On the equilibrium bifurcation of axially deformable holonomic systems: solution of a long-standing enigma

The stability of equilibrium is a fundamental topic in mechanics and applied sciences. Apart from its central role in most engineering fields, it also arises in many natural systems at any scale, from folding/unfolding processes of macromolecules and growth-induced wrinkling in biological tissues to meteorology and celestial mechanics. As such, a few key models represent essential benchmarks in order to gain significant insights into more complex physical phenomena. Among these models, a cornerstone is represented by a structure made of two straight axially deformable bars, connected by an elastic hinge and simply supported at the ends, which are capable of buckling under a compressive axial force. This classical example has been proposed and analysed in some depth by Feodosyev but the attention is here focused on an apparently paradoxical result given by this model, i.e. the existence of a lower bound for the axial-to-flexural stiffness ratio in order for the bifurcation to take place. This enigma is solved theoretically by showing that, differently from other classical stability problems, constitutive and geometric nonlinearities cannot be a priori disconnected and an ideal linearized axial constitutive law cannot be employed in this case. The theory is validated with an experiment, and post-buckling and energy extrema of the proposed solution are discussed as well, highlighting possible snap-back and snap-through phenomena. Finally, the results are extended to the complementary case of tensile buckling.