Analytical expression of elastic rods at equilibrium under 3D strong anchoring boundary conditions

Abstract

A general-purpose method is presented and implemented to express analytically one stationary configuration of an ideal 3D elastic rod when the end-to-end relative position and orientation are imposed. The mechanical equilibrium of such a rod is described by ordinary differential equations and parametrized by six scalar quantities. When one end of the rod is anchored, the analytical integration of these equations lead to one unique solution for given values of these six parameters. When the second end is also anchored, six additional nonlinear equations must be resolved to obtain parameter values that fit the targeted boundary conditions. We find one solution of these equations with a zero-finding algorithm, by taking initial guesses from a grid of potential candidates. We exhibit the symmetries of the problem, which reduces drastically the size of this grid and shortens the time of selection of an initial guess. The six variables used in the search algorithm, forces and moments at one end of the rod, are particularly adapted due to their unbounded definition domain. More than 850 000 tests are performed in a large region of configurational space, and in 99.9% of cases the targeted boundary conditions are reached with short computation time and a precision better than 10−5. We propose extensions of the method to obtain many solutions instead of only one, using numerical continuation or starting from different initial guesses.