We investigate the equilibrium configurations of the ideal 3D elastica, i.e., inextensible, unshearable, isotropic, uniform, and naturally straight and prismatic rods, with linear elastic constitutive relations. Infinite solution trajectories are expressed analytically and classified in terms of only three parameters related to physical quantities. Orientation of sections and mechanical loading are also well described analytically with these parameters. Detailed analysis of solution trajectories yields two main results. First, all particular trajectories are completely characterized and located in the space of these parameters. Second, a general geometric structure is exhibited for every ideal 3D elastic rod, where the trajectory winds around a core helix in a tube-shaped envelope. This remarkable structure leads to a classification of the general case according to three properties called chirality components. In addition, the geometry of the envelope provides another characterization of the ideal 3D elastica. For both results, the domains and the frontiers of every class are plotted in the space of the parameters.
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.
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