Generalized Solutions of Boundary Value Problems of Dynamics of Anisotropic Elastic Media

Abstract

The method of boundary integral equations (BIE) for the solution of non-stationary boundary value problems (BVP) of dynamics of anisotropic elastic mediums is elaborated. The central moment of this method is constructing the fundamental solutions of equations system, kernels of BIE. Here the fundamental solutions in two-and three-dimensional cases (N, M=2,3) are considered and their properties are studied. In the space of generalized functions the solutions of initial BVP are obtained and their integral representations, regular inside a range of definition are given. Generalizing the Green and the Gauss formulas for the generalized solutions of these equations, singular integral equations for the solution of non-stationary BVP are constructed. The uniqueness theorem of the solutions, including for the class of shockwaves, is presented. STATEMENT OF NONSTATIONARY BOUNDARY VALUE PROBLEMSLet u(x,t) be the solution of the system of hyperbolic equations which describes dynamics of anisotropic elastic mediums. We consider it in the cases of plane (N=2) and space (N=3) deformation: L.A. Alexeyeva, SUMMARY Lij(S x A)Uj (x,t) + G,(x,t) = 0, (x,t)eR N+] (2.1) L ij (d x ,d t ) = C l ?'d m d l -S iJ d?, i,j =1,M, m,l =1, Ν ij V m u l s^lm _ ,-ml ij ~ ij ~ ^ ji · (2.2) 259 Brought to you by | Western University