Conservative boundary-value problems of the theory of elasticity and integral representations of solutions

Abstract

The subject of the research is a mathematical model of the equilibrium of a solid body of satisfactory spatial shape under external load. The development of solid mechanics is largely related to practical goals — calculations of the strength of structural elements and machine parts, the violation of which is usually understood as reaching a state where the structural properties of the product change, rendering it unsuitable for use. Many years of research into the system of differential equations in partial derivatives of the laws of equilibrium in mechanics, known as the Navier-Lame system of equations, has not yet, due to its complexity, achieved the level of development that would guarantee the possibility of obtaining analytical solutions. This raises many questions and misunderstandings regarding the solutions of boundary value problems using widely available application packages based on finite difference approaches. The purpose of this article is to present the results of the development of an alternative method of boundary integral equations, which, due to the boundary conditions of deformation of a continuous medium, leads to a system of linear boundary integral equations with the existing set of solutions. Task: to construct integral representations of solutions to the system of differential equations of equilibrium laws of a solid deformable body using the generalized potential theory method for differential operator forms of the corresponding equations. Scientific novelty. The differential operations of vector-tensor analysis are developed and generalized. The generalized integral theorems for differential operators of the second order adequate to the equilibrium laws of a deformable solid are proved. The results obtained. On the basis of the created generalized apparatus of vector-tensor analysis, analytical solutions in the form of integral representations of the main kinematic characteristics of the problem of deformation of a solid of satisfactory spatial form are constructed. Conclusions. The boundary value problem of the equilibrium of a deformed solid in the presence of an external force is reduced to a system of linear boundary integral equations with respect to the kinematic characteristics of the problem. In addition, it is proved for the first time that all the obtained characteristics are related to the vector momentum potential, which greatly simplifies the integral representation of solutions and numerical implementation of solutions of the corresponding integral equations.