﻿ 数学与系统科学研究院

中国矿业大学理学院

Abstract:    Both the bending problem of the elastic plane problem with stress function as its unknown can be reduced to the boundary value problems of bi-harmonic equation, generally they can be solved by the complex variable function method. Although this method gives the Cauchy integral formula of the complex stress function, it is difficult to be used for the numerical calculation because the Cauchy integrals of various problems must be dealt with by means of some artificial interference and by some techniques before they are calculated concretely. On the other hand some numerical method such as the finite element method and the boundary element method, they mainly use the displacement method and the stress method is difficult to use. By the displacement method, the stress solutions are obtained by the numerical differential of the displacement, so the calculation precision is low relatively. Although the boundary element method puts forward the boundary integral formula, it mainly uses the numerical method and can seldom get analytic solution.

In this paper, at first according to the differences in the establishment and expression form of the boundary integral equations of the boundary element method, the relations and the differences between the natural boundary element method and the normal boundary element are expounded, then the mathematical principles of the natural boundary reduction for the boundary value problem of bi-harmonic equation and its related calculation method of the strong singular integral and some related content of the generalized functions are studied and are put into analyses of the bending problems of the elastic thin plates and the elastic plane problems. For the first time a united analytic integral formula of the stress function is provided for the semi-plane elastic body under the all kinds of tangent and normal loads, and the analytic solutions to the semi-plane problems under the various boundary conditions are obtained. For the first time boundary integral formulas of the stress function are produced for the elastic plane problems of interior circular domain and the problems of exterior circular domain with the equilibrium or non-equilibrium loads along the hole’ boundary. By these integral formulas, the analytic solutions to the simple boundary value problems can be obtained directly while for the complicated boundary value problems it is very convenient to get the numerical solutions. The plane problem of exterior elliptic domain loaded only at the infinite boundary is discussed preliminarily, which establishes the study basis by the boundary integral formula for the plane problems of exterior hole domain loaded on the hole’s boundary with various shapes. The above studies on the bi-harmonic elastic problems in the typical domain set up a foundation for exploring the elastic problems in a universal domain by the coupling method.