The effect of a capillary bridge on the crack opening of a penny crack

Young's relation is based on the equilibrium of horizontal components of surface tensions for a liquid droplet on a "rigid" substrate without addressing the substrate deformation induced by the net vertical component of surface tensions. Upon realizing the importance of wetting in controlling the integrity of flexible structures and electronics, the effect of a capillary bridge or a liquid droplet on the crack opening of a penny crack under the action of a far-field tensile stress is analyzed. Closed-form solutions are derived for both the crack opening and the stress intensity factor, which are functions of the size of the capillary bridge or the droplet, surface tension, and the contact angle. Both the capillary bridge and the droplet can introduce the crack closure. The minimum far-field tensile stresses needed for complete crack opening, i.e. no crack closure, are obtained analytically. The net vertical component of the surface tensions introduces the formation of a surface ridge on the crack face at the edge of the droplet for an open crack. The amplitude of the surface ridge increases with the increase of the net vertical component of the surface tensions and the decrease of the breadth width.