Nonclassical Spatial Boundary Value Problems of Statics and Dynamics of Shells and the Asymptotic Method of Their Solution

Abstract

Classical and improved theories of shells consider only one class of boundary value problems - on the facial surfaces of the shell the values of the corresponding stress tensor components are given. If on the facial surfaces other conditions - displacement vector or mixed boundary conditions of theory of elasticity - are given, a nonclassical boundary value problem of the theory of shells arises, which is important in some application cases. It is proved that the Kirchhoff-Love hypotheses of the classical theory of shells are not applicable for the solution of this class of problems. It is shown that such problems can be solved by the asymptotic method of solution of singularly perturbed differential equations. Non-contradictory asymptotic orders of the stress tensor components and a displacement vector are established. The iteration processes for the determination of all sought values with the beforehand given asymptotic accuracy are built. As special cases, the solutions of static and dynamic problems, illustrating the effectiveness of the asymptotic method, are presented. Nonclassical problems related to free and forced vibrations of isotropic and anisotropic shells are considered. The amplitudes of forced vibrations are determined. For layered shells the efficiency of the method is shown.