Eshelby's tensor for embedded inclusions and the Elasto-Capillary phenomenon

Abstract

The elastic state of an embedded inclusion undergoing a stress-free transformation strain was the subject of John Douglas Eshelby's now classical paper in 1957. This paper, the subject of which is now widely known as “Eshelby's inclusion problem”, is arguably one of the most cited papers in solid mechanics and several other branches of physical sciences. Applications have ranged from geophysics, quantum dots to composites. Over the past two decades, due to an interest in all things “small”, attempts have been made to extend Eshelby's elastic analysis to the nanoscale by incorporating capillary or surface energy effects. In this note, we revisit a particular formulation that derives a very general expression for the elasto-capillary state of an embedded inclusion. This approach, that closely mimics that of Eshelby's original paper, appears to have the advantage that it can be readily used for inclusions of arbitrary shape (for numerical calculations) and provides a facile route for approximate solutions when closed-form expressions are not possible. Specifically, in the case of inclusions of constant curvature (sphere, cylinder) subject to some simplifications, closed-form expressions are obtained.