EXPERIMENTAL STUDY OF COMPLEX LOADING WITH ROTATION OF THE STRESS TENSOR AXES UNDER UNIAXIAL COMPRESSION

V. I. MAKSÁK, G. A. DOSHCHINSKII

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

In recent decades, in the field of continuum mechanics, the theory of plastic deformation has undergone intensive development. However, sufficiently reliable results have so far been obtained only for simple loading, i.e., loading in which the components of the stress tensor (or of its deviator) increase proportionally to a single parameter. For the case of complex loading, however, there is no reliable theory that could yield results satisfying the needs of practice.

The study of the regularities of complex loading should probably be divided into the following three lines of investigation:

1) the components of the stress tensor remain constant in magnitude, but the direction of the principal stresses changes;

2) the magnitude of the components of the stress tensor changes nonproportionally to a single parameter, while the direction of the principal stresses remains unchanged;

3) various combinations of these methods.

Investigations show that the rotation of the axes of the principal stresses has a very strong influence on the relationships between stress and strain. The present state of research on complex loading in general is such that, for the most part, experimental facts are being accumulated for a general theory of complex loading. Numerous theoretical and experimental studies are directed toward finding the general laws to which the plastic deformation of a polycrystalline material is subject under arbitrary deformation or loading [1].

The present investigation is an attempt to clarify the qualitative nature of the interrelation between stresses and strains for a linear stress state when the axes rotate but the stress remains constant. In the experiments, linear compression of specimens in the form of cubes and plates was used. Rotation of the axis of the principal stress was accomplished by cutting, from a specimen that had been subjected to plastic compressive deformation up to a certain stress, a new specimen whose faces were rotated relative to the old ones by some angle. The specimen was then again subjected to compression up to the same stress; from it a new specimen with rotated faces was cut out, and so on.

In the experiments, duralumin D16 was investigated. To eliminate the influence of the initial treatment of the material on the results of the experiment, the specimens were annealed before testing.

When new specimens were cut from cubes, geometric similarity was preserved; in the plates their thickness remained constant.

The cutting of specimens was carried out on a planing machine, under conditions that ensured insignificant work hardening during machining. To determine the effect of the planing operation on the experimental results, the following test was performed. After deformation of a specimen, a new one was cut from it in such a way that its faces remained parallel to the former ones. The specimen was then compressed in the same direction and to the same stress as at the beginning. In this case, additional plastic deformation was practically absent.

Deformation was measured with mechanical Huggenberger-type strain gauges with a scale division of 1 micron. Deformation readings were taken half a minute after the load had been applied.

The obvious shortcomings of the experimental method employed are: the nonuniformity of the stress state, arising as a result of the nonuniform distribution of pressure on the face of the cube; friction at the ends of the specimen; and the absence of continuity during rotation because the specimen is unloaded before rotation of the stress axes.

Fig. 1.

However, testing specimens in the form of cubes is used in the study of the laws of the theory of plastic deformation. In particular, it was used in studying the influence of the spherical tensor and the form of the deviator in the works of Vinogradov, Chaplinskii, Yagn [2], [3], and others. Since the purpose of the present investigation is to ascertain only qualitative dependences, this method, despite its shortcomings, could be acceptable.

The deformed state under linear compression is characterized by three principal elongations: \(e_1\), \(e_2\), \(e_3\). As measures of the stressed and deformed states in the deformation theory of plasticity, the intensities of stresses and strains are used, defined by the expressions:

\[ \sigma_i=\sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2-\sigma_1\sigma_2-\sigma_2\sigma_3-\sigma_3\sigma_1}, \]

\[ e_i=\frac{\sqrt{2}}{3}\sqrt{(e_1-e_2)^2+(e_2-e_3)^2+(e_3-e_1)^2}. \]

For a linear stress state they correspond to:

\[ \sigma_i=\sigma,\quad e_i\approx e, \]

Two series of tests were carried out. In the first, 8 specimens in the form of cubes were tested. Two specimens were used to obtain the diagram of linear compression. The remaining specimens were compressed to a stress of \(16000\ \text{kg}/\text{cm}^2\) and unloaded. From specimens Nos. 3, 4, 5, cubes were cut whose faces were rotated through an angle \(\alpha=10^\circ\) relative to the old ones, with two vertical faces remaining perpendicular to the axis of rotation. The specimens were compressed again to a stress of

\(16000\ \text{N}/\text{cm}^2\) and unloaded. This operation was repeated 3 times, i.e., the angle of rotation of the stress reached \(30^\circ\). Figure 2 shows the initial diagram and the plastic-deformation curves of specimen No. 5 for the individual stages of the test. For the other specimens the curves were similar.

From specimens Nos. 6, 7, and 8, small cubes were cut out whose faces were rotated respectively by \(20^\circ\), \(30^\circ\), and \(40^\circ\). They were then compressed to a stress of \(16000\ \text{N}/\text{cm}^2\).

In the second series, 5 plates of the same material, with dimensions \(50 \times 50 \times 9\) (mm), were tested. The plates were compressed to a stress of \(12000\ \text{N}/\text{cm}^2\) and unloaded. From plates Nos. 2 and 6, new ones were cut out with their faces rotated by \(15^\circ\). From plates Nos. 4 and 5, new specimens were cut out with a rotation of \(30^\circ\), and for plate No. 1—with \(45^\circ\). Then all of them were compressed to a stress of \(12000\ \text{N}/\text{cm}^2\) and unloaded. A similar operation was repeated 3 times. The plastic-deformation curves for the individual stages of testing of the plates are analogous to the corresponding curves for the tests of the cubes, and therefore are not given here.

Fig. 2.

For simple loading of an isotropically hardening material, at constant stress intensity no change in strain intensity occurs, i.e., plastic deformation does not increase. In the experiments presented, the following was observed. If, under simple compression, the unloading and reloading processes up to the same stress are elastic processes, then when the axes are rotated, upon reloading, some additional plastic deformation systematically arises. If from specimens predeformed with rotation of the stress axis by an angle \(\alpha_1\), new specimens are cut out with a rotation angle \(\alpha_2 = \alpha_1 + \Delta \alpha\), then upon loading some plastic deformation again occurs. In the experiments, some shortening of the linear portion of elastic deformation during reloading was observed as the angle of rotation increased, i.e., the development of plastic deformation began at lower values of the compressive stress compared with the initial stages. The nature of the increase in strain intensity as a function of the angle of rotation at constant stress intensity is shown in Fig. 3.

In all cases, the increment in strain intensity (the total over successive stages) proved to be approximately proportional to the increase in the angle of rotation of the axis of the principal compressive stress.

Thus, under the conditions of the experiments carried out, it was observed that rotation of the principal stress axes, with the magnitude of the stress intensity unchanged, for a linear stress state is associated with an increase in the degree of plastic deformation.

Fig. 3.

REFERENCES

1. Yu. N. Rabotnov. Mechanics of a rigid body and paths of its development. Izvestiya AN SSSR, Mekhanika i mashinostroenie, No. 2, 1962.

2. N. N. Viktorov, Yu. I. Yagn. On the laws of plasticity. Fizika metallov i metallovedenie, vol. 3, issue 1, 1956.

3. I. A. Chaplygin, Yu. I. Yagn. On the resistance of metals to plastic deformation. Doklady AN SSSR, vol. XC, No. 6, 1953.