Dislocations as a cause of mechanical damping in metals

Abstract

Zener has shown how thermoelastic effects give rise to damping of the mechanical vibrations of a solid. For example, in a vibrating reed opposite sides are alternately compressed and extended. This gives rise to an alternating temperature-difference across the width of the reed, and the resulting flow of heat leads to dissipation of mechanical energy. In a vibrating single crystal of a metal an additional energy loss is observed which is usually attributed to the motion of dislocations. In the present paper the following mechanism is proposed. Dislocations are trapped in 'potential troughs’ at the minima of the internal stress in the crystal. When the crystal vibrates the dislocations oscillate in their potential troughs and the moving stress-system associated with them produces a fluctuating temperature distribution in the material; this leads to damping as in Zener’s case. The rate of loss of energy produced by a dislocation oscillating with given amplitude is calculated and the effect of a collection of them is discussed. An actual estimate of the damping in a vibrating crystal requires (i) a knowledge of the relation between the amplitude of oscillation of a dislocation and the vibrational stress causing it to move, and (ii) a knowledge of the density of dislocations in the material. A tentative discussion of (i) is given. The quantity (ii) is unknown; however, It is shown that the damping depends only on the ratio of the number of dislocations per unit area to the number of potential troughs per unit area. If this ratio is calculated from the theoretical result and the observed damping in copper single crystals, it is found to be of the order of unity. The present theory predicts that the damping should increase with frequency. This is in disagreement with the limited experimental data available. Two subsidiary effects are also investigated, the thermoelastic damping arising from the interaction between the vibrational stresses and the stresses surrounding stationary dislocations, and the damping due to the emission of elastic waves from an oscillating dislocation. Both these effects are shown to be small compared with the thermoelastic damping caused by moving dislocations.