NEW EXPERIMENTAL DATA ON “DISLOCATIONS” IN METAL

In the modern theory of the mechanical properties of metals, an important role is played by the concept of so-called “dislocations.” A dislocation is a linear defect of a lattice; it is characterized by the fact that in one of the rows of the lattice there is an extra ion or, conversely, an ion is missing. Figure 1 shows the arrangement of atoms (lattice sites) in a dislocation. Inside the circle, above the line \(AB\) (along it) there is one more cell (atom) than below it. It is not hard to see that the stress required to move a dislocation by one interatomic distance is very small compared with the stress required for a relative displacement of the planes of a crystal (approximately \(10^6\) times smaller). Indeed, pairs of atoms arranged symmetrically on both sides of the center of the dislocation (Fig. 2, atom б) experience equal and opposite forces from the upper plane. Consequently, if atoms near the center of the dislocation are displaced to the same distances, then half of them experience forces opposing the displacement, while the other half experience forces assisting the displacement (see Fig. 2); thus

Fig. 1.

in the first approximation the work expended on moving dislocations is equal to zero, i.e., dislocations are very mobile. The stressed state in a metal is characterized by the number of dislocations and their arrangement. The theory of dislocations explains a number of regularities in the mechanical properties of metals (see $^{1,2}$).

The change of stresses in a specimen is connected with the movement of dislocations through the metal. During this movement, resistance forces arise (the interaction of dislocations with the atoms of the lattice, with one another, with impurity atoms, etc.); therefore the motion of dislocations is a non-simple process: it leads to the dissipation of energy and to the appearance of internal friction.

Normal arrangement of chains

Dislocation

Fig. 2.

If such an explanation of the effect of internal friction is correct, then under certain conditions (a period of oscillations equal to the time for establishing the stressed state) internal friction must reach a maximum $^{1,3,4}$. However, until very recently such a maximum of internal friction had not been observed. This can be explained by the high mobility of dislocations in pure metals and, consequently, by a very short relaxation time. In alloys, however, dislocations must move more slowly. The reason is that the state in which an impurity atom is located in a dislocation is energetically more favorable than the state in which the impurity atom and the dislocation are separated $^5$. Therefore, on the one hand, dislocations attach impurity atoms to themselves, and on the other, in the places where impurity atoms are located, the motion of free dislocations is considerably hindered. For these two reasons the relaxation time increases, and maxima of internal friction should be expected, even at room temperature, at comparatively low oscillation frequencies. Recently, a member of the Institute of Technical Physics of the Academy of Sciences of the Chinese People’s Republic in Peking, Tsang Ch‘in-sui, succeeded for the first time in observing this maximum on curves of the dependence of internal friction on temperature and on stress $^6$. In his experiments the coefficient of internal friction was defined as $\pi$ divided by the logarithmic decrement of free torsional oscillations of the specimen

$$ \left(\frac{\vartheta}{\pi}=Q^{-1}\right). $$

The specimen studied was prepared by vacuum melting an aluminum bar (Al of 99.991% purity) with balls of electrolytic copper. Chemical analysis showed that the resulting alloy contained 0.5% Cu and less than 0.01% Fe and Si. After the appropriate thermal and mechanical treatment the specimen was given the form of a rod about 30 cm long and 0.83 mm in diameter. Then this rod was annealed at $300^\circ$C (annealing time 1 hour). Metallographic examination after annealing showed that the recrystallization process in the specimen had already begun, but was still far from complete; most of the grains were of irregular shape, with distorted boundaries.

