PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Volume 157                    1970

DEFORMATION UNDER REVERSED “NEUTRAL” LOADING

A. M. KORENEVA, G. A. DOSHCHINSKY

(Presented by the scientific seminar of the Department of Strength of Materials)

In the theory of plasticity, the image of a loading surface (yield surface) is widely used, separating in stress space the elastic and plastic states of a material. The departure of the loading trajectory beyond this closed surface leads to plastic deformation. A loading process characterized by motion along trajectories inside the region bounded by the yield surface is accompanied by a change only in elastic deformations. It is also assumed that motion along the yield surface itself (neutral loading) changes only the components of elastic deformation. The equation of the loading surface is often determined analytically by means of the second invariant of the stress tensor.

However, a number of experiments carried out in recent years show that such a surface is not a surface of neutral loading. Motion on a surface corresponding to the constancy of the loading measure—the stress intensity—is accompanied by a systematic and regular increase in plastic deformation [1], [2]. As previous experiments [1] have shown, the intensity of increments of plastic strains turns out to be approximately proportional to the arc length of the loading path along the surface of a hypersphere of the five-dimensional deviatoric stress space.

The present experimental study was devoted to clarifying what will occur during reversed motion along the loading surface.

The experiments were carried out on thin-walled tubular specimens under conditions of biaxial tension under the action of an axial force and internal pressure on a ZDMU-30 machine (GDR).

The specimens were made by lathe machining in several operations from a solid bar of U8 steel. The working part of the specimen had a length of 130 mm, an (outer) diameter of 30 mm, and a thickness of 1 mm.

Deviations in the outer diameter did not exceed 0.03–0.05 mm, and in thickness 0.02–0.04 mm. Finish boring of the inner surface was performed with a floating cutter at a low machining speed. The manufactured specimens were subjected to recrystallization annealing in a protective medium (sealed containers with cast-iron dust). To measure deformation, strain gauges with dial-type indicators were used. The scale division of the indicators was 0.01 mm for longitudinal deformation and 0.001 mm for transverse deformation.

The conclusion on the isotropy of the material was made after preliminary testing of four specimens by comparing the tensile diagrams in the longitudinal and transverse directions.

Since the deformations under study were small, the change in specimen dimensions during the test was not taken into account. The axial and circumferential stresses and strains were determined from the expressions:

\[ \sigma_z=\frac{P}{F}+\frac{qR}{2t}; \qquad \sigma_\theta=\frac{qR}{t}; \qquad e_z=\frac{\Delta l}{l}; \qquad e_\theta=\frac{\Delta D}{D}. \]

The elastic components of strain were calculated on the basis of the generalized Hooke’s law from the stresses and elastic constants determined in the preliminary tests of four specimens. The components of plastic strain were determined as the difference between the measured strains and the corresponding elastic parts:

\[ \mathcal{E}_z=e_z-\frac{1}{E}(\sigma_z-\mu\sigma_\theta); \qquad \mathcal{E}_\theta=e_\theta-\frac{1}{E}(\sigma_\theta-\mu\sigma_z). \]

The radial plastic strain was determined from the condition of incompressibility. The radial stress was taken to be zero.

The tests were programmed by stresses in the subspace \(S_1—S_2\) of the five-dimensional stress vector of A. A. Ilyushin [3].

\[ \overline{S}=S_1\cdot \overline{i}+S_2\cdot \overline{j}; \qquad S_1=\sqrt{\frac{2}{3}}\left(\sigma_z-\frac{1}{2}\sigma_\theta\right); \qquad S_2=\frac{\sqrt{2}}{2}\sigma_\theta. \]

The components of the plastic-strain vector

\[ \overline{\mathcal{E}}=\mathcal{E}_1\cdot \overline{i}+\mathcal{E}_2\cdot \overline{j} \]

were determined from the expressions:

\[ \mathcal{E}_1=\sqrt{\frac{3}{2}}\,\mathcal{E}_z; \qquad \mathcal{E}_2=\sqrt{2}\left(\mathcal{E}_\theta+\frac{1}{2}\mathcal{E}_z\right). \]

The intensity of stresses and of increments of plastic strains was calculated, respectively, from the formulas:

\[ \sigma_i=\sqrt{\sigma_z^2+\sigma_\theta^2-\sigma_z\cdot\sigma_\theta}; \qquad \Delta \mathcal{E}_i=\frac{2}{\sqrt{3}}\sqrt{\Delta\mathcal{E}_z^2+\Delta\mathcal{E}_\theta^2+\Delta\mathcal{E}_z\cdot\Delta\mathcal{E}_\theta}. \]

The specimens of the series under investigation were brought by axial tension to states corresponding to stress intensities of \(30, 35, 40; 45, 50, 55, 57\ \mathrm{kg/mm^2}\), and then, by changing the internal pressure and axial load, motion was ensured along the surface of the corresponding stress intensity, unchanged for each specimen. After the implementation of “neutral” loading—motion along an arc of constant curvature of considerable length—the main test was carried out with return motion along the previously traversed trajectory to the state of simple tension.

The realized loading paths are shown in Fig. 1. On the segment of direct “neutral” loading, the increment of plastic strain qualitatively corresponds to the regularities noted earlier [1].

Under reverse loading there was even less reason than under direct loading to expect a change in the plastic state of the material; nevertheless, the experiments showed a certain systematic (although considerably smaller than under direct loading) increment of plastic strain. This increment is in direct dependence on the magnitude of the stress intensity.

Figure 2 shows the values of the increment in the intensity of plastic deformation as a function of the path length of reversed loading. No change in the vector properties of the material during reversed motion was observed. The construction, at certain points of the trajectory of reversed motion, of the plastic-deformation vector, as can be seen from Fig. 1, does not show noticeable deviations from the direction obtained at the end of the segment of straight “neutral” loading.

Fig. 1.

As can be seen from Fig. 3, the increment is inversely related to the degree of hardening of the material. The angular coefficient of the increment of plastic deformation as a function of the loading-path length is approximately represented by the relation

\[ \dot K=\frac{\Delta Э_i}{S}\approx A\cdot \frac{1}{m};\qquad m=\frac{d\sigma_i}{de_i};\qquad A\approx \frac{1}{45} \]

(with direct loading \(A\approx \frac{1}{5}\)).

The increment observed in these experiments apparently cannot be explained by the manifestation of creep, since readings in the experiments were taken after a 5 min hold at each stage. This time usually

Fig. 2.

Fig. 3.

sufficient to exclude time-dependent phenomena in steels at room temperatures.

It is hardly possible to explain such an increase from the standpoint of deformation theories and flow theories with isotropic hardening. From the standpoint of translational hardening, it may be assumed that the observed increase is a consequence of an additional translation of the yield surface during the period of direct “neutral” loading, with a substantial change in the plastic state.

REFERENCES

1. G. A. Doshchinskii, V. I. Maksak. Plastic deformation of steel under complex loading with constant stress intensity. Engineering Journal. Solid Mechanics, No. 3, Publishing House of the Academy of Sciences of the USSR, 1966.

2. A. M. Zhukov. Certain features of the curve of neutral stressing. Izvestiya of the Academy of Sciences of the USSR, OTN, No. 8, 1958.

3. A. A. Ilyushin, V. S. Lenskii. Strength of Materials, State Publishing House of Mathematical Literature, Moscow, 1959.