The elastic solutions of separable problems with the applications to multilayered structures

Abstract

This work presents the elastic solutions for the separable problems, where certain coordinate variables in the boundary and continuity conditions in terms of series representations can be eliminated to simplify the systems of equations. The definition of “separable problems” is clarified, which is mainly because the internal displacement or stress fields and the outer boundary conditions share similar patterns of series expansions in the same coordinate systems. Several types of separable problems are illustrated in the introduction to show the applicability and importance of the present theory. The multilayered hollow cylinders are employed to explain the procedure of solving similar types of boundary value problems. A special application of the multilayered cylinders is to embed them into an infinite plate with a hole at the center, where a layer-by-layer functional gradation along the radial direction could lead to the reduction of the stress concentrations. Besides validating the accuracy of the derivations against the solution of a continuously graded cylinder and the original Kirsch solutions, the effects of the geometrical and material properties on the stress distributions are also tested. The idea of solving separable problems can significantly contribute to the theoretical analysis and design of discretely graded structures with various geometries.