Experimental Study of the Neutral Loading Curve
Abstract Generated abstract
This paper reports an experimental investigation of the neutral loading curve, or boundary surface, for annealed copper under a plane stressed state. Tubular specimens were first plastically loaded by combined internal pressure and axial force to prescribed stress intensities, unloaded, and then reloaded along different rectilinear stress paths while circumferential and axial strains were measured. Conditional yield points from the repeated loading curves were used to construct neutral loading curves in normalized stress coordinates and to compare them with the Huber-Mises ellipse. The results suggest that the neutral loading curve lies tangent inside the Huber-Mises ellipse, so that loading along that ellipse would produce plastic deformation, and that increasing preliminary plastic deformation is associated with expansion and displacement of the yield surface in the direction of prior loading.
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EXPERIMENTAL STUDY OF THE NEUTRAL LOADING CURVE
A. M. Koreneva, G. A. Doshchinskii
(Presented by the scientific seminar of the Department of Strength of Materials)
Neutral loading occurs in the case when
\[ \frac{\partial f}{\partial \sigma_{ij}} \dot{\sigma}_{ij} = 0, \qquad f = 0, \]
where \(\dot{\sigma}_{ij}\) is the tensor of the rates of change of stresses,
\(\sigma_{ij}\) is the stress tensor,
\(f\) is the loading function, i.e., when the loading path proceeds tangentially to the loading surface [1]. The function \(f\) in the nine-dimensional stress space defines a surface that is called the loading surface or the boundary surface. This surface separates the region of stressed states which, in an element of a body, can cause only changes in elastic deformations from the region of plastic deformations.
In the space of principal stresses, under a plane stressed state, this surface corresponds to a certain curve—the neutral loading curve. In order to study the form of the neutral loading curve, a number of experiments have been carried out, described in the works of Marin and Hu [2–4], Zhukov [5], Talypov [6, 7], and others. A number of hypotheses have also appeared concerning the change in the form of the neutral loading surface during the deformation process, assuming an expansion of the limiting surface or its translation. The features of material hardening observed in experiments often do not agree with the proposed schemes of change of the indicated surface. Establishing the actual form of the boundary surface and the law of its change is possible only on the basis of extensive experiments carried out under various conditions. The present study is a certain supplement in this respect.
The tests were carried out on tubular specimens made of copper, with a diameter of 34.8 mm, a wall thickness of 1.5 mm, and a working-section length of 130 mm. The specimens were preliminarily annealed. The tests were conducted on a ZMDU-30 complex-loading machine. During the experiment, circumferential and axial deformations were determined by means of mechanical strain gauges. To check the properties of isotropy, two specimens each were tested in tension in the longitudinal and transverse directions. The curves differed by 3–5%, which made it possible to conclude that the degree of isotropy was sufficient.
The experimental procedure consisted in the fact that each specimen was deformed by internal pressure and axial force up to a definite—
...of the specified value of the stress intensity at the ratio \(\sigma_\theta/\sigma_z = 1\), and was then completely unloaded. After preliminary loading and unloading, repeated loading was carried out, with each specimen being loaded along its own rectilinear path in the first quadrant of the plane \((\sigma_\theta,\sigma_z)\).
The first four specimens \((1, 2, 3, 4)\) were loaded to a stress intensity \(\sigma_i = 1000\ \text{kg}/\text{cm}^2\), completely unloaded, and repeated loading was carried out at stress ratios
\[ \frac{\sigma_z}{\sigma_\theta} = 0.5;\ 0.6;\ 0.7;\ 0.8 \]
respectively. The fifth and sixth specimens were initially loaded to intensities of \(1250;\ 1450\ \text{kg}/\text{cm}^2\); repeated loading was carried out along rays at stress ratios
\[ \frac{\sigma_\theta}{\sigma_z}=0.3;\ 0. \]
In Fig. 1 the curves of the dependences \(\sigma_i-e_i\) for repeated loading and the initial tensile diagram of the material \(\sigma_i=\Phi(e_i)\) are given. From the \(\sigma_i-e_i\) curves for repeated loading, the conditional yield limit \(\sigma_{is}\) was determined as the stress at residual strain \(e_{is}=0.172\%\), which corresponds to a tolerance of \(0.2\%\) in tension. For each value of \(\sigma_{is}\), the corresponding values of \(\sigma_\theta\) and \(\sigma_z\) were found from the plots \(\sigma_\theta,\sigma_z-\sigma_i\) constructed for each stress ratio. In Fig. 2 the neutral-loading curves are constructed in relative coordinates
\[ \frac{\sigma_\theta}{\sigma_{is}},\quad \frac{\sigma_z}{\sigma_{is}} \quad \text{and} \quad \frac{S_1}{\sigma_{is}},\quad \frac{S_2}{\sigma_{is}}. \]
\(\sigma_{is}\) is the final value of the stress intensity. The dashed line in this figure represents the Huber—Mises ellipse drawn through the final point of the first loading in the coordinates
\[ \frac{\sigma_\theta}{\sigma_{is}},\quad \frac{\sigma_z}{\sigma_{is}}, \]
and transformed into a circle in the coordinates
\[ \frac{S_1}{\sigma_{is}}-\frac{S_2}{\sigma_{is}};\qquad S_1=\frac{\sqrt{2}}{3}\left(\sigma_z-\frac{1}{2}\sigma_\theta\right),\qquad S_2=\frac{\sigma_\theta}{\sqrt{2}}. \]
Fig. 1
The circle with a cross denotes the final point of the first loading.
The results of this experiment lead to the following conclusion. The neutral-loading curve is tangent inside the Huber—Mises ellipse, which indicates that loading carried out along the Huber—Mises ellipse will cause the appearance of plastic deformations. The symmetry of the preliminary loading, and the equal status of the stresses...
\(\sigma_\theta\) and \(\sigma_z\), and the isotropy of the material make it possible to assume a symmetric change in the neutral-loading curve under these conditions as well. With increasing degree of plastic deformation, the yield surface expands
Fig. 2
and, within the framework of the Huber–Mises plasticity condition, proves to be displaced in the direction of the preliminary loading.
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