In the present review investigation results are presented on the elastic wave propagation in laminated and fibrous unidirectional composite materials modeled by the piecewise-homogeneous medium (the structural model). In contrast to continual theories, the model of such a type does not essentially impose additional restrictions (other than those postulated by solid deformable body mechanics) on frequencies and gradients of the spatial alteration of the wave processes investigated. However, it should be stressed that, within the framework of this approach, it is necessary to solve complicated boundary-value or initial-boundary-value problems, which considerably complicate investigations in this area. Currently, regularly laminated materials are most thoroughly investigated. A detailed, comprehensive analysis is given of bulk, surface, and normal waves. The rule of selection, choice of modes, is formulated for piecewise-inhomogeneous spectra, the structure of the pass bands zones for bulk, shear and longitudinal–transverse waves is described with its dependence of the relative thickness and mechanical properties of layers, and vibration modes on transmission zone boundaries are determined. The theory of Love- and Rayleigh-type surface waves is presented. These waves may propagate in regularly laminated composites at frequencies corresponding to the stop band zones of bulk waves. A highly significant influence of the correspondence of the set of frequency and other composition properties to the pass band or stop band zones for bulk waves in the reflecting materials is noted on the reflection character of shear and longitudinal–transverse plane waves. The existence of frequency intervals and several incidence angles in the cases of the complete internal wave reflection is shown. Penetration of surface disturbances deep into the regularly laminated materials was investigated on plane cylindrical and spherical interface surfaces of properties. The character of the displacements and the stress attenuation is essentially different for pass band and stop band zones of bulk longitudinal and transverse waves in the medium with plane boundaries. Results are also presented for thermoelastic, magnetoelastic and electroelastic (acoustoelectric) waves. The application of the superposition principle and the summation theorems for cylindrical functions for unidirectional fibrous materials with regular packing allows us to construct formal solutions both for doubly periodic media and for the separately situated row of elastic inclusions with periodic location. Solutions of such a type may be extended to smooth inclusions of noncircular cylindrical form. In all cases boundary value problems are reduced to infinite systems of algebraic equations with complex coefficients containing cylindrical functions. For the row of periodically located fibers, the informal character of solutions is shown, and infinite systems of equations are investigated. Specific quantitative results are also obtained. The diffraction of shear and longitudinal-transverse waves on solid and hollow fibers was analyzed. In the discrete frequency spectrum, the existence of resonance effects of the Wood-anomaly type is shown. For shear waves on separately located inclusion, the influence of the noncircular fiber form on the stress distribution was investigated. The prospects for development of wave theory are pointed out within the framework of the structural composite model.

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