Some rigorous results for harmonic holes with surface tension

Abstract

Harmonic holes are designed to leave the mean stress as a constant in the surrounding material. When surface tension is imposed on the boundary of the holes, the existence of harmonic holes within an infinite elastic plane subjected to plane deformation was verified roughly in the literature by numerical techniques. However, a rigorous proof for the existence of harmonic holes has still been absent in the literature for any of the cases involving surface tension. In this paper, we perform an accurate analysis for the case of a single harmonic hole with constant surface tension in an infinite elastic plane under a uniform remote (in-plane) shear loading. We show that the harmonic hole exists strictly if and only if a certain combination of the surface tension, shear loading and the size of the hole does not exceed a critical value. Explicit exact formulae are obtained for describing the shape of the harmonic hole in both deformed and undeformed configurations. These formulae may find applications in the design of functional porous materials, in validating relevant numerical methods and in elucidating the preferred shapes of fluid-elastic membranes and cell membranes.