Axisymmetric equilibrium of an isotropic elastic solid circular finite cylinder

Abstract

Axisymmetric equilibrium of an elastic solid circular finite cylinder is one of the oldest problems in the theory of linear elasticity. Crosswise superposition is a well-known method that is used to solve this boundary value problem (BVP); however, technical realization of its underlying ideas still seems to be vague and the solution obtained by this method suffers from some convergence issues. In this study, we follow two main objectives via analyzing a benchmark problem where an isotropic elastic solid circular cylinder of finite length is subjected to normal lateral loading. The first goal is to add more insight into the method of crosswise superposition by extending the ideas that are used for solving classical BVPs via the superposition principle. For this purpose, a new unified approach that naturally gives rise to the subtle concept of corner conditions (CCs) in the context of crosswise superposition method is introduced. Another goal is to demonstrate the influence of CCs on the convergence of the solution obtained by the method of crosswise superposition. In this avenue, the Love function approach is used to convert the Navier equations for an isotropic elastic material to a single axisymmetric biharmonic equation. Next, a general solution for the axisymmetric biharmonic equation consisting of separable and non-separable solutions is presented in cylindrical coordinates. These two classes of solutions are used to construct the Love function and associated elastic fields through the unified approach. Numerical results reveal that considering the CCs can significantly affect the convergence rate of the solution on the boundaries of the cylinder. Furthermore, it is observed that the solution does not converge to the boundary data at the rims without considering the CCs. Far enough from the boundaries of the cylinder, the solution does not seem to be much different with or without taking the CCs into account.