Displacement field around a rigid sphere in a compressible elastic environment, corresponding higher-order Faxén 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 Faxén relations of next-higher order beyond the stresslet are calculated explicitly for compressible media. They quantify higher-order moments involving the force distribution on a rigid spherical particle in a deformed elastic medium. As a consequence, additional contributions to the distortions of the deformed elastic medium are identified that are absent to lower order. Furthermore, the displaceability and rotateability matrices for an ensemble of rigid spheres 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. In this way, additional coupling effects are identified that are absent to lower order, particularly when rotations and torques are involved. 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.