Nonlinear Localized Strain Waves in a 2D Medium with Microstructure

A two-dimensional model of the 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 the partial derivatives equations have been derived that describe propagation of longitudinal, transverse and rotational waves in such a medium. In the field of low frequencies, when the rotational degree of freedom of particles can be neglected, the obtained nonlinear three-mode system degenerates into a two-mode system. Analytical dependencies 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 for both nonlinear models. Due to these dependencies, numerical estimations of the nonlinearity coefficients are performed. The two-mode system is shown to be reduced by the multi-scale method to Kadomtsev–Petviashvili evolutionary equation for transverse deformation, which has a soliton solution. For some crystals with a cubic symmetry it is found out, whether soliton is steady and what kind of polarity it has.