Generalization of the method of perturbation of the form of the boundary in the mechanics of deformable bodies

Abstract

Introduction. Under the title "A Method of Pertubation of the Form of the Boundary" there,appeared in the scientific literature an approximate analytical method of solution of boundary value problems of continuum mechanics for noncanonical domains (domains not allowing a solution by the method of direct separation of variables). Its conceptual basis was contained in a paper by Guz' [4], published in 1962, in which, for the first time, an effective, approximate analytical method was proposed for studying stress concentration around curvilinear holes in shells. This idea proved to be sufficiently fruitful, and the developed method so general, that with no principal difficulties it was extended not only to threedimensional boundary value problems of elasticity theory for bodies of revolution [6, 13] and noncircular cylinders [7, 15], but also to broad classes of three-dimensional boundary value problems of the mechanics of deformable b6dies [12] and related problems of continuum mechanics [5, 9] for noncanonical domains. The recursion relationships and differential operators constructed in an arbitrary approximation [16] make it possible to solve stated problems in principle with specified accuracy. A broadening of the possibilities of applying the method proposed in [4] (subsequently referred to in [8] as a first variant of the method of perturbation of the form of the boundary) was enhanced through its application, along with other analytical methods, to the solution of boundary value problems for media with complicated elastic properties. Thus, along with the methods of successive approximations and the perturbation of elastic properties, it was applied for the first time to the solution of planar [I0] and spatial [14] nonlinear physical problems of elasticity theory and also to the solution of boundary value problems for curvilinear-orthotropic bodies of noncanonical form [18]. The indicated first version of the method of perturbation of the form of the boundary, along with an integral Laplace transform with respect to the time, was also applied to the solution of spatial [27] and planar [23] related problems of the mechanics of saturated porous media with noncanonical processing.