  On a sufficiently-soft substrate, a resting fluid droplet will cause
significant deformation of the substrate. This deformation is driven by a
combination of capillary forces at the contact line and the fluid pressure at
the solid surface. These forces are balanced at the surface by the solid
traction stress induced by the substrate deformation. Young's Law, which
predicts the equilibrium contact angle of the droplet, also indicates an a
priori radial force balance for rigid substrates, but not necessarily for soft
substrates which deform under loading. It remains an open question whether the
contact line transmits a non-zero radial force to the substrate surface in
addition to the conventional vertical force. We present an analytic Fourier
transform solution technique that includes general interfacial energy
conditions which govern the contact angle of a 2D droplet. This includes
evaluating the effect of gravity on the droplet shape in order to determine the
correct fluid pressure at the substrate surface for larger droplets.
Importantly, we find that in order to avoid a strain singularity at the contact
line under a nonzero radial contact line force, it is necessary to include a
previously-neglected radial traction boundary condition. To quantify the
effects of the contact line and identify key quantities that will be
experimentally-accessible for testing the model, we evaluate solutions for the
substrate surface displacement field as a function of Poisson's ratio and
zero/non-zero radial contact line forces.
