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In this paper we present a fully-coupled, two-scale homogenization method fordynamic loading in the spirit of FE$^2$ methods. The framework considers thebalance of linear momentum including inertia at the microscale to capturepossible dynamic effects arising from micro heterogeneities. A finite-strainformulation is adapted to account for geometrical nonlinearities enabling thestudy of e.g. plasticity or fiber pullout, which may be associated with largedeformations. A consistent kinematic scale link is established as displacementconstraint on the whole representative volume element. The consistentmacroscopic material tangent moduli are derived including micro inertia inclosed form. These can easily be calculated with a loop over all microscopicfinite elements, only applying existing assembly and solving procedures. Thus,making it suitable for standard finite element program architectures. Numericalexamples of a layered periodic material are presented and compared to directnumerical simulations to demonstrate the capability of the proposed framework.

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