WKB Special Expansion and the Solutions of Neo-Hookean and Varga Elastic Cylinder, Sphere and Cube

Abstract

In this paper, our main objective is to apply the following specific form of WKB expansion i.e.

to the solution of the eigenvalue problems of Varga and Neo-Hookean elastic cylinder, everted cylinder, sphere, everted sphere and cube. The governing equations first formulated as a fourth order eigenvalue problems and then the analytic findings are compared with the counterpart numerical solutions. By applying this form of WKB expansion, we are showing that at the leading order the roots of  are repeated for Varga but un-repeated for Neo-Hookean materials. We also conclude that the differential equations of , , , … become second order for Varga and first order for Neo-Hookean materials.