Nonlinear resonances in a general class of granular dimers with no pre-compression

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In this work we study the existence of nonlinear resonances in a general 1: $$N$$ one-dimensional granular dimer chains, i.e., granular chains consisting of periodic sets of ‘heavy’ beads followed and preceded by $$N$$ ‘light’ beads. Each bead is assumed to be spherical and purely elastic and to interact with its neighbors through Hertzian contact law. In a previous work (Jayaprakash et al. in 112(3):1–17, 2012) we proved the existence of countably infinite families of solitary waves in these systems; these are localized pulses that propagate without distortion of their waveforms through these highly inhomogeneous nonlinear media. We attributed these waves to nonlinear anti-resonance that led to complete elimination of radiating waves in the trail of the propagating localized pulse. Anti-resonances were associated with certain symmetries of the velocity waveforms of the beads of the dimer. In this work we report on the opposite phenomenon, that is, of the breakup of waveform symmetries of the bead responses leading to drastic attenuation of propagating pulses due to energy radiation to the far field by means of nonlinear traveling waves. We use the connotation of resonance to describe this dynamical phenomenon resulting in the maximum amplification of the amplitudes of radiated waves that emanate from the propagating pulse. We study numerically and analytically the nonlinear resonance mechanism in this class of strongly nonlinear periodic media, and demonstrate that it can lead to drastic attenuation of shock-induced pulses propagating in the dimers.