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An efficient route to the displacement field around a rigid sphericalinclusion in an infinitely extended homogeneous elastic medium is presented ina slightly alternative way when compared to some common textbook methods.Moreover, two Fax\'en relations of next-higher order beyond the stresslet arecalculated explicitly for compressible media. They quantify higher-ordermoments involving the force distribution on rigid particles in a deformedelastic medium. Additionally, the displaceability and rotateability matricesare calculated up to (including) sixth order in inverse particle separationdistance. These matrices describe the interactions mediated between the rigidembedded particles by the elastic environment. All methods and results canformally be transferred to the corresponding case of incompressiblehydrodynamic low-Reynolds-number Stokes flow by considering the limit of anincompressible environment. The roles of compressibility of the embeddingmedium and of the here additionally derived higher-order contributions arehighlighted by some selected example configurations.

physical review. e

Displacement field around a rigid sphere in a compressible elastic  environment, corresponding higher-order Fax\'en relations, as well as  higher-order displaceability and rotateability matrices