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NONLINEAR RESONANCE IN CuAlNi ALLOYS

-The phenomenological theory of resonance of a nonlinear anelastic reed is presented. It predicts three modes of oscillation, one in which amplitude and phase are constant and two in which they vary in time. The automodulations corresponding to the latter two can either be sinusoidal or have square wave character. Nonlinear resonances and both types of automodulations have been observed in CuAlNi alloys in the vicinity of the martensitic transformation. They are attributed to the motion of interfaces. Introduction.-The classical theory of the standard or linear anelastic solid (1,2) is not adequate for anelastic studies performed at finite deformations. This is particularly true in investigations of first order martensitic transformations or interface controlled deformations in general where finite deformations are of the essence. In these cases the theory of the standard anelastic solid must be extended to include nonlinear phenomena. In this paper the theory of the nonlinear anelastic solid will be formulated for the case of resonance. The theoretical predictions are qualitatively compared to nonlinear resonances and the accompanying automodulations observed in Cu-Al-Ni alloys and other metals. Equation of Motion of a Nonlinear Anelastic Reed.-Both interface controlled deformations in general and martensitic transformations in particular constitute responses of a solid to a shear stress. The experiments in progress are being performed by bending a reed periodically. It is thus of interest to ascertain how compression and shear deformations enter into the flexural oscillations of a reed. Basically, flexural oscillations of a reed represent standing antisymmetric Lamb waves (3). While the velocity of the Lamb wave is dependent on the thickness of the reed the velocity of the bulk shear wave is solely dependent on the shear elastic constants and the density of the specimen. Hence, it may appear that the relationship between the bulk shear wave and the flexural vibration of a reed is Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19815158 C5-1026 JOURNAL DE PHYSIQUE rather complex. This is not so, however. When the wave length of a deformation wave is comparable to the thickness, the bulk deformation waves can no longer propagate along the direction perpendicular to the surface of the reed. In order to satisfy the boundary condition at the surface, the wave must be reflected at the surface of the reed. The angle of the reflection must satisfy a Bragg condition so that the multiply reflected wave can be guided through the reed. Hence, the deformation wave with a wave length longer than 2d cannot propagate in the reed with a thickness d. As the wave length of the compression wave of a given frequency is longer than the wave length of the shear wave of the same frequency, the vibration of a reed with the lowest possible frequency must consist mainly of the multiply reflected bulk shear wave. It is therefore justified to use the deflection of the reed from its equilibrium position, x, as a shear deformation parameter. The differential equation of motion of a nonlinear anelastic reed has been derived (4) and is given by where w o is the eigenfrequency of the reed and In equation (2) the constants Cri and CUi denote the relaxed and unrelaxed deformation parameters of order i, and E is the external driving force. The equation can be solved by the Krylow-Bogoliubov averaging technique (5). This technique is based on the assumption that the function f represents a small pertubation of the equation of motion of the undamped linear oscillator. The constant amplitude and phase of the solution of the latter may thus be expected to be slowly varying functions of time. Hence, a solution of the form x (t) = a (t) cos (wot+O (t) ) + bsinwlt (3