Asymptotic analysis and design of linear elastic shell lattice metamaterials

Di Zhang; Ligang Liu

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arXiv · 2506.18910

Asymptotic analysis and design of linear elastic shell lattice metamaterials

Di Zhang·Ligang Liu

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Abstract

We present an asymptotic analysis of shell lattice metamaterials based on Ciarlet's shell theory, introducing a new metric--asymptotic directional stiffness (ADS)--to quantify how the geometry of the middle surface governs the effective stiffness. We prove a convergence theorem that rigorously characterizes ADS and establishes its upper bound, along with necessary and sufficient condition for achieving it. As a key result, our theory provides the first rigorous explanation for the high bulk modulus observed in Triply Periodic Minimal Surfaces (TPMS)-based shell lattices. To optimize ADS on general periodic surfaces, we propose a triangular-mesh-based discretization and shape optimization framework. Numerical experiments validate the theoretical findings and demonstrate the effectiveness of the optimization under various design objectives. Our implementation is available at https://github.com/lavenklau/minisurf.

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Di Zhang, Ligang Liu. 2025-05-19. Asymptotic analysis and design of linear elastic shell lattice metamaterials. https://arxiv.org/abs/2506.18910

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