Nonlinear dynamics of granular assemblies
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In this work we investigate granular chains, which are one-dimensional systems of discrete macroscopic particles interacting via the intrinsically nonlinear Hertz law. Such systems support the propagation of solitary waves (SWs), which are non-dispersive, mobile bundles of energy. A comprehensive analysis into the dynamical behaviour of these systems and the properties of SW propagation is presented, and several interesting new results are obtained. First, we find that the transition to the quasi-equilibrium (QEQ) phase in granular chains can be manipulated by altering the material properties of the system. We further use these results to develop a novel shock absorption device with a predictable and tunable frequency response, making it useful also for energy harvesting applications. Second, we show for the first time that granular chains with various nonlinearities of the contact potential can achieve thermal equilibrium at sufficiently long times, and thus QEQ is an intermediate phase of these systems. We characterize the equilibrium phase by deriving approximate distribution functions for grain velocity and kinetic energy and system kinetic energy in a microcanonical ensemble of interacting particles. As a by-product, we derive the equilibrium specific heat, and a size-dependent correction term, for such systems. We also show how these ideas extend to heterogeneous systems such as diatomic, tapered, and random-mass chains. Furthermore, we look closely at the transition to equilibrium by using statistical tests to show that the long-term dynamics is ergodic, and by examining the behaviour of various correlation functions close to the onset of the transition. Third, we solve a highly nonlinear, fourth-order wave equation that models the continuum theory of long-wavelength pulses in weakly compressed, homogeneous granular chains with a general power-law contact interaction, to characterize all travelling wave solutions admitted by the equation. This involves deriving conservation laws admitted by the wave equation, followed by a modified energy analysis. We find that the wave equation admits various types of travelling wave solutions, including SW solutions as well as nonlinear periodic wave solutions. Not only have the SW solutions not appeared before in the literature on granular chains, but they are also a new addition to the literature on SWs in general.