Consistency of the boundary value problem of an elastic body involving surface tension in small deformations

Surface tension has a significant influence on the mechanical response of solids at small scales to external loadings. Conventional small deformation theories remain effective in capturing such an influence provided the corresponding boundary condition incorporates some measure of the deformed configuration of the solid’s surface in evaluating the surface tension–induced traction acting on the solid. Since the governing equations arising from conventional small deformation theories for the bulk of the solid are defined strictly in the undeformed configuration, they may be sometimes incompatible with the surface tension–related boundary condition incorporating certain measure of the deformed configuration, leading to an inconsistent boundary value problem. To clarify how to pose a self-consistent boundary value problem involving surface tension, we consider the plane deformation of a perforated medium with its internal boundary endowed with surface tension and surface elasticity and its external boundary subjected to a mechanical loading. We first review three kinds of related boundary conditions on the internal boundary of the medium: the Gurtin–Murdoch-type boundary condition and two modified boundary conditions based on the change-in-curvature formula. We then derive the resultant force and moment of the traction caused by surface tension and surface elasticity for each of the boundary conditions. We show that for the Gurtin–Murdoch-type boundary condition, the resultant force vanishes unconditionally but the resultant moment depends on the rigid-body rotation of the medium, while for the two modified boundary conditions, both the resultant force and resultant moment (about a certain point) rely on local deformation gradients and the curvature of the internal boundary of the medium. Consequently, for the Gurtin–Murdoch-type boundary condition, one may easily establish a self-consistent boundary value problem by specifying an appropriate mechanical loading, while for the two modified boundary conditions, the boundary value problem is only occasionally self-consistent for some special and compatible combinations of the geometry of the internal boundary and the mechanical loading. Nevertheless, if the medium becomes infinitely large with its external boundary extremely distant from the internal boundary, then the boundary value problem corresponding to either of the three boundary conditions is always self-consistent for an arbitrary configuration of the internal boundary and an arbitrary far-field loading.