Elasticity is a fundamental property of all substances occurring in nature. The theory of elasticity is concerned with the study of the response of elastic bodies to the action of forces. The importance of the application of the theory of elasticity in physics and technical mechanics involved in the stated problems from the viewpoint of mathematical analysis attracted the attention of the greatest investigators of the 20th centuries to this branch of science. This thesis is devoted to study some problems in the mechanics of elastic bodies, which might have some interesting applications in the Earthquake science, Geophysics, ..., etc. or in the industry, under certain boundary conditions. The topics discussed here are classified into two parts. The first and second chapters are concerned with the determination of frequency in orthotropic non-homogeneous cylinder and sphere. The third and fourth chapters are concerned with the determination of stress in a dynamical non-homogeneous cylinder containing core, using the stresses-strains relationship in which the elastic constants and density are taken as functions of the radial coordinates in the absence of the body forces