Some dynamic problems of the theory of elasticity—a review

Abstract

On the basis of functionally-invariant solution of the wave equation, suggested by Smirnov and Sobolev, a complete solution for one class of self-similar problems of dynamic theory of elasticity is given. This class covers the following problems: 1. (a) a half-plate arbitrary loaded at the boundary including the case when the ends of the loaded area are moving at any constant velocities; 2. (b) the contact problem for a half-plane when the ends of the contact area are moving at any constant velocities; 3. (c) a combination of arbitrary loaded cuts along one and the same straight line which are moving at constant velocities, the velocities of different cut ends being different. The solutions of the above mentioned problems are reduced in the simplest cases to those of Dirichlet problems or of Keldysh-Sedov mixed problems of the theory of analytical functions of a complex variable. The process of finding the solution appears principally not more complicated than that for analogous problems of statics and stationary dynamics (the solutions of the latter problems were obtained, mainly by Kolosov, Muskhelishvili, Galin and Radok). At first the general representations for the solution are derived in terms of analytical functions of complex variable for an arbitrary index of self-similarity; description of the general solution method is given. Then the proposed method is demonstrated for several concrete problems belonging to the class under consideration. We confine ourselves to the plane problems for a homogeneous and isotropic body, yet the method is not difficult to generalize for a case of anisotropic piecewise homogeneous body when the upper and lower half-planes have different elastic constants.