  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 (or
a string with an arbitrary Poisson's ratio) supported by a rigid prism (at
limiting-equilibrium case and in steady-equilibrium case) or supported by a
rigid general cone (at limiting-equilibrium case only). Our models indicate
that the stress profile of the elastic body depends on the change in curvature
of the rigid obstacle at the contact region. As a comparison, we extend Kikuchi
and Oden's model for Coulomb's law of static friction to curvilinear
coordinates, and conduct numerical experiments to examine the properties of
Coulomb's law of static friction implied by Kikuchi and Oden's model in
curvilinear coordinates and implied by our generalised capstan equation. Our
numerical results indicate that for a fixed coefficient of friction, the
frictional force is independent of the Young's modulus of the elastic body, and
increasing the curvature, increasing the Poisson's ratio and decreasing the
thickness of the elastic body increases the frictional force. Our analysis also
shows that for given a tension ratio, the coefficient of friction inferred by
each model is different, indicating that the implied-coefficient of friction is
model dependent.

Classical Physics