Band gaps and vibration of strongly heterogeneous Reissner–Mindlin elastic plates

We consider an elastic plate governed by the Reissner–Mindlinʼs model, i.e., whose equilibrium equations introduce a coupling between the vertical displacement and the rotation of the normal. This structure is made of a composite with a periodic arrangement of strongly heterogeneous materials and some characteristics of the heterogeneities are comparable to the size of the microstructures. 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. This means that there exist intervals of frequencies – the band gaps – for which wave propagation is suppressed, or restricted to certain polarizations.

RésuméOn étudie lʼinfluence de fortes hétérogénéités sur la propagation des ondes dans une plaque de Reissner–Mindlin. On montre que lorsque certaines caractéristiques élastiques sont comparables à la taille des microstructures du composite il apparaît des bandes interdites, i.e. des intervalles de fréquences – les bandes interdites – pour lesquels la propagation des ondes ne peut pas se faire ou bien est restreinte à certaines polarisations.