VARIATION OF THE NORMAL ELASTIC MODULUS WITH INCREASING DEGREE OF PLASTIC DEFORMATION

G. A. Doshchinskii, N. V. Pogozheva, A. M. Pogozhev

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

The theory of plasticity regards the unloading process as an elastic process; therefore, the components of deformation in a plastically deformed body are determined by means of the elastic constants of the material, which are considered unchanged also in the plastic region.

But beginning in the late 1950s, works by various researchers appeared in print indicating that the elastic characteristics of materials do not remain constant during plastic deformation. In 1958 A. M. Zhukov [1] pointed out this phenomenon. In work on the investigation of the neutral-loading curve, as a secondary result it was found that a preliminary axial plastic deformation of approximately 2% caused a decrease in the shear modulus by almost 19%, while a preliminary plastic shear deformation of about 4% gave a decrease in \(E\) of almost 20%.

This fact required the formulation of a special experiment, the purpose of which was a more careful study of the behavior of the moduli of normal and tangential elasticity during plastic deformation [2]. In the first series of the experiment, the influence of plastic tension on the shear modulus was studied. The maximum decrease in \(G\) was 16.5%. The most intensive decrease occurred near the yield plateau (steel specimens were tested). But no definite regularity of this decrease was obtained.

Another series of tests was carried out to determine the influence of plastic shear on the elastic moduli. The maximum decrease in \(E\) in this case was 8%.

In 1961, a similar experiment was carried out by O. A. Shishmarev and E. Ya. Kuzmin [3]. The results of this experiment qualitatively confirmed the conclusions obtained in [1], [2], but the degree of decrease in \(E\) and \(G\) did not exceed 6–8%.

In 1967, Yu. A. Sazonov carried out an experiment to determine the change in the elastic moduli of nickel and copper during plastic deformation [4]. The experiment was distinguished by particular thoroughness. As a result it was established that the normal elastic modulus of copper drops sharply at the beginning of plastic deformation, and with further deformation slowly increases. In nickel, after a sharp drop at 2% plastic deformation, no further decrease in the modulus occurred. Quantitatively, these changes amounted to about 5%.

From the works listed it becomes clear that, as plastic deformation increases, elastic characteristics may change noticeably; however, the question of the regularity of their change and of its quantitative limits remains unresolved.

At the same time, obtaining such a regularity would be of great importance in the theory of plasticity when determining the components of plastic deformation. Acceptance of the fact that elastic constants vary in the process of deformation essentially leads to the need to revise certain propositions of the modern theory of plasticity, which is based on conceptions of absolute constants of the elastic behavior of a material.

In particular, in the work of A. A. Ilyushin [5] it was shown theoretically that such consequences of Drucker’s postulate as the proposition that the vector of the increment of plastic deformation coincides with the normal to the yield surface, and the proposition on the convexity of this surface, are valid only under the assumption that plastic deformation of the body under any loading is not accompanied by a noticeable change in elastic properties.

In the present work, the results of experiments are set forth whose purpose was: to confirm (or not to confirm) the fact of a change in elastic characteristics during plastic deformation; to refine the regularities of this change; and to establish the quantitative limits of this phenomenon.

The article presents the results of a study of the change in the normal elastic modulus \(E\) as a function of the degree of preliminary plastic extension.

At first, the experiment was carried out on Gagarin’s press, the main advantage of which over other testing machines is that it provides automatic recording of the loading diagram on a large scale. A broad series of materials was tested: several grades of steels, including low-carbon, high-carbon, and alloy steels, aluminum, duralumin, copper, brass, bronze, zinc, tin, lead, and cadmium. In all, 42 specimens were tested. The specimens were preliminarily subjected to recrystallization annealing. The unloading–loading cycle was repeated after certain stages of plastic deformation. The modulus of elasticity at each stage of plastic deformation was determined as the angle of inclination to the unloading straight line.

To exclude the influence of secondary factors, a relative comparison of the angular coefficients of the unloading straight lines was carried out. The modulus of elasticity in each cycle was determined with allowance for the changing geometric dimensions of the specimens.

