Analysis of the three‐dimensional elasticity theory problem for a radially inhomogeneous cylinder

Abstract

In this paper an axisymmetric problem of elasticity theory for a radially inhomogeneous cylinder of small thickness is studied. It is assumed that homogeneous mixed boundary conditions are specified on lateral surfaces of the cylinder, and boundary conditions that leave the cylinder in equilibrium are specified on the ends of the cylinder. It is considered that elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thinness of the cylinder walls. To determine its solution, the method of asymptotic integration is used, based on three iteration processes. All solutions of the equilibrium equation are constructed, satisfying the specified homogeneous boundary conditions on lateral surfaces. It is obtained that the solution of the problem consists of a penetrating solution and a solution of the nature of the boundary layer. An analysis of stress‐strain states corresponding to different types of solutions is carried out. Based on the constructed asymptotic solutions, a connection is found between the solutions obtained by the equation of elasticity theory and by the applied theory of shells. It is shown that the penetrating solution determines the internal stress‐strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the principal vector of stress acting in a cross section perpendicular to the axis of the cylinder.The first terms of asymptotic expansions of the penetrating solution in a small parameter coincide with solutions in the applied theory of shells. New classes of solutions are defined as the character of a boundary layer, which are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in the applied theories of shells. The first terms of the asymptotic expansion of the solution, having the nature of a boundary layer, are equivalent to the Saint‐Venant edge effect in the theory of heterogeneous plates. Asymptotic formulas for displacement and stresses are constructed, allowing one to calculate the three‐dimensional stress‐strain state of a radially heterogeneous cylinder of small thickness with any predetermined accuracy. Based on the obtained asymptotic expansions, one can estimate the applicability areas of applied theories and construct a refined applied theory for radially heterogeneous cylindrical shells.