Snapping and Fluttering of Elastic Rods

The exact solutions for planar rods undergoing large rotations and subject to kinematically controlled ends are presented in the first part of the thesis. In particular, the equilibrium equations for a rod subject to Dirichlet boundary conditions and to isoperimetric constraints are derived through variational principles for both the Euler's elastica and the Reissner beam, while the related closed-form solutions are obtained in terms of the Jacobi elliptic functions. The study of stability of the Euler's elastica is addressed in the second part of the thesis through a modified version of the conjugate points method, thus disclosing the existence of a universal snap surface that represents the whole set of "saddle points" of the total potential energy, and therefore corresponding to snapping configurations. These theoretical findings allow for the prediction of snapping instabilities along any equilibrium path involving variations in the boundary conditions and are confirmed by numerical and experimental data. The universal snap surface is also exploited towards the realization of the elastica catastrophe machine, as the first extension of the classical Zeeman's machine to continuous elastic elements. Two families of the elastica catastrophe machine are presented and the theoretical model is fully validated through a prototype designed and tested at the Instability Lab of the University of Trento. Finally, the equations of motion of a pre-stressed planar rod and of its discretized counterpart subject to non-holonomic constraints are obtained in the last part. The analysis of the linearized stability surprisingly proves the existence of flutter instabilities despite the conservative nature of the considered systems. Moreover, Hopf bifurcations and destabilization paradoxes in the presence of dissipative forces are found. The non-linear equations of the proposed discretized model are also numerically solved, thus confirming the predicted stability properties and revealing the birth of periodic stable solutions.