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

An efficient route to the displacement field around a rigid spherical inclusion in an infinitely extended homogeneous elastic medium is presented in a slightly alternative way when compared to some common textbook methods. Moreover, two Fax\'en relations of next-higher order beyond the stresslet are calculated explicitly for compressible media. They quantify higher-order moments involving the force distribution on rigid particles in a deformed elastic medium. Additionally, the displaceability and rotateability matrices are calculated up to (including) sixth order in inverse particle separation distance. These matrices describe the interactions mediated between the rigid embedded particles by the elastic environment. All methods and results can formally be transferred to the corresponding case of incompressible hydrodynamic low-Reynolds-number Stokes flow by considering the limit of an incompressible environment. The roles of compressibility of the embedding medium and of the here additionally derived higher-order contributions are highlighted by some selected example configurations.