On vector block elements in mechanics problems

Abstract

A plane dynamic problem of the second kind for the Lame equation is considered in the first quadrant. For the first time the exact solution of this boundary value problem in the form of a Packed vector block element is constructed by the block element method. A system of Two lame differential equations is considered in the boundary value problem. To solve it by the block element method, operations of external algebra, external analysis are performed and a vector block element consisting of two components is constructed. For its construction there is a problem of differential factorization of a matrix-function of the second order-coefficient of the functional equation necessary for the correct construction of pseudo-differential equations. Their solution allows you to build components of the external form and the packaged block element itself. The solution of the considered boundary value problem is of interest because it serves the purpose of substantiating the existence and investigation of the properties of cracks of a new type, where previously the antiplane problem was used for these purposes. Also, the solution of the problem is important in the development of methods for designing materials based on block elements, in the analysis of landslides, in the problems of seismology in the analysis preparation of crustal earthquakes.