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

TL;DR

This paper develops a theoretical framework to calculate how rigid inclusions embedded in elastic materials respond to forces and torques, revealing long-range interactions mediated by the elastic matrix, with applications in composite materials and microrheology.

Contribution

It introduces an analytical method adapted from hydrodynamics to explicitly compute inclusion displacements and rotations in elastic media, accounting for compressibility and matrix-mediated interactions.

Findings

Derived explicit formulas for inclusion displacements and rotations.

Included effects of medium compressibility in the analysis.

Provided a basis for characterizing elastic composite behaviors.

Abstract

Embedding rigid inclusions into elastic matrix materials is a procedure of high practical relevance, for instance for the fabrication of elastic composite materials. We theoretically analyze the following situation. Rigid spherical inclusions are enclosed by a homogeneous elastic medium under stick boundary conditions. Forces and torques are directly imposed from outside onto the inclusions, or are externally induced between them. The inclusions respond to these forces and torques by translations and rotations against the surrounding elastic matrix. This leads to elastic matrix deformations, and in turn results in mutual long-ranged matrix-mediated interactions between the inclusions. Adapting a well-known approach from low-Reynolds-number hydrodynamics, we explicitly calculate the displacements and rotations of the inclusions from the externally imposed or induced forces and torques. Analytical expressions are presented as a function of the inclusion configuration in terms of displaceability and rotateability matrices. The role of the elastic environment is implicitly included in these relations. That is, the resulting expressions allow a calculation of the induced displacements and rotations directly from the inclusion configuration, without having to explicitly determine the deformations of the elastic environment. In contrast to the hydrodynamic case, compressibility of the surrounding medium is readily taken into account. We present the complete derivation based on the underlying equations of linear elasticity theory. In the future, the method will, for example, be helpful to characterize the behavior of externally tunable elastic composite materials, to accelerate numerical approaches, as well as to improve the quantitative interpretation of microrheological results.