Fundamental solutions of the dynamical equations of elasticity for nonhomogeneous media

Abstract

The fundamental tensor for the dynamical equations of the theory of elasticity for nonhomogeneous isotropic media is constructed. This problem was posed in [1]. The construction is carried out along the lines of the developments in [2]. The immediate application of the results of [2] is not possible here, because the three-dimensional equations of elasticity possess multiple characteristics. In problems in the theory of elasticity vibrations an important role is played by the so-called point sources of vibrations: concentrated forces in infinite space, centers of expansion, double forces, concentrated couples, concentrated moments, and so forth. The known fundamental solution of Voltera for the equations of elasticity represents, as may be easily shown, a combination of a center of expansion and concentrated moments, with corresponding moment axes along the coordinate axes. For a homogeneous elastic medium, the problem of the determination of the effect of a concentrated force varying in an arbitrary manner but always directed along the x-axis has been solved in finite form in [3]. The corresponding problem for a nonhomogeneous medium is considered below.