  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 be very close to that of Shanahan and de Gennes, but
shift 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.
