A General, Implicit, Large-Strain FE$^2$ Framework for the Simulation of Dynamic Problems on Two Scales

TL;DR

This paper introduces a comprehensive two-scale finite element framework for simulating dynamic problems involving large strains, micro-inertia, and nonlinear geometries, applicable to complex heterogeneous materials.

Contribution

It develops a fully-coupled FE$^2$ homogenization method incorporating micro inertia and finite strains, compatible with standard finite element software architectures.

Findings

Accurately captures micro-inertia effects in dynamic large-strain simulations.

Demonstrates effectiveness through layered material examples compared to direct simulations.

Provides explicit formulas for macroscopic tangent moduli including micro inertia.

Abstract

In this paper we present a fully-coupled, two-scale homogenization method for dynamic loading in the spirit of FE$^2$ methods. The framework considers the balance of linear momentum including inertia at the microscale to capture possible dynamic effects arising from micro heterogeneities. A finite-strain formulation is adapted to account for geometrical nonlinearities enabling the study of e.g. plasticity or fiber pullout, which may be associated with large deformations. A consistent kinematic scale link is established as displacement constraint on the whole representative volume element. The consistent macroscopic material tangent moduli are derived including micro inertia in closed form. These can easily be calculated with a loop over all microscopic finite elements, only applying existing assembly and solving procedures. Thus, making it suitable for standard finite element program architectures. Numerical examples of a layered periodic material are presented and compared to direct numerical simulations to demonstrate the capability of the proposed framework.