Forces and torques on rigid inclusions in an elastic environment: resulting matrix-mediated interactions, displacements, and rotations

摘要

Embedding rigid inclusions into elastic matrix materials is a procedure ofhigh practical relevance, for instance for the fabrication of elastic compositematerials. We theoretically analyze the following situation. Rigid sphericalinclusions are enclosed by a homogeneous elastic medium under stick boundaryconditions. Forces and torques are directly imposed from outside onto theinclusions, or are externally induced between them. The inclusions respond tothese forces and torques by translations and rotations against the surroundingelastic matrix. This leads to elastic matrix deformations, and in turn resultsin mutual long-ranged matrix-mediated interactions between the inclusions.Adapting a well-known approach from low-Reynolds-number hydrodynamics, weexplicitly calculate the displacements and rotations of the inclusions from theexternally imposed or induced forces and torques. Analytical expressions arepresented as a function of the inclusion configuration in terms ofdisplaceability and rotateability matrices. The role of the elastic environmentis implicitly included in these relations. That is, the resulting expressionsallow a calculation of the induced displacements and rotations directly fromthe inclusion configuration, without having to explicitly determine thedeformations of the elastic environment. In contrast to the hydrodynamic case,compressibility of the surrounding medium is readily taken into account. Wepresent the complete derivation based on the underlying equations of linearelasticity theory. In the future, the method will, for example, be helpful tocharacterize the behavior of externally tunable elastic composite materials, toaccelerate numerical approaches, as well as to improve the quantitativeinterpretation of microrheological results.