determined from the change in the radii of the kerfs made on the end face of the specimen of Kh18N9T steel before upsetting. The radial strain was determined from the relation
\[ \frac{d e_\varphi}{d r} = \frac{e_r - e_\varphi}{r}. \tag{4} \]
Fig. 2. Components of the stress deviator and strains on the end face of the cylinder being upset
In Fig. 2 the diagrams of the strains \(e_r\) and \(e_\varphi\) are shown. The diagrams of the stress-deviator components calculated from the relations of deformation theories [2] are also shown there.
As is evident from Figs. 1 and 2, the error in determining stresses under the Haar—Karman assumption is greater at the end faces than on the horizontal axis of symmetry.
To verify the Haar—Karman assumption on a strain-softening material, the investigation was carried out on a composite lead specimen. A grid consisting of a system of circles 3 mm in diameter was located in the meridional section of the specimen. The specimen was composed of two semicylinders soldered together in the meridional plane, pressed with a slight interference fit into a lead holder 107 mm in diameter.
Subsequently the holder with the composite specimen was upset to a deformation equal to 10%.
Along the horizontal axis of symmetry, the circumferential deformation and the radial deformation were determined as
\[ e_{\varphi}=\ln \frac{R}{R_0}, \qquad e_r=\ln \frac{b}{d_0}, \tag{5} \]
where \(R_0, R\) are the radius of the point before and after deformation, \(b\) is the major axis of the ellipse, and \(d_0\) is the initial diameter of the imprint. The radial deformation was also checked by the relation [4].
To calculate the stresses it is necessary to know the angle \(\varphi\) between the direction of the principal deformations and the horizontal axis. The angle \(\varphi\) was measured (as were the deformations) at various points of the meridional plane with an instrumental microscope. The stress state was calculated in a quarter of the meridional section.
According to the deformation theory,
\[ \tau_{rz}=\frac{\sigma_i}{3e_i}\,g_{rz}. \tag{6} \]
The strain intensity and the shear are respectively
\[ \left. \begin{aligned} e_i&=\frac{\sqrt{2}}{3}\sqrt{(e_1-e_2)^2+(e_1-e_3)^2+(e_2-e_3)^2},\\ g_{rz}&=\pm(e_1-e_2)\sin 2\varphi, \end{aligned} \right\} \tag{7} \]
where \(e_1, e_2\) are the principal deformations in the meridional section \((e_1>e_2)\). Taking into account that \(e_3=-e_1-e_2\), we find
\[ \tau_{rz}=\frac{\sqrt{3}\sigma_i}{6}\, \frac{e_1-e_2}{\sqrt{e_1^2+e_1e_2+e_2^2}}\sin 2\varphi \tag{8} \]
or
\[ \tau_{rz}=\frac{\sigma_i}{2}(M_1-M_2)\sin 2\varphi, \tag{9} \]
where
\[ M_1=\frac{1}{\sqrt{3}\sqrt{1+\dfrac{e_2}{e_1}+\left(\dfrac{e_2}{e_1}\right)^2}}, \qquad M_2=M_1\frac{e_2}{e_1}. \tag{10} \]
The radial component of the stress deviator is
\[ s_r=\sigma_r-\sigma=\frac{2}{3}\frac{\sigma_i}{e_i}e_r. \]
The corresponding deformation is
\[ e_r=e_1\cos^2\varphi+e_2\sin^2\varphi. \tag{11} \]
Taking (10) into account, we find
\[ s_r=(M_1\cos^2\varphi+M_2\sin^2\varphi)\sigma_i. \tag{12} \]
The circumferential deformation is
\[ e_{\varphi}=-e_1-e_2. \]
Fig. 3. Stresses during settlement of a cylinder.
In the figure:
\(1\) — calculation by deformation theories
\(2\) — calculation along slip lines under the Haar–Karman assumption
After simple transformations we obtain
\[ s_\varphi=\sigma_\varphi-\sigma =\frac{2}{3}\,\frac{\sigma_i}{e_i}\,e_\varphi =-(M_1+M_2)\sigma_i. \tag{13} \]
In the axial direction
\[ e_z=e_1\sin^2\varphi+e_2\cos^2\varphi, \tag{14} \]
\[ s_z=\sigma_z-\sigma=(M_1\sin^2\varphi+M_2\cos^2\varphi)\sigma_i. \tag{15} \]
To determine the hydrostatic pressure, we integrate one of the differential equilibrium equations
\[ \frac{\partial\sigma_r}{\partial r} +\frac{\partial\tau_{rz}}{\partial z} +\frac{\sigma_r-\sigma_\varphi}{r}=0. \tag{16} \]
We begin the calculation of stresses from the free boundary.
\[ (\sigma_r)_B=(\sigma_r)_A- \int_A^B \left[ \frac{\partial\tau_{rz}}{\partial z\sigma_i} +\frac{s_r-s_\varphi}{r\sigma_i} \right]\,dr. \tag{17} \]
Having determined the stresses \(\dfrac{\sigma_r}{\sigma_i}\) along the horizontal axis of symmetry, we subtract from them the corresponding components of the stress deviator, referred to \(\sigma_i\), and find the hydrostatic pressure \(\dfrac{\sigma}{\sigma_i}\). Adding it to the remaining components of the deviator, we find the corresponding normal stresses.
The stresses were also calculated by the slip-line method under the Haar—Karman assumption, using the apparatus of the theory of ideally plastic bodies [6].
As a result, diagrams of the stress distribution along the radius of the barrel were obtained, calculated by the two methods. The diagrams are presented in Fig. 3. As follows from the graphs, the stress components \(\dfrac{\sigma_r}{\sigma_i}\) and \(\dfrac{\sigma_\varphi}{\sigma_i}\) do not coincide. Near the free boundary, close agreement is observed between the diagrams of the axial stresses obtained by the two methods. Toward the center of the specimen the discrepancy becomes significant; apparently, this is explained by the increase in the error associated with integration of the differential equation of equilibrium.
The results of the study show that, under conditions of axisymmetric settlement, the Haar—Karman assumption is not confirmed; consequently, determination of the stress state under this assumption is associated with large errors. At the same time, the errors in determining the circumferential and radial stresses are greater than the errors associated with determining the axial stress.
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