Harmonic holes with surface tension in an elastic plane under uniform remote loading

Harmonic holes are designed to leave undisturbed the mean stress in an uncut body subjected to a system of prescribed remote loadings. The role of residual surface tension in the design of harmonic holes is an important consideration, which is usually neglected at the macroscale but remains a significant factor in the design of such holes at the nanoscale. We consider the identification of the geometry of a single harmonic hole in an elastic plane subjected to uniform remote loading when residual surface tension is incorporated into the model of deformation. The geometry of the hole is defined by a conformal mapping with certain unknown coefficients determined from a system of non-linear equations. We illustrate our results with several examples. In particular, we show that for a given remote loading and surface tension, the shapes obtained exhibit strong size-dependency. Moreover, we find that the incorporation of the effect of surface tension greatly extends the range of admissible uniform remote loadings that guarantee the existence of harmonic holes.