Straight contact lines on a soft, incompressible solid.

The deformation of a soft substrate by a straight contact line is calculated, and the result applied to a static rivulet between two parallel contact lines. The substrate is supposed to be incompressible (Stokes-like description of elasticity), and having a non-zero surface tension, that eventually differs depending on whether its surface is dry or wet. For a single straight line separating two domains with the same substrate surface tension, the ridge profile is shown to be very close to that of Shanahan and de Gennes, but shifts from the contact line of a distance equal to the elastocapillary length built upon substrate surface tension and shear modulus. As a result, the divergence near contact line disappears and is replaced by a balance of surface tensions at the contact line (Neumann equilibrium), though the profile remains nearly logarithmic. In the rivulet case, using the previous solution as a Green function allows one to calculate analytically the geometry of the distorted substrate, and in particular its slope on each side (wet and dry) of the contact lines. These two slopes are shown to be nearly proportional to the inverse of substrate surface tensions, though the respective weight of each side (wet and dry) in the final expressions is difficult to establish because of the linear nature of standard elasticity. A simple argument combining Neumann and Young equations is however provided to overcome this limitation. The result may have surprising implications for the modelling of hysteresis on systems having both plastic and elastic properties, as initiated long ago by Extrand and Kumagai.