A general, implicit, finite-strain FE $$^2$$ framework for the simulation of dynamic problems on two scales

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. In addition, a simulation of a split Hopkinson tension test showcases the applicability of the framework to engineering problems.