Supplementary material from 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 our 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 <i>a priori</i> 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.

平衡稳定性是力学与应用科学中的基础性课题。其不仅在绝大多数工程领域中占据核心地位，更广泛存在于各类尺度的自然系统之中——从大分子的折叠/解折叠过程、生物组织内生长诱导的褶皱现象，到气象学与天体力学，均有涉及。为此，若干关键模型可作为基准框架，为理解更为复杂的物理现象提供重要洞见。其中一类基石性模型由两根直轴向可变形杆件构成：两杆通过弹性铰相连，两端均为简支约束，可在轴向压缩荷载作用下发生屈曲（buckling）。该经典模型已由费奥多谢夫（Feodosyev）提出并进行了深入分析，但本文的研究重点在于该模型所呈现的一项看似悖论的结论：即分岔（bifurcation）发生的前提是轴向刚度与弯曲刚度比存在下界。这一谜题通过理论分析得以破解，研究表明，与其他经典稳定性问题不同，本构非线性与几何非线性无法先验地相互解耦，且本场景下无法采用理想的线性化轴向本构关系。随后通过实验验证了该理论，并对所提解的后屈曲（post-buckling）特性与能量极值展开讨论，揭示了可能出现的回跳（snap-back）与跳越（snap-through）现象。最后将研究结果推广至拉伸屈曲的互补情形。