Localized Breathing Modes in Granular Crystals with Defects

TL;DR

This paper explores nonlinear localized vibrational modes at defects in a one-dimensional granular chain, analyzing their formation, stability, and bifurcations, revealing differences based on contact conditions of the impurities.

Contribution

It introduces a detailed analysis of nonlinear defect modes in granular crystals with defects, highlighting bifurcation behaviors and stability characteristics under different contact scenarios.

Findings

Localized defect modes exist at light-mass impurities.

Bifurcation and stability depend on impurity contact conditions.

Symmetry-breaking bifurcations occur in non-contact impurity cases.

Abstract

We investigate nonlinear localized modes at light-mass impurities in a one-dimensional, strongly-compressed chain of beads under Hertzian contacts. Focusing on the case of one or two such "defects", we analyze the problem's linear limit to identify the system eigenfrequencies and the linear defect modes. We then examine the bifurcation of nonlinear defect modes from their linear counterparts and study their linear stability in detail. We identify intriguing differences between the case of impurities in contact and ones that are not in contact. We find that the former bears similarities to the single defect case, whereas the latter features symmetry-breaking bifurcations with interesting static and dynamic implications.