Non-stationary problem of elasticity for a quarter-plane

Abstract

The plane problem for an elastic quarter-plane under the non-stationary loading is solved in the article. The method for solving was proposed in the previous authors’ papers, but it was used for the stationary case of the problem there. The initial problem is reduced to the one-dimensional problem by using the Laplace and Fourier integral transforms. The one-dimensional problem in transform space is written in vector form. Its solution is constructed as the superposition of the general solution for the homogeneous equation and the partial solution for the inhomogeneous equation. The general solution for the homogeneous vector equation is derived using the matrix differential calculations. The partial solution is found through Green’s matrix-function. The derived expressions for displacements and stresses are inverted by using of mutual inversion of Laplace-Fourier transforms. The solving of the initial problem is reduced to the solving of the singular integral equation regarding the displacement function at the one of the boundary of the quarter-plane. The time discretization is used, and the singular integral equation is solved using the orthogonal polynomials method at the fixed time moments. Based on numerical research some important mechanical characteristics depending on the time and loading types were derived.