Dynamic homogenization of resonant elastic metamaterials with space/time modulation

We present a variational multiscale strategy in the spirit of FE ^2 method for simulating the real time evolution of resonant elastic metamaterials with space/time modulation. The implicit time discretization used guarantees the stability of the numerical solution, while the accuracy is quantified by direct comparison with the response obtained via direct numerical simulation. The results showcase a remarkable accuracy of the method, even when the dynamic material properties drastically change over length scales that are equal to the macroscopic mesh size and time scales that are of the same order of magnitude than the external excitations. We further propose a strategy to identify the band gap structure of these metamaterials based on their real time response. The methodology is verified for spatially periodic time invariant systems via classical unit cell analyses in the frequency domain.