Self-similar problems in elastodynmics

Abstract

This paper presents a formulation of self-similar mixed boundary-value problems of elastodynamics that is a natural extension of one already developed by the writer for elastostatic problems. By thus exposing the analytical structure that is common to both the dynamic and static problems, the existence of properties common to certain static and dynamic problems is explained, and further such properties are derived. Common features of both two-dimensional and three-dimensional problems are brought out by reducing them to Hilbert problems, directly in two dimensions and by introducing the Radon transform in three dimensions. Several applications of the theory are presented, typical problems involving the indentation of a half-space by a conical or wedge-shaped indentor, and cracks expanding under the influence of a non-uniform applied stress. More difficult problems, that have not before been formulated, include dynamic indentation problems with adhesion, and problems of cracks expanding on interfaces between dissimilar materials. A method of solution of such problems is presented and an example of each type is worked out in detail. The method of analysis hinges upon representations of the solutions of ' unmixed ’ self-similar problems for half-spaces, which are obtained by use of an alternative to Cagniard’s technique whose application is routine, even for an anisotropic half-space. The representations provide more general solutions of the unmixed problems than were available previously. The main singularities, or ' arrivals ’, of the stress fields are extracted from the representations; these expressions are new and should be useful for certain problems in seismology. It is predicted, for instance, that a crack expanding on an interface can generate a ‘conical wave’, that is, a region in which the singularity has a logarithmic component as well as a step function, even in its P -wave arrival, which could not occur for a crack in a homogeneous medium. The properties of the equations of elastodynamics that are employed are that they are linear, homogeneous and self-adjoint and the methods that are developed are equally applicable to any other system with these properties.