No abstract

A system of six two−dimensional plate equations is derived for motions of small−amplitude waves or vibrations superimposed on finite, elastic deformations due to static, initial stresses. In the stress−strain relations, the nonliner terms associated to the third−order elastic stiffness coefficients are included. These equations accomodate the coupling of the six lowest modes of vibration, i.e., the flexure, extension, face−shear, thickness−shear, thickness−twist, and thickness−stretch modes, and all their anharmonic overtones. The new equations are applied to the rotated Y cuts of quartz in studying the thickness−shear and flexural vibrations. The changes in the resonance frequencies of the fundamental thickness−shear vibrations are computed as functions of the direction of initially applied force and of the angle of rotated Y cuts. The predicted results are compared with experimental data and with existing computed results. An explicit formula is obtained for the change of fundamental thickness−shear frequencies in terms of initial deformations and the second− and third−order elastic stiffness coefficients. Subject Classification: 40.24.

Motions of a sandwich plate with symmetric facings are studied in the framework of the three-dimensional equations of elasticity. Both the core and facings are assumed to be isotropic and linearly elastic.Harmonic wave solutions, which satisfy traction-free face conditions and continuity conditions of tractions and displacements at the interfaces, are obtained for four cases: symmetric plane strain solutions for extensional m6tion, antisymmetric plane strain solutions for flexural motion, and solutions for the symmetric and antisymmetric SH-waves. The dispersion relation for each of these cases is obtained and computed. In order to exhibit the effect of the ratios of facing to core thicknesses, elastic stiffnesses and densities, on the dynamic behavior of sandwich plates, dispersion curves are computed and compared for plates with "thick, light, and soft" facings as well as for plates with "thin, heavy, and stiff" facings. Asymptotic expressions of dispersion relations for extensional, flexural, and symmetric SH-waves are obtained in explicit form, as the frequencies and wave numbers approach zero.The thickness vibrations in sandwich plates are studied in detail. The resonance frequencies and modal functions of the thickness-shear and thickness-stretch motions are obtained. Simple algebraic formulas for predicting the lowest thickness-shear and the lowest thickness-stretch frequencies are deduced. The orthogonality of the thickness modal functions is established.

Exact solution of the three-dimensional linear equations of piezoelectromagnetism is obtained for doubly rotated piezoelectric crystal plates surrounded by vacuum and excited by face traction. A generalized Poynting’s theorem is derived for general media in which electromagnetic and mechanical fields interact with each other. For linear piezoelectric crystals it is shown that the generalized theorem may still be interpreted as an energy theorem, and hence densities of energy stored in the electric, magnetic, and elastic strain fields can be identified. Radiated power, per unit surface area and averaged over the period, and induced strain and electric fields in the middle plane of the plate are calculated for doubly rotated quartz plates whose cut orientations follow the upper and lower loci of zeros of the first-order temperature coefficient of frequency of the x1 -thickness-shear mode. Quality factors and partition of stored energies are also examined.

The equations of the dynamical theory of polarizable diatomic lattices, for the long-wavelength approximation, are shown to coincide with Mindlin's continuum equations for diatomic elastic dielectrics. The governing equations of the coupled mechanical and electrical fields account for the surface effects due to elastic deformations and electronic and atomic polarizations and accommodate the optical as well as the acoustical branches in dispersion relations. The material coef6cients in the constitutive relations of the continuum theory are related to the lattice properties via the lattice formulation and their numerical values are computed for NaI, NaC1, KI, and KC1. The long-wavelength»~its of the transverseand longitudinal-optical branches of dispersion relations are predicted and compared with experimental values with close agreement.