Dispersion properties of the phononic crystal consisting of ellipse-shaped particles

A two-dimensional model is considered in the form of a phononic crystal having a rectangular lattice with elastically interacting ellipse-shaped particles possessing two translational and one rotational degrees of freedom. The linear differential-difference equations are obtained by the method of structural modeling to describe propagation of longitudinal, transverse and rotational waves in the medium. It is found analytically how the coefficients of the equations depend on the sizes of the particle and on the parameters of interactions between them. The dispersion properties of the model are analyzed. Existence of a backward wave is established. The threshold frequencies of acoustic and rotational waves in some crystalline materials with cubic symmetry are estimated.