Internal Friction in Solids V. General Theory of Macroscopic Eddy Currents

Abstract

In vibrating ferromagnetic metals macroscopic eddy currents tend to shield the interior of the metal from changes in magnetic induction. The dissipation of energy by these eddy currents contributes to the internal friction of the metal. This internal friction has previously been investigated theoretically only for longitudinal vibrations, and only the asymptotic formulae for high and low frequencies have been found. In this paper this internal friction is calculated for all frequencies for both longitudinal and transverse vibrations. The theory of internal friction due to macroscopic eddy currents is shown to be formally identical with the theory of internal friction due to macroscopic thermal currents. The methods developed by the author for the study of thermoelastic internal friction are thus directly applicable to the study of the macroscopic electric eddy currents. §1. INTRODUCTION P ART of the internal friction of ferromagnetic metals arises from the magnetic-elastic coupling. Under certain conditions this part is of a larger order of magnitude than the internal friction of nonferromagnetic origin. In spite of numerous experimental investigations, 1 only one source of ferromagnetic internal friction has been theoretically investigated 2 for small strains, namely that due to the macroscopic eddy currents. These eddy currents tend to shield the interior of the sample from changes in magnetic induction. Further, this effect has only been investigated for longitudinal vibrations, and only the asymptotic values for high and low frequencies have been obtained. In this paper the internal friction due to macroscopic eddy currents is calculated at all frequencies for the longitudinal and transverse vibrations of circular rods, and for the transverse vibrations of reeds.It is found that the internal friction due to the eddy currents may be written in the form .tively. The first factor may be calculated by means of Eq. (14). The second factor is given in §3 for the three cases, and is plotted in Fig. 1. The asymptotic expressions for / at high and low frequencies are shown in Fig. 1 by broken lines, and are likewise given explicitly in §3. The frequency of maximum internal friction is determined by the magnetic diffusion constant DH-This is given bywhere /J, is the differential magnetic permeability, and the electric conductivity a is in ohm -1 cm -1 . The area beneath all three curves is the same, namely %ir logio £ = 0.682. This is a direct consequence of the general formulavalid for a specimen of arbitrary cross section in which the only stress associated with the vibration is a tensile stress along the axis of the specimen. The method used in the present investigation is that developed by the author for studying the internal friction due to macroscopic thermal currents. 3 The differential equations for the two problems become identical when T and S are replaced by H and B/4:ir, respectively. The two problems would be identical if the adiabatic boundary condition for thermal flow were replaced by an isothermal boun...

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