I am attempting to make a Mechanics of materials approach to describe non-linear deformations of thin lines/wires on a torus. A mathematical modeling of its probable geometry is required at start of formulation to bring in bending and twisting moments of Euler and St. Venant with mechanical rigidity constants ( EI and GJ).
What geometric quantity is conserved before and after the twist?
The picture (sorry, poor quality) shows closed geodesic lines or filaments on an imaginary circular torus (girls’ plastic hair-band) before twisting.
I gave one twist by hand to a single flexible plastic filament coil to see how the deformations settle off on the torus by moment/torque equilibrium. Locally it makes a reaction twist in the opposite direction for one or two coils, but along with it a Figure of 8 forms globally, i.e., the deformation occurs/ gets transferred throughout the length of torus.
Hope someone helps me describe new deformed geometry/configuration using Knot theory, winding numbers etc., or variations of twist and curvature by means of any parametrization to connect to a twisting moment/torque introduced at a single given point on the torus.
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EDIT1:
It is better to consider all the above aspects on a common rubber O-Ring . Illustrated images are of the same seal 5.5 inch diameter and $ \frac 14^{"}$ section diameter deformed as shown.The modeling to obtain an ODE would be simpler and would be a subset of toroidal winding case.
A curiosity ( topological?) that may be related I find, is that an O-Ring [or a V-Belt used to drive electric motors] can be coiled into odd number of 1 (undeformed), 3, 5 continuous rings but never into even 2,4 etc.number of close packed smaller circular rings.
