A comprehensive study on numerical homogenization of re-entrant honeycomb lattice and analytical model assessment

Re-entrant honeycomb lattice structures offer significant advantages, such as a high strength-to-weight ratio, superior energy absorption capabilities, and efficient material usage. However, for design optimization, the computational challenges posed by simulating extensive structures featuring lattice cells demand an exponential increase in degrees of freedom, inevitably prolonging computational time. The present work addresses some of the challenges in simulating these structures by employing numerical homogenization and investigating the effective elastic properties, aiming to reduce computational costs while maintaining accuracy. The homogenization approach is validated using experiments on lattice structures made in polymer. The study also evaluates the applicability of existing analytical models by comparing their predictions with numerical homogenization results. A comprehensive analysis considering the structural parameters, namely re-entrant angle (theta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document}) and relative density (rho rel\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\text {rel}}$$\end{document}), is presented to understand the prediction accuracy of different analytical models. Both uncertainty and sensitivity analyses were conducted to quantify the influence of these structural parameters on the effective properties and to assess the variability due to probable geometric uncertainties introduced in manufacturing. Additionally, the influence of cell density on the homogenization model's accuracy is also examined. The findings reveal good agreement between the lattice simulation, the homogenized model, and the experimental result within the linear elastic limit. The study infers that amongst the analytical models, Malek's model is highly accurate for predicting E11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{11}$$\end{document}, E33\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{33}$$\end{document}, and Poisson's ratio mu 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{12}$$\end{document} for higher theta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document} and up to 30% rho rel\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\text {rel}}$$\end{document} but shows significant deviations for G12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{12}$$\end{document}, G23\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{23}$$\end{document}, and mu 23\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{23}$$\end{document}, necessitating numerical homogenization for higher accuracy beyond these ranges. The uncertainty analysis indicates that elastic moduli such as E11/Es\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{11}/E_{s}$$\end{document} and E33/Es\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{33}/E_{s}$$\end{document} exhibit the highest sensitivity to variations in strut thickness and angle, highlighting the need for precise manufacturing control to mitigate variability in effective elastic properties. The sensitivity analysis revealed that strut thickness (t) significantly influences the elastic and shear moduli, while the interaction of theta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document} and t plays a crucial role in determining Poisson's ratios for the auxetic honeycomb lattice. Additionally, numerical homogenization effectively predicts the elastic properties of re-entrant honeycomb structures with higher accuracy and lower computational costs. This comprehensive analysis enhances the understanding and practical application of both analytical and numerical methods in lattice structure design.