Evolution of the Primary Pulse in 1D Granular Crystals Subject to On-Site Perturbations: Analytical Study

TL;DR

This paper develops an analytical model to predict how primary pulses evolve in 1D granular chains under various on-site perturbations, including dissipative and energy source effects, validated by numerical simulations.

Contribution

The study introduces a general analytical approach for predicting pulse evolution in perturbed granular chains, including novel cases with dissipation and energy input.

Findings

Analytical predictions closely match numerical simulations.

Stable stationary shocks can form under specific perturbations.

The model accurately estimates amplitudes and stability zones of shocks.

Abstract

Propagation of primary pulse through an un-compressed granular chain subject to external on-site perturbation is studied. Analytical procedure predicting the evolution of the primary pulse is devised for the general form of the on-site perturbation applied on the chain. The validity of the analytical model is confirmed with several specific granular setups such as, chains mounted on the nonlinear elastic foundation, chains perturbed by the dissipative forces as well as randomly perturbed chains. Additional interesting finding made in the present study corresponds to the chains subject to a special type of perturbations including the terms leading to dissipation and those acting as an energy source. It is shown in the study that application of such perturbation may lead to formation of stable stationary shocks acting as attractors for the initially unperturbed, propagating Nesterenko solitary waves. Interestingly enough the developed analytical procedure provides an extremely close estimations for the amplitudes of these stationary shocks as well as predicts zones of their stability. In conclusion we would like to stress that the developed analytical model have demonstrated spectacular correspondence to the results of direct numerical simulations for all the setups considered in the study.