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On a sufficiently-soft substrate, a resting fluid droplet will causesignificant deformation of the substrate. This deformation is driven by acombination of capillary forces at the contact line and the fluid pressure atthe solid surface. These forces are balanced at the surface by the solidtraction stress induced by the substrate deformation. Young's Law, whichpredicts the equilibrium contact angle of the droplet, also indicates an apriori radial force balance for rigid substrates, but not necessarily for softsubstrates which deform under loading. It remains an open question whether thecontact line transmits a non-zero radial force to the substrate surface inaddition to the conventional vertical force. We present an analytic Fouriertransform solution technique that includes general interfacial energyconditions which govern the contact angle of a 2D droplet. This includesevaluating the effect of gravity on the droplet shape in order to determine thecorrect fluid pressure at the substrate surface for larger droplets.Importantly, we find that in order to avoid a strain singularity at the contactline under a nonzero radial contact line force, it is necessary to include apreviously-neglected radial traction boundary condition. To quantify theeffects of the contact line and identify key quantities that will beexperimentally-accessible for testing the model, we evaluate solutions for thesubstrate surface displacement field as a function of Poisson's ratio andzero/non-zero radial contact line forces.

Deformation of an Elastic Substrate Due to a Resting Sessile Droplet