Homogenized Models of Mechanical Metamaterials

TL;DR

This paper develops and validates homogenized continuum models for mechanical metamaterials, capturing their complex lattice behaviors efficiently and providing open-source tools for simulation.

Contribution

It rigorously derives micropolar-type homogenized models from variational principles and demonstrates their effectiveness through numerical convergence and examples.

Findings

Micropolar models accurately capture lattice kinematics.

Numerical convergence of discrete to continuum energy is shown.

Open-source implementations are provided for reproducibility.

Abstract

Direct numerical simulations of mechanical metamaterials are prohibitively expensive due to the separation of scales between the lattice and the macrostructural size. Hence, multiscale continuum analysis plays a pivotal role in the computational modeling of metastructures at macroscopic scales. In the present work, we assess the continuum limit of mechanical metamaterials via homogenized models derived rigorously from variational methods. It is shown through multiple examples that micropolar-type effective energies, derived naturally from analysis, properly capture the kinematics of discrete lattices in two and three dimensions. Moreover, the convergence of the discrete energy to the continuum limit is shown numerically. We provide open-source computational implementations for all examples, including both discrete and homogenized models.