Computational frameworks for homogenization and multiscale stability analyses of nonlinear periodic metamaterials

TL;DR

This paper develops a multiscale computational framework for homogenization and stability analysis of nonlinear periodic metamaterials, enabling accurate prediction of bifurcation points and buckling phenomena at micro and macro scales.

Contribution

It introduces a consistent multiscale finite strain homogenization method combined with stability analysis techniques including Bloch wave analysis and convexity checks.

Findings

Homogenization results are valid until the first bifurcation point.

Microscale buckling can be detected via rank-1 convexity loss.

Numerical validation confirms the framework's effectiveness on hyperelastic and elastoplastic metamaterials.

Abstract

This paper presents a consistent computational framework for multiscale 1st order finite strain homogenization and stability analyses of rate-independent solids with periodic microstructures. Based on the principle of multiscale virtual power, the homogenization formulation is built on a priori discretized microstructure, and algorithms for computing the matrix representations of the homogenized stresses and tangent moduli are consistently derived. The homogenization results lose their validity at the onset of 1st bifurcation, which can be computed from multiscale stability analysis. The multiscale instabilities include: a) microscale structural instability which is calculated by Bloch wave analysis; and b) macroscale material instability which is calculated by rank-1 convexity checks on the homogenized tangent moduli. Details on the implementation of the Bloch wave analysis are provided, including the selection of the wave vector space and the retrieval of the real-valued buckling mode from the complex-valued Bloch wave. Three methods are detailed for solving the resulted constrained eigenvalue problem - two condensation methods and a null-space based projection method. Both implementations of the homogenization and stability analyses are validated using numerical examples including hyperelastic and elastoplastic metamaterials. Various microscale buckling phenomena are also demonstrated by examining several representative metamaterial examples. Aligned with theoretical results, the numerical results show that the microscopic long wavelength buckling can be equivalently detected by the loss of rank-1 convexity of the homogenized tangent moduli.