Direct connection between Navier and spherical harmonic kernels in elasticity

Dimitra Labropoulou; Panayiotis Vafeas; George Dassios; Dimitra Labropoulou; Panayiotis Vafeas; George Dassios

doi:10.3934/math.2023158

AIMS Mathematics

2023, Volume 8, Issue 2: 3064-3082. doi: 10.3934/math.2023158

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Research article Special Issues

Direct connection between Navier and spherical harmonic kernels in elasticity

Department of Chemical Engineering, University of Patras, 26504 Patras, Greece

Received: 12 July 2022 Revised: 22 October 2022 Accepted: 03 November 2022 Published: 15 November 2022
MSC : 74H10, 74B05, 35Q74, 35J05, 33C55

Linear isotropic elasticity is an interesting branch of continuum mechanics, described by the fundamental laws of Hooke and Newton, which are combined in order to construct the governing generalized Navier equation of the displacement within any material. Implying time-independence and in the absence of external body forces, the latter is reduced to the corresponding form of a homogeneous second-order partial differential equation, whose solution is given via the Papkovich differential representation, which expresses the displacement field in terms of harmonic functions. On the other hand, spherical geometry provides the most widely used framework in real-life applications, concerning interior and exterior problems in elasticity. The present work aims to provide a little progress, by producing ready-to-use basic functions for linear isotropic elasticity in spherical coordinates. Hence, we calculate the Papkovich eigensolutions, generated by the spherical harmonic eigenfunctions, obtaining connections between Navier and spherical harmonic kernels. A set of useful results are provided at the end of the paper in the form of examples, regarding the evaluation of displacement field inside and outside a sphere.
- solid mechanics,
- isotropic elasticity,
- Navier equation,
- analytical solutions,
- Papkovich representation,
- spherical harmonics
Citation: Dimitra Labropoulou, Panayiotis Vafeas, George Dassios. Direct connection between Navier and spherical harmonic kernels in elasticity[J]. AIMS Mathematics, 2023, 8(2): 3064-3082. doi: 10.3934/math.2023158

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Abstract

Linear isotropic elasticity is an interesting branch of continuum mechanics, described by the fundamental laws of Hooke and Newton, which are combined in order to construct the governing generalized Navier equation of the displacement within any material. Implying time-independence and in the absence of external body forces, the latter is reduced to the corresponding form of a homogeneous second-order partial differential equation, whose solution is given via the Papkovich differential representation, which expresses the displacement field in terms of harmonic functions. On the other hand, spherical geometry provides the most widely used framework in real-life applications, concerning interior and exterior problems in elasticity. The present work aims to provide a little progress, by producing ready-to-use basic functions for linear isotropic elasticity in spherical coordinates. Hence, we calculate the Papkovich eigensolutions, generated by the spherical harmonic eigenfunctions, obtaining connections between Navier and spherical harmonic kernels. A set of useful results are provided at the end of the paper in the form of examples, regarding the evaluation of displacement field inside and outside a sphere.

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