Nonlinear Waves in Granular Lattices

We present a numerical study of solitary waves in one dimensional (1D) granular lattices. Our system consists of an array of deformable spheres which we model using Hertzian interactions between neighbouring bodies. A general discussion of the origin of solitary waves is presented. We then provide an analysis of the Hertz force. For the case of a uniform chain of spheres, we derive an approximate solution in the asymptotic limit to the system equations. The solution is a solitary wave and numerical simulations verify this result. A scaling analysis is used to determine a relation between wave speed and amplitude. We find vs ∝ F 1/6 peak, where vs is the wave speed and Fpeak the amplitude and this is corroborated using simulations. We then proceed from uniform chains to those containing a defect. A study of the effect of defect size and material properties (i.e. Young’s modulus E and Poisson ratio ν) on a propagating solitary wave is performed. Finally, we outline the origin of intrinsic localised modes in granular lattices. These objects cause the scattering of incoming waves which can lead to interesting resonance phenomena. We probe the interaction of both solitary and plane waves with these modes. The transmission coefficient for plane waves is measured as a function of wavenumber q in order to observe whether the system permits behaviour analogous to Fano resonance. The results obtained do not clearly correspond to the Fano resonances observed in other systems. Thus, further studies are required to explain the mechanism of a plane wave’s interaction with a localised mode.