Bending/shear wave dispersion analysis of granular chains – Discrete and enriched continuous Cosserat modelling

Abstract

The current study focuses on the wave dispersion analysis of granular systems by investigating a paradigmatic one-dimensional discrete chain. Most of the analysis of the discrete problem are based on the discrete granular chain without additional confinement. This work is motivated by the detection of standing waves and negative velocity of acoustic and optical waves observed in discrete granular models. This study allows to understand better the incorporation between the theoretical models, granular materials and the included effective parameters. Concerning wave propagation, granular elements create a structured wave-guide network through which mechanical energy is transferred. In the presence of internal (microstructural) length scales, the elastic wave propagation problem involves an interplay between wave dispersion and structural features. The unidimensional granular chain is composed of uniform spherical grains elastically connected with both shear and rotational springs. The presented structural system can be considered as an elastic lattice model or a Cosserat chain model with shear interaction. We show how the elastic foundation affects the wave dispersion response of the system. The length scale (grain diameter) at which the system is probed, is an important issue bridging together multi-scale behavior and heterogeneity. In a first step, the wave dispersion of the discrete system is derived from the uncoupled mixed differential-difference equation of deflection. We further show that for infinite number of grains the dispersion equation of the discrete model converges toward the continuum model of Bresse-Timoshenko.The wave dispersive properties of this discrete model are investigated also in the Brillouin zone. Next, the discrete to continuous model is approximated by the methods of Taylor polynomial development and Padé rational expansion. Finally, a comprehensive dispersive analysis is done through the dispersion equations of the discrete model, the discrete to continuous approximation beam and the continuous one. Some experimental results for Carbon Nanotubes (CNTs) are replicated by the model. We show that the model is able to predict accurately the positive and negative group of wave velocity as well as the standing wave. Scale effects and wave dispersive characteristics of the granular chain are captured by the continuous gradientelasticity model.