Dynamic problems of the theory of elasticity for layers and semilayers with cavities

Abstract

We present a solution methodology for dynamic problems of the theory of elasticity based on the fundamental (F)-solutions approach for layers and semilayers containing cavities. Under the proposed solution framework boundary-value problems for three-dimensional cylindrical bodies are reduced to well-studied systems of one-dimensional singular integral equations. With the aid of the integral Fourier transform in time, we study the problem of impulse loading at the sides of cavities. We also demonstrate how the combination of the proposed methodology with the approach of reflections can be used for the solution of analogous problems for semi-infinite layers.

An exact solution of the dynamical problem for the infinite elastic layer with a cylindrical cavity

“…A typical and sufficiently investigated problem of this class is the axisymmetric elasticity problem on the stress state of a layer, weakened by a cylindrical defect, when different boundary conditions are set on layer's faces and defect's surface. Existing research can be divided into three approaches: 1) a construction of an analytic solution of the problem in an explicit form [10], [2]; 2) a construction of an analytical-numerical solution, when the problem is reduced either to an integral equation or to an infinite system of algebraic equations [3], [4]; 3) a numerical solving of the problem [5], [6].…”

“…Dynamic statement of the mentioned problem was considered at the papers [9], [10]. The theory of harmonic oscillations and wave propagation in elastic bodies was widely investigated in the monograph [11].…”

“…The problem of elasticity for an infinite layer with a cylindrical cavity in a static statement was considered by G. Ya. Popov [10], where an exact solution was obtained. In this paper this method was extended on the analogical problem in the dynamic statement.…”

“…where (1) ( ) is the Hankel first order function, = 0, 1, an important relationship has been proven [10] 2 H( ,…”

“…The reality of the solution's values (15) is guaranteed by the special choice of constants 0 , 1 , which can be found from the boundary conditions (10). It leads to the linear system of equations…”