Application of the perturbation method in mechanics of deformable solids

Abstract

Although the solutions of the classical problems of continuum mechanics have been studied sufficiently well, the smallest deviations, for example, of the body boundary or of the material characteristics from the traditional values prevent one from obtaining exact solutions of these problems. In this case, one has to use approximate methods, the most common of which is the perturbation method. The problems studied in [1–6] belong to classical problems in which the perturbation method is used to study the behavior of deformable bodies. A wide survey of studies analyzing the perturbations of the body boundary shape caused by variations in its stress-strain state is given in [5, 6].

In numerous studies, it was noted that the problem on the convergence of approximate solutions and hence the studies of the continuous dependence of the solution of the original problem on the characteristics of perturbations (“imperfections”) play an important role. In the present paper, we analyze the forms of mathematical models of deformable bodies by studying whether the solution of the original problem continuously depends on the characteristics of the perturbed shape of the body boundary on which the boundary conditions are posed in terms of stresses and on the characteristics of the material properties. We use the results of this analysis to conclude that, when using the perturbation method, one should state the boundary conditions in terms of stresses on the boundary of the real body in stressed state.