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<p>The linear model of elastic theory for an isotropic body is considered. The Navier equation solution is used in the curvilinear orthogonal coordinate system. It is proven that the complete three-dimensional solution of the system of equations of elasticity theory 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 ϑ are given for the first time. It is shown that when the displacement functions do not depend on the azimuth angular coordinate&#xa0;φ, the stress-strain state is divided into a&#xa0;spherical axisymmetric 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 one stress state in the sphere, which corresponds to the effect of uniform pressure on its surface. The basic solutions, which correspond to the main force vector components, are built for an elastic sphere.</p>

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Study of solutions of the elasticity theory equations in a spherical coordinate system

  • 期刊论文

  • 发表日期：

    2025年11月17日

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  • V. P. Revenko

摘要

The linear model of elastic theory for an isotropic body is considered. The Navier equation solution is used in the curvilinear orthogonal coordinate system. It is proven that the complete three-dimensional solution of the system of equations of elasticity theory 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 ϑ are given for the first time. It is shown that when the displacement functions do not depend on the azimuth angular coordinate φ, the stress-strain state is divided into a spherical axisymmetric 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 one stress state in the sphere, which corresponds to the effect of uniform pressure on its surface. The basic solutions, which correspond to the main force vector components, are built for an elastic sphere.