The propagation of harmonic elastic waves through composite media with a periodic structure is analyzed. Methods utilizing the Floquet or Bloch theory common in the study of the quantum mechanics of crystal lattices are applied. Variational principles in the form of integrals over a single cell of the composite are developed, and applied in some simple illustrative cases. This approach covers waves moving in any direction relative to the lattice structure, and applies to structures of the Bravais lattice groups which include, for example, parallel rods in a square or hexagonal pattern, and an arbitrary parallelepiped cell. More than one type of inclusion can be considered, and the elastic properties and density of the inclusion and matrix can vary with position, as long as they are periodic from cell to cell. The Rayleigh-Ritz procedure can be applied to the solution of the variational equations, which provides a means of calculating dispersion relations and elastic properties of specific composite materials. Detailed calculations carried out on layered composites confirm the effectiveness of the method.
Variational Methods for Dispersion Relations and Elastic Properties of Composite Materials
Kohn, W., Krumhansl, J. A., and Lee, E. H. (June 1, 1972). "Variational Methods for Dispersion Relations and Elastic Properties of Composite Materials." ASME. J. Appl. Mech. June 1972; 39(2): 327–336. https://doi.org/10.1115/1.3422679
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