Analysis of Stress-Strain State of a Cylinder with Variable Elasticity Moduli Based on Three-Dimensional Equations of Elasticity Theory

Abstract

Introduction. Functionally graded materials are of great use, because heterogeneity of properties enables to control the strength and rigidity of structures. This has caused great interest in the topic in the world scientific literature. The construction of solutions to such problems depends significantly on the type of boundary conditions. In this paper, we consider the equilibrium of a thin-walled circular cylinder whose mechanical properties change along the radius. Homogeneous boundary conditions were set on cylindrical surfaces that had not been considered before, the effect was on the ends. The mathematical formulation of the problem was carried out in the linear theory of elasticity in the framework of axisymmetric deformation. Expressions were constructed for the components of the stress-strain state of the cylinder, in which some coefficients were found from the solution to the resulting system of linear algebraic equations.Materials and Methods. The material of the cylinder was linearly elastic, the elastic modulus of which depended linearly on the radial coordinate. The basic research method was the asymptotic method, in which half the logarithm of the ratio of the outer and inner radii acted as a small parameter. Iterative processes were used to construct the characteristics of the stress-strain state of the cylinder.Results. Homogeneous solutions to the boundary value problem were obtained for a linearly elastic functionally gradient hollow thin-walled cylinder. An analysis of these solutions made it possible to reveal the nature of the stress-strain state in the cylinder wall. For this purpose, an asymptotic analysis of the solutions was carried out, relations for displacements and stresses were obtained. It was determined that those solutions corresponded to the boundary layer, while their first terms determined Saint-Venant edge effect similar to the plate theory.Discussion and Conclusion. The analytical solution to the equilibrium problem of a thin-walled cylinder inhomogeneous in radius constructed by asymptotic expansion can be used for numerical solution to a specific problem. For this, it is required to solve the obtained systems of linear algebraic equations and determine the corresponding coefficients. The resulting asymptotic representations provide analyzing the three-dimensional stress-strain state. The selection of the number of expansion terms makes it possible to calculate displacements and stresses with a given degree of accuracy. This analysis can be useful in assessing the adequacy of applied calculation methods used in engineering practice.