Study of the Elastic Properties of Metal under Large Plastic Deformations
Abstract Generated abstract
This paper examines whether the elastic moduli of metals remain constant during large plastic deformation, a common assumption in classical plasticity theory. Solid cylindrical specimens of low-carbon steel, copper, and brass were plastically twisted to fracture after annealing, with shear and normal elastic moduli measured after successive unloading stages using torsion and tensile testing methods. The experiments indicate that noticeable changes in elastic properties occur mainly during the first 1 to 2 percent of plastic deformation, after which the moduli tend to stabilize despite physical scatter. The authors suggest that, for large plastic deformations, elastic constants may be treated as deformation-independent if determined after preliminary plastic deformation.
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INVESTIGATION OF THE ELASTIC PROPERTIES OF METAL UNDER LARGE PLASTIC DEFORMATIONS
G. A. Doshchinsky, S. Kh. Abdrakhmanov, S. A. Palvinskaya,
T. I. Goloburdo
(Presented by the scientific seminar of the Department of Strength of Materials)
Classical plasticity theory proceeds from the condition that the elastic part of the deformation, even beyond the elastic limit, is determined by the same laws as in the initial elastic stage. It is also assumed that the unloading process obeys the laws of elastic deformation (Gerster’s law). Plastic deformations are usually determined by the difference between the total and elastic deviatoric parts of the strain tensor
\[ Э_{ij}=e'_{ij}-\frac{\sigma'_{ij}}{2G}. \]
It is believed that the elastic properties of a material, even under plastic deformation, are characterized by the same “constants” as under purely elastic behavior.
However, in real metals plastic deformation develops through very complex mechanisms of intra- and intercrystalline rearrangements of the metal structure. In this case, it may be expected that the characteristics of elasticity, which are an integral expression of the properties of a polycrystalline body, may become different from the initial ones.
Studies of recent years, carried out by Zhukov [1], [2], Sazonov [3], and others, show the possibility of changes in the process of plastic deformation of the moduli of normal \((E)\) and shear \((G)\) elasticity.
In experiments [4], [5], the regularities of changes in the elastic “constants” with increasing degree of plastic deformation were studied. It was noted that in low-carbon steels the elastic characteristics change noticeably (by 10–15%) in the initial stages (1–2%) of plastic deformation. Subsequently, a tendency toward stabilization of the elastic characteristics of the metal was observed.
All the studies noted, however, were carried out in the region of small elastoplastic deformations (not exceeding 10%). They do not provide information on the possible limits of change in the elastic “constants” under large plastic deformations implemented in various technological operations in metal forming. Information of this kind makes it possible to refine the general form of the regularity of changes in the elastic properties of metal under plastic deformations. Below are presented the results of experiments with determination of the values of the moduli of el-
stiffness under large plastic deformations up to fracture of the metal.
Large deformations were achieved by torsion of solid cylindrical specimens of low-carbon steel (St. 3), copper, and brass. The elastic moduli were determined from the angular coefficients of direct proportionality under elastic loading of specimens unloaded after the next stage of plastic twisting. Plastic twisting was performed on an Amsler-system torsion testing machine (150 kgm). The angle of twist was determined from the machine dial, recalculated to a working length of 10 cm. Measurement of the elastic angle of twist for determining the shear modulus was carried out by the mirror-and-scale method with a magnification of 3000. To determine the modulus of normal elasticity, after each stage of plastic twisting the specimens were mounted in a Mohr–Federhaff-system tensile testing machine (30 t). In this case, Gutenberg extensometers with a magnification of 1000 were used to measure elastic deformations. Before the tests, all specimens were subjected to recrystallization annealing (St. 3 — \(T_{\text{ann.}} = 700^\circ\text{C}\), copper, brass — \(T_{\text{ann.}} = 500^\circ\text{C}\)). The annealing time was 2.5 hours, with cooling in the furnace. Each value of the modulus of elasticity was determined as the average of several repeated elastic loadings. All tests ended with fracture of the specimens.
Figure 1 presents the relative values \(E/E_0\) and \(G/G_0\) as functions of the angle of plastic twisting for steel specimens. \(E_0, G_0\) are the initial values of the elastic moduli of the metal after annealing. For copper and brass an analogous picture was obtained, only with a greater physical scatter.
It should be noted that the observed scatter of the experimental points is associated with nonmonotonic changes in the elastic properties of polycrystalline metal during plastic deformation, i.e. it is a physical scatter and not a consequence of experimental errors. This conclusion follows from the good reproducibility of the results under several elastic loadings and from the barely noticeable deviations of the experimental points from the straight line of proportionality under each elastic loading for a given stage.
Fig. 1
The results obtained confirm the previously observed result [4], [5], namely that a noticeable change in the elastic properties of a metal occurs only in the initial stages of plastic deformation. As the degree of plastic deformation increases, the elastic properties stabilize. Therefore, for large plastic deformations it appears possible to regard the elastic characteristics of the metal as independent of the degree of deformation. It can only be recommended that the values of the elastic constants of the metal be determined after some preliminary plastic deformation exceeding 1–2%.
It may also be assumed that the corrected gradient principle according to A. A. Ilyushin [6] applies only to the initial stages of plastic deformation,
\[ \frac{d \mathcal{E}}{dS} + (E)'\bar{\sigma} = \lambda \operatorname{grad} f . \]
For the greater part of the plastic region, however, the gradient principle will be acceptable in its simplest form,
\[ d\bar{\mathcal{E}} = d\lambda \operatorname{grad} f . \]
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