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In this article, we extend the capstan equation to non-circular geometries.We derive a closed form solution for a membrane with a zero Poisson's ratio (ora string with an arbitrary Poisson's ratio) supported by a rigid prism (atlimiting-equilibrium case and in steady-equilibrium case) or supported by arigid general cone (at limiting-equilibrium case only). Our models indicatethat the stress profile of the elastic body depends on the change in curvatureof the rigid obstacle at the contact region. As a comparison, we extend Kikuchiand Oden's model for Coulomb's law of static friction to curvilinearcoordinates, and conduct numerical experiments to examine the properties ofCoulomb's law of static friction implied by Kikuchi and Oden's model incurvilinear coordinates and implied by our generalised capstan equation. Ournumerical results indicate that for a fixed coefficient of friction, thefrictional force is independent of the Young's modulus of the elastic body, andincreasing the curvature, increasing the Poisson's ratio and decreasing thethickness of the elastic body increases the frictional force. Our analysis alsoshows that for given a tension ratio, the coefficient of friction inferred byeach model is different, indicating that the implied-coefficient of friction ismodel dependent.

A Generalisation of the Capstan Equation and a Comparison Against Kikuchi and Oden's Model for Coulomb's Law of Static Friction