Elastocapillary contact between droplets and highly bendable membranes

Recently, there has been great interest in how the classic picture of Young contact is modified for soft solids: horizontal force balance must be supplemented by a vertical force balance. Previous theoretical work has largely focussed on the contact of droplets with semi-infinite soft solids or flexible elastic beams. However, several recent experiments have studied how the contact of a droplet on a thin membrane may be used experimentally to infer the state of stress within a membrane (under the assumption that a small droplet does not change this stress significantly). We consider the mechanics of an elastic membrane subject to a pre-stress and wetted by a droplet, neglecting the effects of bending stiffness. We find that for `strong' pre-stress the tension at the contact line is only weakly perturbed from the value of the pre-stress. However, for `weak' pre-stress, the presence of the droplet significantly modifies the stress within the membrane: the tension at the contact line is significantly larger than the value prior to wetting. We define what is meant by `strong' and `weak' in these cases, and how these two cases may be distinguished experimentally when the level of pre-stress is not known a priori.