LOCALISED STRAIN WAVES IN A TWO-DIMENSIONAL CRYSTALLINE MEDIUM WITH A NON-DENSE PACKING OF THE PARTICLES

Abstract

A two-dimensional model of a crystalline (granular) medium is considered that represents a square lattice consisting of elastically interacting particles, which possess translational and rotational degrees of freedom. In the long-wavelength approximation a set of nonlinear equations in partial derivatives has been derived that describes propagation of longitudinal, transverse and rotational waves in such a medium. Dependences of the velocities of elastic waves and the nonlinearity coefficients on the sizes of particles and the parameters of interactions between them have been found in the analytical form. In the field of low frequencies, when the rotational degree of freedom of particles can be neglected, the obtained three-mode system degenerates into a two-mode system, which is reduced by the multi-scale method to Kadomtsev–Petviashvili evolutionary equation for shear deformation, which has a soliton-type solution. For some crystals with cubic symmetry it is found out, whether the soliton is steady and what kind of polarity it has.

Keywords: microstructured medium, the plane localized strain waves, stability and polarity of soliton.