The annealed specimen was placed in an apparatus constructed by the author7 for torsional oscillations, and then the amplitudes of successive small deflections were measured. A small mirror was fastened to the rod, as usual; the beam of light reflected from the mirror fell on a scale, the distance between the mirror and the scale being 3 m. At the maximum amplitude of the oscillations, the displacement of the light spot along the scale did not exceed 3 cm. Fig. 3 (curve a) shows the successive amplitudes of the free torsional oscillations of the specimen at \(t = 50^\circ\)C. The abscissa axis gives the ordinal numbers of the oscillations; the ordinate axis gives the corresponding displacements of the light beam along the scale. The tangent of the angle of inclination of the tangent to this curve is equal, for small damping, to the logarithmic decrement of the oscillations. It is seen from the graph that the coefficient of internal friction \((Q^{-1})\) is small at first, then increases, passes through a maximum, and then falls. The author calls internal friction that obeys such a law anomalous. It is, however, not always observed. In pure Al without an admixture of copper no such anomaly of internal friction was found8. If the alloy studied is subjected to a secondary anneal (for 2 hours at 400°C), the effect of anomalous internal friction also disappears. During such annealing the specimen undergoes complete recrystallization and the grains acquire a regular structure. The law of damping of the oscillations changes (Fig. 3, curve b): the slope of the curve, and therefore the coefficient of internal friction as well, is highly constant.

The amplitude of torsional oscillations is connected with the magnitude of the stresses in the specimen, so that the dependence of internal friction on amplitude can be reduced to the dependence of internal friction on stress. Fig. 4 shows the dependence of internal friction on the magnitude of the stresses in the specimen at 50°C, obtained in this way from curve a of Fig. 3. It should be noted that curves of this type also depend on the “history” of the specimen—on the stresses applied to it in the past and on the time elapsed after annealing. Nevertheless, a large number of experiments carried out by the author under very different conditions show that the very existence of a maximum on the curves of the dependence of internal friction on stress and on temperature is beyond any doubt. This is in full agreement with the theory of dislocations. The number of “pinned” dislocations that have lost their mobility depends on the number of impurity atoms (in the present case, copper) and on the cold working to which the specimen has been subjected. When the stress is still very small, most of the “pinned” dislocations do not manage in one period of oscillation to shift through any appreciable distance, so that the losses due to internal friction are small. When the stress in the specimen, on the contrary, is very large, the “pinned” dislocations break away from the copper atoms surrounding them. Having thus acquired considerable mobility, they scatter little energy in their motion, so that the internal friction is again small. In the intermediate case (at intermediate stresses), when the relaxation time is of the same order as the period of the oscillations, the coefficient of internal friction is considerable.

The author also investigated, at several prescribed stresses, the dependence of internal friction on temperature, studying the temperature dependence of the coefficient of anomalous internal friction, calculated as the difference between the total internal friction (curve a, Fig. 3) and the normal internal friction (curve b, Fig. 3). The dependence obtained is shown in Fig. 5. The characteristic feature is the presence on the curve of a resonance maximum. The author likewise explains this maximum by the pinning of dislocations by copper atoms. At low temperatures the mobility of copper atoms is small, and small mobility is also associated with the dislocations themselves. During a period the dislocations shift very little. With time,

Fig. 3.

Fig. 4.

Fig. 5.

relaxation comparable with the period of oscillation, the internal friction is maximal. At these temperatures the dislocation manages, within one period, to shift by a considerable distance, and at the same time the resistance to its motion on the part of the copper atoms is still large. At higher temperatures the distance traversed by the dislocation during a period increases, but the resistance to its motion on the part of the copper atoms captured by it drops sharply. The losses per period decrease, and the internal friction again diminishes.

M. G.

CITED LITERATURE

1. Ya. I. Frenkel, Introduction to the Theory of Metals, ch. 22, 2nd ed., Gostekhizdat, 1950.
2. F. Seitz, The Physics of Metals, chs. 6 and 10, Gostekhizdat, 1947.
3. Zener, Elasticity and Anelasticity of Metals, Univ. of Chicago Press, 1948.
4. T. S. Kê, Trans. Amer. Inst. Min. and Met. Engrs. 176, 448 (1948).
5. Cottrell, Report on the Strength of Solids, Phys. Soc. London, 1948, p. 30.
6. T. S. Kê, Science Report of Academia Sinica, Peking, 3, 61 (1950).
7. T. S. Kê, Phys. Rev. 71, 533 (1947).
8. T. S. Kê, Trans. Amer. Inst. Min. and Met. Engrs. 188, 575 (1950).