Ivestigation of solutions of the elasticity theory equations in the spherical coordinate system

Keywords

elastic stresses and displacements, spherical coordinate system, displacement functions, solutions of Navier equations, basic solutions.

Cite as

Revenko V. P. Ivestigation of solutions of the elasticity theory equations in the spherical coordinate system. Physicochemical Mechanics of Materials. 2025. 61(3), 084-090.

https://doi.org/10.15407/pcmm2025.03.084

Abstract

The linear model of elastic theory for an isotropic body is considered. The Navier equation solution was used in the curvilinear orthogonal coordinate system. It is proved that the complete three dimensional solution of the system of equations of elasticity in the spherical coordinate system is expressed through three harmonic displacement functions. Analytical formulas for expressing displacements and stresses, which are written relative to the elevation angle υ are given for the first time. It is shown that when the displacement functions do not depend on the angular coordinate φ, the stress-strain state is divided into a spherical oxisymmetric stress, which is described by two functions and the stress state of pure twisting, which is described by one function. It is established that if elastic displacements and stresses depend on the radial variable only, they describe in the sphere one stress state which corresponds to the effect of uniform pressure on its surface. The basic solutions are built for a solid sphere.

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