Elastic waves in noncohesive frictionless granular crystals

An ordered structure of noncohesive spherical beads constitutes a phononic crystal. This type of media combines the properties of wave propagation in phononic crystals (dispersion due to the geometrical periodicity) with the properties of wave propagation in granular media (nonlinearities, rotational degree of freedom) and gives the opportunity to have interesting features as tunable frequency band gaps for example. In this work, the acoustic bulk modes of a hexagonal close packed (hcp) structure of beads, considered as rigid masses connected by springs, are theoretically evaluated and their associated resonance frequencies are compared to experimental results. When friction is neglected, the elastic interaction between the beads are reduced to a normal spring interaction given by the Hertz theory. According to this theory, the rigidity of the contact depends on its static loading. The theory predicts the existence of elastic transverse and longitudinal acoustical-type modes and transverse and longitudinal optical-type modes. The acoustic transfer function of a hcp crystal slab built with stainless steel beads is measured and its resonance frequencies are compared to the theoretical predictions. Despite some differences between theory and experiments, which could come for instance from the disordered character of the contact loads, the developed theory and the experimental results show relatively good agreement.