Study of the Stress State of Solid Cylinders with Inhomogeneous Structure Under Various Boundary Conditions at the Ends

Abstract

Solving the problems of the theory of elasticity on the stress state of continuous-inhomogeneous bodies requires the improvement of existing and the development of new numerical-analytical methods. This is due to the need to fully take into account arbitrary dependencies of the mechanical properties of the material on the coordinates and nature of the applied load. The paper is devoted to the solution of the axisymmetric problem of the linear theory of elasticity on the equilibrium of the solid inhomogeneous cylinders of the finite length with the different boundary conditions at the ends. The polymeric continuous-inhomogeneous material with a gradient profile corresponding to the quadratic variants of change of the Young's modulus along the radial coordinate is considered. The solution of the problem is based on the application of the method of spline-approximation of functions in the direction of the longitudinal coordinate and the numerical method of discrete orthogonalization along the radial coordinate. The boundary conditions at singular point r=0 of continuous-inhomogeneous solid cylinder are formulated. An analysis of the stress state of the solid cylinders depending on the variant of the change of elastic characteristics of the material and the different boundary conditions is carried out. It is shown that the greatest influence of the law of change of Young's modulus on the stress state of cylinders is observed for circumferential stresses on the outer surface in the average length section for both methods of the ends fixing. In addition, the influence of the material occurs for both circumferential and radial stresses on the ends for short cylinders (l=6l₀) with the rigidly fixed ends. The comparative analysis of the stress distribution for the different variants of the mechanical properties of the continuously inhomogeneous solid cylinder of the finite length is carried out. There are edge effects at the ends, which depend on the length of the cylinder with conditions of rigid fastening at the ends. The given results can be used in the strength calculations of the cylindrical elements of the modern structures.