Uniqueness of solutions for constrained Elastica

We address the number of solutions in constrained Elastica, i.e. the number of forms and tensions possibly adopted by steady rods or sheets enclosed in a prescribed box. Our main result refers to sheets making contact with the compressing plates at their extremities only. For these so-called folds, we provide the first demonstration of uniqueness of solution. This result is obtained by combining direct methods that are focused on definite families of solutions and global methods that address the structure of the set of solutions in the parameter space. While the first methods are specific to Elastica, the latter are not and may be applied to other kinds of systems. We then address the origins of the multiplicity of solutions in more general configurations and find two of them: geometrical non-linearity; freedom in the distribution of the length of flat contacts. The former gives rise to two free-standing fold solutions beyond buckling; the latter yields multiple buckling thresholds and thus multiple elastic responses to compression. Altogether, these results improve the non-local analysis of Elastica and clarify the reasons for unique or multiple buckling thresholds in constrained configurations. Moreover, fold uniqueness appears very useful for saving computational times in the numerical quest for solutions, for detecting parasitic solutions in simulations, for establishing the robustness of the Euler's model and for clarifying the role of friction. On a more general ground, our study provides non-local methods for determining the conditions for bifurcation in constrained systems involving a free energy.