# A Spectral Theory Formulation for Elastostatics by Means of Tensor Spherical Harmonics

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## Abstract

Consider a set of (*N*+1)-phase concentric spherical ensemble consisting of a core region encased by a sequence of nested spherical layers. Each phase is spherically isotropic and is functionally graded (FG) in the radial direction. Determination of the elastic fields when the outermost spherical surface is subjected to a nonuniform loading and the constituent phases are subjected to some prescribed nonuniform body force and eigenstrain fields is of interest. When the outermost layer is an unbounded medium with zero eigenstrain and body force fields, then an *N*-phase multi-inhomogeneous inclusion problem is realized. Based on higher-order spherical harmonics, presenting a three-dimensional strain formulation with a robust form of compatibility equations, a spectral theory of elasticity in the spherical coordinate system is developed. Application of the established spectral theory leads to the exact closed-form solution when the elastic moduli of each phase vary as power-law functions of radius.

## Keywords

3D spectral analysis Tensor spherical harmonics Multi-inhomogeneous inclusion Spherically isotropic Functionally graded materials (FGMs) Exact closed-form elastic fields## Mathematics Subject Classification

74B05 74B10 35Q74## Notes

### Acknowledgements

The authors acknowledge the support of Iran National Science Foundation (INSF).

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