Explicit forms of the first-order approximate boundary conditions are derived for a 2D problem of SH waves scattering by a thin, curvilinear, elastic, rigidly supported inclusion in a uniform background. The effects of varying elastic modulus and geometrical forms of the inclusion on the stress and strain states of the body near and far from the ends of the inhomogeneity are examined. The method of investigation is based on the matching of asymptotic expansions with the thickness-to-length ratio as the perturbation parameter.
IntroductionThe study of wave propagation in inhomogeneous media consisting of thin-walled layers, confronts with the problem of appropriate approximations for boundary conditions and equilibrium equations of thin-walled inhomogeneities. The general purpose of the approximated boundary conditions, usually referred to as effective boundary conditions, [1, 2], or imperfect interface conditions, [3], is to simplify the analytical or numerical solutions of the wave scattering problem involving complex structures by, e.g., converting a three-media problem into a two-media problem. Such boundary conditions have been widely used in problems of wave propagation and diffraction in inhomogeneous layered media, [4, 5, 6], antenna and radar cross-section studies, [2,7], nondestructive testing of materials, [8-10], modeling of interphases in fiber-reinforced composites, [3,11,12], etc. In the latter case, the interphases constitute thin elastic layers between the fibers and the matrix, where the fiber is usually much stiffer that the matrix material. For this class of structures, only cylindrical or plane unit cell models have been used, see, e.g., [11,12].The goal of the present investigation is to obtain effective boundary conditions for elastodynamic interactions between a thin-walled, curvilinear, elastic, rigidly supported inclusion and the surrounding matrix that would hold for arbitrary values of mechanical parameters of the matrix and the inclusion in the 2D case of SH-waves scattering. The displacement asymptotic near the ends of the scatterer and the approximation of displacements over the scatterer thickness are also considered.
