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Published December 2009 | Published
Journal Article Open

Localized breathing modes in granular crystals with defects


We study localized modes in uniform one-dimensional chains of tightly packed and uniaxially compressed elastic beads in the presence of one or two light-mass impurities. For chains composed of beads of the same type, the intrinsic nonlinearity, which is caused by the Hertzian interaction of the beads, appears not to support localized, breathing modes. Consequently, the inclusion of light-mass impurities is crucial for their appearance. By analyzing the problem's linear limit, we identify the system's eigenfrequencies and the linear defect modes. Using continuation techniques, we find the solutions that bifurcate from their linear counterparts and study their linear stability in detail. We observe that the nonlinearity leads to a frequency dependence in the amplitude of the oscillations, a static mutual displacement of the parts of the chain separated by a defect, and for chains with two defects that are not in contact, it induces symmetry-breaking bifurcations.

Additional Information

© 2009 The American Physical Society. Received 22 June 2009; published 1 December 2009. P.G.K. gratefully acknowledges support from NSF through Grants No. NSF-DMS-0349023 (CAREER) and No. NSF-DMS-0806762, as well as from the Alexander von Humboldt Foundation. The work of IGM and MK was partially supported by NSF and ACS(PRF). C.D. thanks NSF CAREER and NSF Grant No. CMMI-0825345 for support of this project.

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