Construction of a homogeneous solution to the elasticity theory problem for an inhomogeneous truncated transversally isotropic cone

Abstract

Using an asymptotic integration method, this paper investigates the axisymmetric problem of elasticity theory for an inhomogeneous transversal-isotropic truncated cone of small thickness. It is believed that the moduli of elasticity are arbitrary continuous functions of the cone opening angle. It is assumed that the lateral part of the cone is free of stresses, and arbitrary boundary conditions are set at the ends of the cone, leaving it in equilibrium. Homogeneous solutions have been constructed, that is, all solutions to equilibrium equations that satisfy the condition of absence of stresses on the lateral surfaces of the cone. Three groups of solutions were derived: a penetrating solution, solutions such as a simple edge effect, as well as the boundary layer solutions. An analysis of the stressed-strained state was carried out. It is shown that the penetrating solution and solutions having the character of a boundary effect determine the internal stressed-strained state of the cone. Solutions having the character of a boundary layer are localized at the ends of the cone and its first terms are equivalent to the edge effect of the Saint-Venant inhomogeneous plate. A particular type of inhomogeneous transversal-isotropic cone of small thickness with the degeneration of its median surface into a plane has been studied. It is shown that this case of degeneration is special, and the solutions consist of a penetrating solution and a solution of the nature of the boundary layer. Asymptotic formulas have been derived that make it possible to calculate the stressed-strained state of an inhomogeneous transversal-isotropic cone of small thickness. On the basis of the obtained solutions, it is possible to build a new refined applied theory and determine the applicability of existing applied theories for conical shells. New classes of solutions have been identified that no applied theory can describe