But as a result of the study it became clear that an experiment of this kind, i.e., determination of the normal elastic modulus, in principle cannot be carried out on Gagarin’s press. The reason is that crushing of the specimen heads, which is included in the total deformation of the specimen and is reflected in the diagram, can distort the true picture of deformation. This circumstance was pointed out by A. M. Zhukov. Since determining the influence of the heads and separating it from the total deformation is practically impossible, the results obtained by us on Gagarin’s press proved doubtful, although they did indicate the possibility of different laws of variation of \(E\) for different metals.

Therefore it was decided to carry out a series of experiments with the same aim on universal testing machines. Round specimens 15 mm in diameter and 320 mm long were used, made of grade St. 3 steel and several nonferrous metals: copper, brass, and bronze. In all, 14 specimens were tested. The specimens were preliminarily subjected to recrystallization annealing. The forces were measured on the scale of the machine. Elastic deforma-

—by mechanical strain gauges, while plastic deformations were measured by an indicator system. The unloading–loading cycle was also repeated after certain stages of plastic deformation. The modulus of elasticity for each stage was determined as the angular coefficient of the loading line, and, in order to exclude the influence of secondary factors, the results were carefully compared. Figure 1 presents curves of the change in the modulus of normal elasticity for the specimens tested. The moduli were determined both with allowance for the changing geometrical dimensions of the specimens and without it. For specimens of steel 3, the curves of the change in

Fig. 1

$E$ were constructed with allowance for the changing cross-sectional area. The curves determined without allowance for the influence of the geometrical dimensions do not differ qualitatively from the former; they pass only somewhat lower and are not shown in the figure. That is, for the steel specimens the physical changes occurring with the growth of plastic deformation predominate over the geometrical influence; therefore the latter does not introduce a qualitatively different picture into the regularity of the change in $E$. As can be seen from Fig. 1, the modulus of normal elasticity for steel 3 decreases sharply at the beginning of plastic deformation, by 12–18%, and, with further growth of deformation, becomes noticeably stabilized.

For nonferrous metals, in Fig. 2 the solid lines show the curves of the change in $E$ with allowance for the changing cross-sectional area, while the dashed lines show the curves without allowance for the change in geometry. If for the steel specimens there was no qualitative difference between the two such approaches to determining $E$, then in nonferrous metals, evidently, the physical phenomena occurring in the process of plastic deformation are not so pronounced, and therefore the correction for the changing cross-sectional area significantly changes the character of the curve. If the geometrical dimensions of the specimens are taken as constant, the modulus of normal elasticity remains approximately constant and decreases somewhat only at large degrees of deformation. If a correction is introduced for the change in

sectional dimensions, then the modulus determined in this way increases gradually for brass and bronze from the beginning of deformation, while for copper—after an insignificant decrease in the initial stage of deformation.

As a result of the investigations carried out, it was possible to trace the regularities in the change of this elastic characteristic for the tested materials at degrees of plastic deformation up to 10%. These regularities are quantitatively and qualitatively different for each material and, apparently, are predetermined by the structural features of the metal.

In the plots: \(E_n/E_0\); copper; brass; bronze; \(\delta\%\).

Fig. 2

LITERATURE

1. A. M. Zhukov. Certain features of the neutral loading curve. Izv. AN SSSR, OTN, No. 8, 1958.

2. A. M. Zhukov. Elastic properties of plastically deformed metal and complex loading. Inzhenernyi sbornik. Publishing House of the Academy of Sciences of the USSR, vol. 30, 1960.

3. O. A. Shishmarev, E. Ya. Kuzmin. On the dependence of the elastic constants of a metal on plastic deformation. Izv. AN SSSR, OTN, “Mechanics and Machine Building”, No. 3, 1961.

4. Yu. A. Sazonov. On measurements of the elastic moduli of copper and nickel in the process of plastic deformation. Trudy [[unclear: abbreviated institution]], Leningrad, 1967.

5. A. A. Ilyushin. On the nature of plastic deformation and the surface of flow. Applied Mathematics and Mechanics. Publishing House of the Academy of Sciences of the USSR, vol. XXIV, issue 4, 1960.