Micromorphic Computational Homogenization for Mechanical Metamaterials with Patterning Fluctuation Fields

TL;DR

This paper develops a micromorphic homogenization method for elastomeric metamaterials that accounts for long-range fluctuation fields, enabling efficient multiscale modeling and accurate prediction of material behavior.

Contribution

It introduces a novel homogenization framework incorporating patterning fluctuation fields and an efficient computational approach for elastomeric metamaterials.

Findings

The framework accurately predicts material response compared to full-scale simulations.

The method effectively captures long-range correlated microstructural fluctuations.

Computational efficiency is improved through local energy density approximations.

Abstract

This paper presents a homogenization framework for elastomeric metamaterials exhibiting long-range correlated fluctuation fields. Based on full-scale numerical simulations on a class of such materials, an ansatz is proposed that allows to decompose the kinematics into three parts, i.e. a smooth mean displacement field, a long-range correlated fluctuating field, and a local microfluctuation part. With this decomposition, a homogenized solution is defined by ensemble averaging the solutions obtained from a family of translated microstructural realizations. Minimizing the resulting homogenized energy, a micromorphic continuum emerges in terms of the average displacement and the amplitude of the patterning long-range microstructural fluctuation fields. Since full integration of the ensemble averaged global energy (and hence also the corresponding Euler--Lagrange equations) is computationally prohibitive, a more efficient approximative computational framework is developed. The framework relies on local energy density approximations in the neighbourhood of the considered Gauss integration points, while taking into account the smoothness properties of the effective fields and periodicity of the microfluctuation pattern. Finally, the implementation of the proposed methodology is briefly outlined and its performance is demonstrated by comparing its predictions against full scale simulations of a representative example.