A Comparative Study of Enriched Computational Homogenization Schemes Applied to Two-Dimensional Pattern-Transforming Elastomeric Mechanical Metamaterials

TL;DR

This study compares various computational homogenization methods for predicting the nonlinear and pattern-transforming behavior of 2D elastomeric metamaterials, highlighting the strengths and limitations of each scheme.

Contribution

It provides a systematic comparison of classical and enriched homogenization schemes, identifying the most accurate method for different microstructural behaviors in elastomeric metamaterials.

Findings

Second-order scheme qualitatively predicts responses but overestimates bifurcation strains.

Micromorphic method yields the most accurate predictions overall.

First-order scheme performs well at high scale separations but has convergence issues.

Abstract

Elastomeric mechanical metamaterials exhibit unconventional behaviour, emerging from their microstructures often deforming in a highly nonlinear and unstable manner. Such microstructural pattern transformations lead to non-local behaviour and induce abrupt changes in the effective properties, beneficial for engineering applications. To avoid expensive simulations fully resolving the underlying microstructure, homogenization methods are employed. In this contribution, a systematic comparative study is performed, assessing the predictive capability of several computational homogenization schemes in the realm of two-dimensional elastomeric metamaterials with a square stacking of circular holes. In particular, classical first-order and two enriched schemes of second-order and micromorphic computational homogenization type are compared with ensemble-averaged full direct numerical simulations on three examples: uniform compression and bending of an infinite specimen, and compression of a finite specimen. It is shown that although the second-order scheme provides good qualitative predictions, it fails in accurately capturing bifurcation strains and slightly over-predicts the homogenized response. The micromorphic method provides the most accurate prediction for tested examples, although soft boundary layers induce large errors at small scale ratios. The first-order scheme yields good predictions for high separations of scales, but suffers from convergence issues, especially when localization occurs.