In this study, impact behavior and deformation mechanism of triply periodic minimal surface (TPMS) sheet (Schwarz’ P, Schoen’s I-WP, and Neovius) were numerically investigated. The convergence study of mesh size, sensitivity of cell number, and parametric investigation of shell thickness were performed. Three different collapse modes “ $\bigvee +\bigwedge =\mathrm{X},$” “ $/+\backslash =\mathrm{X},$” and “Whole-X” were experienced corresponding to Schwarz’ P, Schoen’s I-WP, and Neovius sheets, respectively. Comparison between the anisotropic 2D circular honeycomb and isotropic 3D TPMS were conducted. The results show that TPMS sheets perform excellent mechanical properties due to their isotropic smooth surface transitions and open-cell connectivity.

This study examines a hierarchical vertex-based structure that improves the crashworthiness of the conventional multi-cell square, a biological hierarchy of natural origin with exceptional mechanical properties. The vertex-based hierarchical square structure (VHS) is explored for its geometric properties, including infinite repetition and self-similarity. The cut-and-patch method is used to derive an equation for the material thicknesses of different orders of the VHS based on the principle of the same weight. A thorough parametric study of VHS was conducted using LS-DYNA, which examined the effects of material thickness, orders, and various structural ratios. The results were evaluated based on common crashworthiness criteria and demonstrated that the total energy absorption (TEA), specific energy absorption (SEA), and mean crushing force ($P_m$) of VHS exhibited similar monotonicity concerning the orders. SEA of the first-order VHS with ${\lambda }_1=0.3$ and the second-order VHS with ${\lambda }_1=0.3$ and ${\lambda }_2=0.1$ are improved by at most 59.9% and 102.4% respectively; the second-order VHS with $0.2\le {\lambda }_1\le 0.4$ and $0.1\le {\lambda }_2\le 0.15$ have the better overall performance of crashworthiness. Then, the half-wavelength equation of VHS and $P_m$ of each fold was established based on the Super-Folding Element method. Meanwhile, a comparative analysis with the simulation results reveals three different out-of-plane deformation mechanisms of VHS. The study indicated that material thickness had a greater impact on crashworthiness. Finally, the comparison with conventional honeycombs demonstrated that VHS holds great promise as a structure for crashworthiness. These results provide a solid foundation for further research and development of new bionic energy-absorbing devices.

Inspired by the hierarchical organization of natural cellular solids, this study presents a stiffness-equivalence framework for predicting the in-plane effective Young’s modulus and Poisson’s ratio of bioinspired vertex-based hierarchical square honeycombs (VHS). Based on structural symmetry and Saint-Venant’s principle, simplified equivalent models are established for one- and two-order VHS unit cells, and closed-form expressions for the effective elastic constants are derived using Castigliano’s second theorem. A recursive stiffness-equivalence framework is then proposed to extend the predictions to higher hierarchical orders, revealing the convergence behavior of the elastic constants with increasing hierarchy. The analytical results are validated against quasi-static compression experiments on fused filament fabrication (FFF) 3D-printed PLA specimens and finite element simulations using Abaqus with Beam22 elements. The numerical model agrees with the experiments within 4% for the hierarchical configurations and 19% for the origin structure, while theoretical predictions show errors within 20% over the valid geometric range. The effective Young’s modulus decreases monotonically with hierarchical order while Poisson’s ratio increases, governed by the transition from axial-dominated to bending-dominated deformation as hierarchical refinement introduces additional load-redistribution paths. The framework provides design-oriented rules for bioinspired thin-walled cellular materials with prescribed in-plane elastic properties.