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Confinement of filamentary objects is ubiquitous in numerous biological,medical, and engineering scenarios. Quantitatively determining the mechanicalinteraction between flexible filaments and surface confinement is particularlychallenging due to the unknown contact force induced by elasticity interactingwith geometric constraints. Here, we consider a simplified model of confinedfilamentary object: an elastic curve under surface confinement. Local force andmoment balance equation incorporating the role of contact is utilized to derivethe contact force exerted by surface on the confined elastic curve. It revealsthe relation between contact force and local geometry at balanced state andprovides a route to obtain the contact force from local confined geometrydirectly. Examples are provided to illustrate how to calculate contact forcefrom obtained geometries. We believe that our results contribute to futureefforts in the mechanics of filamentary objects under surface confinement.

Relation between the contact force and local geometry for an elastic curve under general surface confinement