Nonlinear Dynamics of Two-Dimensional Lattices with Complex Structure

Abstract

The dynamics of two-dimensional lattice structures is studied. The complexity of the structure includes non-neighboring interactions between the lattice masses, consideration of both translational and rotational interactions and also their nonlinear character (physical nonlinearity). The asymptotic procedures are developed to obtain the governing nonlinear equations of motion in the continuum limit. The equations obtained are studied both analytically and numerically. Of special interest are the propagation and transverse instability of the plane solitary strain waves. It is shown that the dynamics of longitudinal and shear waves is different in various two-dimensional lattices. The relationships for the elastic constants are obtained to characterize the type of the localized strain waves (tensile or compression), their transverse instability and possible auxetic behavior. Numerical solutions are obtained that describe unstable and stable dynamics of the plane longitudinal and shear waves.