STUDY OF CHANGES IN THE ELASTIC CONSTANTS OF A MATERIAL UNDER PLASTIC DEFORMATION

G. A. Doshchinskii, N. V. Pogozheva

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

Since the late 1950s, works have periodically appeared in print [1, 2, 3, 4] noting the possibility of changes in the moduli of normal and shear elasticity in the process of plastic deformation. The results of these investigations have shown that the named constants for metals may change substantially, but no definite qualitative and quantitative regularity of their change with increasing degree of plastic deformation has been established.

The aim of the present work was to investigate changes in the modulus of normal elasticity and Poisson’s ratio, as well as the quantities derived from them—the shear modulus and the coefficient of volumetric elastic deformation—in the process of plastic drawing. To increase the reliability of strain measurements and, accordingly, of determining the elastic constants, large-size specimens of 20Kh steel, 700 mm long and 45 mm in diameter, were used in this work. The investigation was carried out on a 100-ton Olsen’s testing machine. Loads were measured by the machine scale; longitudinal elastic strains by mechanical Huggenberger tensometers; longitudinal plastic strains by an indicator system; and transverse strains by a micron indicator. The unloading–loading cycle was repeated after increments of plastic deformation approximately equal to one percent. For each specimen the elastic characteristics were determined at 10–12 stages, so that the total deformation of the specimen reached 9–10%.

The modulus of normal elasticity for each stage of plastic deformation was determined as the angular coefficient of a straight line constructed from points corresponding to a series of loading stages in the coordinates: stress—longitudinal relative elastic strain; and Poisson’s ratio—as the angular coefficient of a straight line constructed analogously to that noted above in the coordinates: elastic longitudinal—elastic transverse relative strain. Both characteristics were determined taking into account the dimensions of the specimen changing in the course of plastic deformation. Then the modulus of normal elasticity \(E_n\) and Poisson’s ratio \(\mu_n\), determined in this way for each stage, were related, respectively, to the initial values \(E_0\) and \(\mu_0\), obtained on the basis of several preliminary elastic loadings. From these ratios, curves of change of \(E\) and \(\mu\) were constructed in the coordinates \(\dfrac{E_n}{E_0} - \sigma'\%\) and \(\dfrac{\mu_n}{\mu_0} - \sigma'\%\), which are shown on...

Fig. 1 for three specimens, with different designations of the experimental points.

Using the values of \(E_n\) and \(\mu_n\) obtained for each degree of plastic deformation, according to the formulas \(K=\dfrac{E}{3(1-2\mu)}\) and \(G=\dfrac{E}{2(1+\mu)}\), the values of the derived elastic characteristics were determined: the coefficient of bulk elasticity, \(K_n\), and the shear modulus, \(G_n\), respectively, for the same degrees of plastic deformation. The curves of change of \(K\) and \(G\) were then plotted from the ratios of these quantities to their initial values, in the coordinates \(\dfrac{K_n}{K_0}\)—\(\delta\%\) and \(\dfrac{G_n}{G_0}\)—\(\delta\%\), shown in Fig. 2.

Fig. 1

The results of the study showed that the modulus of normal elasticity \(E\) and of tangential elasticity \(G\), with increasing axial plastic deformation, decrease for this material by 15–18%; moreover, the most intense change in these “constants” occurs in the initial stage of plastic deformation (during the yield period). With further growth of plastic deformation, these characteristics become noticeably stabilized.

Fig. 2

The coefficient of elastic transverse deformation \(\mu\) for the tested metal increases by 12–15% in the process of plastic deformation; moreover, for this characteristic the same feature is observed, i.e., it changes intensively within the range of 1–1.5% plastic deformation, and subsequently changes only insignificantly.

The coefficient of volumetric elastic deformation remains practically unchanged in the process of plastic deformation, equal to the original value obtained under the initial loading.

It should be noted that the noticeable scatter of the points (Figs. 1, 2) is not a consequence of experimental errors. The experimental points on the basis of which the angular coefficients corresponding to the elastic constants were determined lie quite clearly on straight lines, testifying to the reliability of the measurements. The fluctuations in the values of the constants are due to nonmonotonic changes in the structural constitution of the polycrystalline aggregate of the metal in the process of plastic deformation and, correspondingly, of its integral elastic qualities, whose expression is the elastic constants.

Fig. 3

Also deserving attention is the circumstance that the character of the changes in the elastic constants has a certain correspondence with the change in the magnitude of the Bauschinger effect. In particular, in studies [5] (for this same steel) a substantial change in the Bauschinger effect was noted in the initial stages of plastic deformation, followed by stabilization of the effect. Apparently, this analogy is predetermined by the fact that at the basis of these changes in both cases lies the influence of residual microstresses, which are the initial cause of the occurrence of anisotropy of strengthening.

LITERATURE

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

2. A. M. Zhukov. Elastic properties of plastically deformed metal and complex loading. Engineering Collection, Vol. 30, Moscow, 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, Moscow, 1961.

4. Yu. A. Sazonov. On the change in the elastic moduli of copper and nickel in the process of plastic deformation. Proceedings of LPI, No. 278, Leningrad, 1967.

5. Z. P. Kamentseva. Study of the Bauschinger effect under small plastic deformations. Collection Studies in Elasticity and Plasticity, No. 6, LSU Publishing House, Leningrad, 1967.