A Convergence-Based Correction Framework for Finite Effects in Tensile Lattice Structures

Abstract

The effective Young's modulus of periodic lattice structures is contingent not solely on topology and relative density, but also on the number of unit cells incorporated within the computational domain or specimen. This issue assumes particular significance in uniaxial tension, for which representativeness criteria are less well-defined than in compression. The present work proposes a convergence-based numerical framework for the quantitative description of finite effects and the correction of the effective Young's modulus determined for non-representative domains. The analysis was conducted for six periodic topologies, including three rod structures (BCC, Fluorite, Kelvin) and three TPMS-type structures (Fischer–Koch, Gyroid shell, Schwarz shell). The linear-elastic FEM simulations were performed for various relative densities and for cubic domains of N × N × N unit cells, with N from 1 to 12. It has been demonstrated that the convergence of the effective Young's modulus to the asymptotic value can be adequately described by an exponential model governed by a pair of parameters: the convergence length τ, which sets the decay rate, and the normalized amplitude α, which sets the magnitude of the initial deviation. Together they determine the minimum number of cells required for a given error level. On this basis, representativeness error maps and correction factor maps C(N, ρ_rel) were developed. These maps constitute a calibrated correction framework that allows the response of an underrepresented domain to be rescaled toward the asymptotic stiffness within the calibrated topology–density range, rather than an independent predictive model. A comparison with experimental tensile tests shows that the corrected predictions move closer to the measurements for most configurations. Direct simulations of elongated domains reveal, however, that a correction calibrated on isotropically scaled domains captures only a small part of the stiffness change produced by extending a specimen along the loading axis alone, so the framework is presented as a correction for isotropically scaled domains. At lower relative densities geometric deviations and manufacturing inaccuracies additionally contribute to the remaining discrepancy.