Bioinspired Vertex-Based Hierarchical Square Honeycombs: A Stiffness-Equivalence Framework with Convergence Analysis

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.