Modulated phononic crystals: Non-reciprocal wave propagation and Willis materials

Research on breaking time-reversal symmetry in wave phenomena is a growing area of interest in the field of phononic crystals and metamaterials aiming to realize one-way propagation devices which have many potential technological applications. Here we investigate wave propagation in phononic crystals, periodic laminates in particular, where both elastic moduli and mass density are modulated in space and time in a wave-like fashion. The modulation introduces a bias which breaks time-reversal symmetry and reciprocity. A full characterization of how the dispersion curve transforms due to wave-like modulations is given in analytical and geometrical terms for both low (subsonic) and high (supersonic) modulation speeds. Theoretical findings are supported by numerical simulations. More specific to low frequencies, the macroscopic constitutive law of 1, 2 and 3D modulated laminates is proven to be of the Willis type with a non-negligible Willis coupling in the strictly scale-separated homogenization limit. The existence of a macroscopic stress-velocity and momentum-strain Willis coupling is in fact directly related to the breaking of reciprocity. Finally, closed form expressions of the macroscopic constitutive parameters are obtained and some elementary yet insightful energy bounds are derived and discussed.