Triply periodic minimal surface metamaterials stiffness prediction via the variational asymptotic method for unit cell homogenization

This work presents a comparative study on the mechanical homogenisation of Triply Periodic Minimal Surface (TPMS) lattice structures, which have attracted significant interest for their unique ability to combine lightweight design with tailored mechanical, thermal, and acoustic properties. The study investigates the effective mechanical behaviour of Representative Unit Cells (RUCs) generated using the open-source Python tool Microgen. Two homogenisation strategies are considered: (i) Finite Element (FE)-based homogenisation carried out in Abaqus, and (ii) the Mechanics of Structure Genome (MSG), a unified theory for multiscale constitutive modelling, implemented in a specialized software framework. The comparison encompasses multiple TPMS topologies, including well-studied cases used for validation as well as less-explored ones to provide new insights, namely gyroid, diamond, PMY, and F-Rhombic Dodecahedron (F-RD). RUCs are analysed across relative densities ranging from 10% to 50%. Equivalent linear elastic properties (Young’s moduli, shear moduli, and Poisson’s ratios) are derived and compared to assess the consistency, accuracy, and computational efficiency of the two approaches. Furthermore, the anisotropy of each TPMS topology across the range of relative densities is examined through the directional distribution of Young’s moduli. The outcomes are expected to clarify the strengths and limitations of FE versus MSG in capturing the effective behaviour of architected cellular solids, thus supporting the selection of homogenisation strategies for the design of lattice-based lightweight structures.