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By Gerald Holton, Stephen G. Brush

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Two cases are considered: perfectly bonded inclusions (displacement and tractions are continuous at the interface) and inclusions separated from the matrix by an imperfect linear interface. 1 Inclusions with perfect bonding at the 0 0 interface Let R be the inclusions radius, and A , fj, , A, /x be the Lame elastic moduli of materials of the inclusion and the matrix. Circular elastic inclusions 29 1. For the case of out-of-plane shear (Laplace's operator), the dipole matrix can be written as follows m = 27T/U / * - W > # T M + Mo where I is the identity matrix.

The analysis uses the multipole method (for a classical scalar version see Lord Rayleigh, 1892) applied to circular inclusions in the two-dimensional linear elasticity (plane strain). 1 Governing equations We study a plane-strain problem for an isotropic elastic body, with the Lame moduli A, /x, containing a set Q, — l j i f2j of two circular inclusions fli, i = 1,2,... 1,712) is the unit normal vector on the boundary of the inclusion. The following conditions are prescribed at infinity an(x) -> oft, of2 C I 2 ( : E ) —» 0, as \x\ —> 00.

The dispersion diagram constructed in accordance with the above formulae exhibits band gaps for the cases of bi-atomic structures of particles of different mass. It can also be shown that a correspondence can be established between this simple lattice structure and a continuum model describing fields in highly porous media. Square array of circular voids. 8). 183) Discrete lattice approximations 53 0 G © 0 O 0 © O O Fig. 8 where kf, = UJ/VI, A square array of circular holes. and v\ = —, the material density is p, and shear modulus is fi.

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