An asymptotic approach to the problems of the theory of elasticity of bodies of finite dimensions

Abstract

A method is developed to solve the equilibrium problems of elastic bodies of fixed dimensions, based on a separation of the boundary-layer part of the solution by considering the problem for a half-strip. A closed solution in quadratures is constructed for the half-strip with a free lateral face and with given normal displaced longitudinal boundaries, using both symmetric and antisymmetric loading. When the normal stresses on the front boundaries are specified, the problem reduces to an integral equation of the first kind in a semi-infinite interval, the inversion of which is obtained by reduction to an infinite system of algebraic equations. The approach considered for problems of bodies of finite dimensions is asymptotic with respect to the small parameter characterizing the body's thickness. Testing of the method on a plane problem for an elastic rectangle enables the range of variation of this parameter to be investigated, in which this procedure is fairly accurate. In the example considered, for the case of a rectangular area, the stresses are found and compared with the results obtained earlier by other methods. The nature of the influence of the boundary layer on the stress distribution inside the body is investiaged. Asymptotic methods, used for bodies of slab configuration, one of whose characteristic dimensions (thickness) is significantly less than the other two /1–4/, can obviously be classified into three types. The first of them /5,6/ is characterized by the application of joined asymptotic expansions to a certain class of solutions of the equations of the theory of elasticity, namely uniform solutions. The second type /7,8/ is distinguished by an asymptotic analysis of the equations of the theory of elasticity. From this it is clear that, to separate the boundary-layer part of the solution, it is enough, to a first approximation, to examine the two-dimensional problem and the problem of torsion for the half-strip, the lateral face of which is combined with the generating lateral surface of the plate at a given point. Finally, the third class includes the Vekua-Poniatovskii theory /9,10/, in which asymptotic methods are also developed /11/. In this sense this paper relates to the second of these methods.