Geometrically nonlinear dynamic model for a hexagonal lattice.

It is shown that angular stiffness in the hexagonal lattice model plays a significant role in the geometrical nonlinear terms in the equations of the continuum limit. A geometrically nonlinear discrete model is formulated for the hexagonal lattice by considering the interaction of two sublattices. An asymptotic procedure is developed in order to obtain the nonlinear coupled equations of motion in the continuum limit of the discrete model. An interaction of longitudinal and shear plane strain waves is studied by using the solutions of the obtained equations.

Kink and solitary waves may propagate together.

It is shown numerically that kink-shaped and bell-shaped localized moving defects may coexist in a biatomic crystalline lattice. The shape and velocity of these waves are defined from a corresponding single wave exact traveling wave solution to the governing coupled nonlinear equations. The features of the initial conditions, or an external loading, are found that provide simultaneous propagation of these nonlinear moving defects, inputs, and their amplitudes.

Two approaches to study essentially nonlinear and dispersive properties of the internal structure of materials.

It is shown that essentially nonlinear models for solids with complex internal structure may be studied using phenomenological and proper structural approaches. It is found that both approaches give rise to the same nonlinear equation for traveling longitudinal macrostrain waves. However, presence of the connection between macro- and microfields in the proper structural model prevents a realization of some important solutions. First, it is obtained that simultaneous existence of compression and tensile waves is impossible in contrast to the phenomenological approach. Then, it is found that correlation between macro- and internal strains gives rise to the crucial influence of the velocity of the macrostrain wave on the existence of either compression or tensile localized strain waves. Also, it is shown that similar profiles of the macrostrain solitary waves may be accompanied by distinct profiles of the microstrain waves. Finally, the dispersion curves for the waves belong to the different branches. This is important for internal structural deviations caused by the dynamical loading due to the localized macrostrain wave propagation and demonstrates a need in the development of the proper structural approaches relative to the phenomenological ones.

Propagation of localized longitudinal strain waves in a plate in the presence of cubic nonlinearity.

Possible propagating longitudinal strain solitary waves in a plate are shown to be seriously altered when physical cubic nonlinearity is taken into account in the modeling. This also affects an amplification of the wave due to the transverse instability of plane-localized waves and due to the plane-wave interaction.