Supplementary material from "Macro-micro synergistic mechanisms of Curculio rostrum against buckling"

Slender biological structures, which are widely observed in nature, exhibit exceptional properties and biological functions. A notable example is the Curculio rostrum that can drill into fruits for feeding and egg-laying without buckling. However, it remains unknown how the slender and curved Curculio rostrum resists buckling. In this paper, we propose an elastic rod model for the rostrum that has multi-layered microstructures reinforced by helical chitin fibers. Its buckling behavior is investigated through the combination of theoretical analysis and numerical simulations. The critical buckling load is derived and verified by experiments. It is found that despite the coupled twisting deformation indued by the helical microstructure, the originally curved rostrum exhibits a two-dimensional (2D) buckling mode. The critical buckling load for this 2D buckling is much higher than for three-dimensional buckling, owing to the relatively large stiffness of the outermost layer. The macroscopic straight-curved shape helps the rostra to achieve similar buckling resistance capability among individuals, thereby enabling them to sustain the same feeding habits. Our results reveal that the macroscopic shape and helical microstructure synergistically contribute to the buckling resistance of Curculio rostra. This work could also inspire the bionic design of highperformance slender structures and soft robots.