Nanomechanics, and biomedical nanomechanics: Eshelby's inclusion and inhomogeneity problems at the discrete/continuum interface

The solution of the problem of a large elastostatic matrix, with an embedded eigenstraining inclusion is presented. The inclusion is modeled as an array of discrete points, in accordance with the theory of doublet mechanics (DM), while the matrix is viewed as a conventional continuum. The integration of the two representations affords the simultaneous access to atomic-scale stress and deformation analysis, and retention of the modeling benefits associated with the macroscopic continuum treatment of non-critical material regions. Thus, the theory presented appears suitable for the analysis of the mechanical states in nanotechnological devices, embedded within constraining matrices, biological and otherwise. The linear eigenstrain model employed is applicable to the analysis of nanomechanical states arising from thermal expansion, phase changes, crystallization, differential hygral absorption, and environmental effects such as changes in pH, all occurring within a small subdomain of an homogeneous medium. In parallel with the celebrated approach employed by J. Eshelby in the theory of linear continuum elastostatics, the solution to the nanomechanical inclusion problem is also used to solve the nanomechanical inhomogeneity problem. This consists of a nanoscale material domain embedded within a large volume of a different material, treated herein as a continuum-level matrix. The assembly is loaded by homogeneous displacement conditions at the outer boundary. Modeling the inhomogeneity as an array of discrete nodes, again following the methods of DM, and employing a stress equivalence approach, the nanomechanical states within the inhomogeneity are determined.