The solution of two-dimensional problems of mechanical diffusion using the Volterra integral equation of the 1st kind

Abstract

Calculation of stress-strain state of structures and their elements, working in conditions of unsteady effects of different physical nature, in the general case reduces to the solution of boundary initial value problems of mechanics-related fields. But the solution of unsteady problems in continuum mechanics along with the elastic diffusion problems is associated with serious mathematical challenges. On the on hand, these are due to the need of Laplace transform conversion used to solve problems of this type. On the other hand, the complexity of solving the unsteady problem significantly increases with its dimension. Depending on certain types of boundary conditions, the solution for these problems may be obtained using the Fourier trigonometric series (or sine and cosine transforms), which significantly simplifies the originals’ finding algorithm. The disadvantage of this method is the restricted application area, which is due to the specifics of boundary conditions. We propose a method to solve the initial value problems of elastic diffusion, based on the construction of a system of Volterra integral equations of the 1st kind. These equations connect the right-hand sides of the boundary conditions of two different tasks of the same dimension and geometry. Kernels of integral operators are the Green's functions of a solved problem. The method is demonstrated on the example of two-dimensional elastic diffusion problem for the orthotropic layer. For the solution of the integral equations the quadrature formulas of medium rectangles are used. As quadrature formulas used rectangles formula. These solutions are presented in the form of graphs.