On the investigation of elasticity equation for orthotropic materials and the solution of the associated scattering problem

The present work concerns the investigation of the two-dimensional direct scattering problem of time-harmonic elastic waves from bounded anisotropic components of isotropic media. We obtain a Fourier series expansion for the elastic field in the interior of the anisotropic inclusion based on a suitable diagonalization applied to the underlying differential system and a plane wave expansion of the sought field, provided that the inclusion exhibits orthotropic symmetry. This expansion is then exploited to acquire a semi-analytical solution to the associated elastic transmission scattering problem. Numerical results for several geometric configurations and varying degree of anisotropy are presented revealing the pronounced effect of the specific anisotropic character on the scattering mechanism.