Nonlinear wave scattering at the flexible interface of a granular dimer chain

Uncompressed granular dimer chains composed of repetitive pairs of heavy-light spherical, linearly elastic beads exhibit interesting intrinsic responses. The dynamics of these highly discontinuous nonlinear media is governed by the mass ratio scaling the mass disparity of each heavy-light pair of beads. In particular, it has been theoretically and experimentally shown that they support countable infinities of anti-resonances at a discrete set of mass ratios leading to solitary pulses propagating through the dimers with no attenuation or distortion. Conversely, they support countable infinities of resonances at a different discrete set of mass ratios, leading to substantial and rapid attenuation of propagating pulses due to energy scattering from low-to-high frequencies and wavenumbers by means of radiating traveling waves. In this work we computationally study nonlinear scattering of impeding pulses at the interface of an impulsively excited dimer chain with a dispersive elastic boundary, namely, a finite linear string resting on an elastic foundation. We develop a computational algorithm which, through iteration and interpolation at successive time steps, accurately computes (and ensures convergence of) the highly discontinuous contact forces and displacements at the flexible interface of the granular medium. This enables accurate computation of wave transmission, reflection, localization or multi-scale nonlinear scattering at the flexible interface for varying mass ratios of the dimer and the interface parameters. We show that, depending on the mass ratio of the dimer and the stiffness of the elastic foundation, the nonlinear scattering at the flexible interface may lead to significant reduction of the maximum contact force at the interface, and, thus, drastically affect the transmitted and reflected energy at the flexible boundary. In fact, an inverse relation between the stiffness of the elastic foundation and the residual energy transferred from the dimer chain to the flexible boundary is found. Moreover, for sufficiently small mass ratios of the dimer chain transient breathers are realized close to the interface in the form of localized “fast” oscillations of light granules of the dimer that entrap shock energy and then release in a slow time scale back to the chain and the flexible boundary. This work paves the way for studying highly discontinuous and nonlinear scattering phenomena at interfaces of granular media with flexible continua.