A Refined Gibson-Ashby Model for Functionally Graded Honeycombs with Random Irregularities

Honeycomb lattices are widely used as lightweight architected materials, and their mechanical behavior is strongly influenced by unit-cell wall thickness and geometric configuration. Inspired by natural cellular structures, predicting the elastic response of honeycombs with wall-thickness grading and geometric irregularities remains challenging. This study refines the Gibson-Ashby equation for predicting the elastic modulus of honeycombs by introducing two correction factors, 𝜅 and 𝜂, accounting for functionally graded (FG) wall thickness variations and random irregularities, respectively, i.e., (relative modulus) = 𝜅𝜂𝐶 (relative density)n. Analytical expressions based on Timoshenko beam theory are extended to FG architectures, providing a stepwise formulation for relative density and elastic modulus variations along the FG direction. This yields a closed-form expression for the effective relative elastic modulus of FG honeycombs, validated by finite element (FE) simulations. Here, 𝜅 is calibrated over a broad range of linear and nonlinear wall thickness gradients. Since closed-form formulations for periodic lattices cannot capture random irregularities, 𝜂 is fitted to an extensive set of FE simulations with random designs. These correction factors are presented as design charts covering hexagonal, square, and triangular honeycombs. Additively manufactured samples produced by 3D printing are tested to validate the predictions and quantify real-world imperfections. This framework provides a direct and computationally efficient means to estimate the elastic modulus of graded and irregular 2D lattices without requiring full numerical homogenization or experiments. The study introduces a unified correction-based approach that bridges analytical cellular-material models with the complexity of natural and fabricated honeycombs.