Energy-minimizing configurations for an elastic rod with self-contact energy close to the inextensible–unshearable and hard-contact limits

We study self-contact configurations of elastic rods by adding a repulsive energy to the bend, twist, shear, and stretch energies of a classical elastic rod. We use a discretized approach, with the parameters of the discrete rod calibrated to approach the continuous case as the number N of rod segments goes to infinity, and use a quasi-Newton approach to minimize the discretized energy. We explore convergence of the results as N→∞, and also as we vary shear, stretch, and contact parameters to approach the inextensible–unshearable and hard-contact limits. For a straight and isotropic rod, we present results for three sets of boundary conditions – loop, buckled strut, and hairpin, in each case as an imposed twist angle is increased – and compare energies and configurations for the non-contact and contact problems. Finally, we show how our results change when isotropy is broken in two ways: by the rod being bent in its undeformed state, or by the rod having unequal bending stiffnesses.