Asymptotic analysis and design of linear elastic shell lattice metamaterials

TL;DR

This paper introduces an asymptotic analysis framework for shell lattice metamaterials, defining a new metric called asymptotic directional stiffness (ADS), and demonstrates its application in optimizing and understanding the stiffness properties of complex periodic shell structures.

Contribution

It provides the first rigorous explanation for high bulk modulus in TPMS-based shell lattices and offers a novel optimization framework for ADS on periodic surfaces.

Findings

ADS effectively quantifies geometry-driven stiffness

Optimization improves ADS for various surface designs

Theoretical results match numerical experiments

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.