On Derivations of Basic Governing Equations for Solid Bodies in Bi-spherical Coordinate System

Abstract

Traditionally solid mechanics problems are formulated in Cartesian, cylindrical and spherical coordinate system. Using such formulation and coordinate system, solutions of solid mechanics problems are obtained for specific geometries such as straight boundary, circular, cylindrical and spherical boundaries. However, such available coordinate system cannot describe many geometries in spherical or cylindrical coordinate system with inclusion of eccentricities, two spherical or cylindrical bodies in contact and parallel. In this article, author address this issue by giving complete original formulations and derivations of basic governing partial differential equations in Bi-spherical coordinate system. Bi-spherical coordinate system allow us to solve problems such as eccentric spheres, two parallel spheres (intersecting or non-intersecting) in solid mechanics which traditional spherical coordinate system cannot handle. Author develop the original equations in Bi-spherical coordinates in terms of useful quantities of stresses and strains components in three dimensions. The paper is limited to basic formulations and practical applications of such formulations can easily be expanded for getting solutions using any known analytical or numerical methods.