Multiscale Modeling of Elastic Waves: Theoretical Justification and Numerical Simulation of Band Gaps

We consider a three-dimensional composite material made of small inclusions periodically embedded into an elastic matrix; the whole structure presents strong heterogeneities between its different components. In the general framework of linearized elasticity we show that, when the size of the microstructures tends to zero, the limit homogeneous structure presents, for some wavelengths, a negative “mass density” tensor. Hence we are able to rigorously justify the existence of forbidden bands, i.e., intervals of frequencies in which there is no propagation of elastic waves. In particular, we show how to compute these band gaps and illustrate the theoretical results with some numerical simulations.