Time scale disparity yielding acoustic nonreciprocity in a two-dimensional granular-elastic solid interface with asymmetry

We study nonreciprocal wave transmission across the interface of two dissimilar granular media separated by an elastic solid medium. Specifically, a left, larger-scale and a right smaller-scale granular media composed of two-dimensional, initially uncompressed hexagonally packed granules are interfacing with an intermediate linearly elastic solid, modeled either as a thin elastic plate or a linear Euler-Bernoulli beam. The granular media are modeled by discrete elements and the elastic solid by finite elements assuming a plane stress approximation for the thin plate. Accounting for the combined effects of Hertzian, frictional and rotational interactions in the granular media, as well as the highly discontinuous interfacial effects between the (discrete) granular media and the (continuous) intermediate elastic solid, the nonlinear acoustics of the integrated system is computationally studied subject to a half-sine shock excitation applied to a boundary granule of either the left or right granular medium. The highly discontinuous and nonlinear interaction forces coupling the granular media to the elastic solid are accurately computed through an algorithm with interrelated iteration and interpolation at successive adaptive time steps. Numerical convergence is ensured by monitoring the (linearized) eigenvalues of a nonlinear map of interface forces at each (variable) time step. Due to the strong nonlinearity and hierarchical asymmetry of the left and right granular media, time scale disparity occurs in the response of the interface which breaks acoustic reciprocity. Specifically, depending on the location and intensity of the applied shock, propagating wavefronts are excited in the granular media, which, in turn, excite either (slow) low-frequency vibrations or (fast) high-frequency acoustics in the intermediate elastic medium. This scale disparity is due to the size disparity of the left and right granular media, which yields drastically different wave speeds in the resulting propagating wavefronts. As a result, the continuum part of the interface responds with either low-frequency vibrations—when the shock is applied to the larger-scale granular medium, or high-frequency waves—when the shock is applied to the smaller-scale granular medium. This provides the fundamental mechanism for breaking reciprocity in the interface. The nonreciprocal interfacial acoustics studied here apply to a broad class of asymmetric hybrid (discrete-continuum) nonlinear systems and can inform predictive designs of highly effective granular shock protectors or granular acoustic diodes.

Figure 1

Schematics of the granular-elastic solid interface subject to shock excitation: (a) 3D isometric view, (b) 2D planar model with boundary conditions, (c) granule-granule and granule-elastic solid contact points, corresponding free body diagrams and notations for normal and tangential interaction forces.

Figure 2

Computational modeling of the granular-elastic solid interfaces: (a) FE mesh when the elastic solid is modeled as a slender Euler-Bernoulli beam—note that the contact forces are applied at nodes of the FE mesh; (b) FE mesh when the elastic solid is modeled as a thin plate—note the denser mesh close to each contact point with a granule; small rigid layer at a contact point between a granule of (c) the LSGM and (d) the SSGM and the thin plate model.

Figure 3

Acoustic nonreciprocity in the granular-elastic solid (modeled as a plate) interface of Fig. 1: (a) The Strong shock excitation and (b) velocities of granule 3 of the SSGM, $v_{3C}$, and granule C of the LSGM, $v_{C3}$, for excitations as described in the text and in the inset of (a).

Figure 4

Instantaneous normalized energies (with respect to impulsive energy) of the LSGM, the SSGM, the elastic solid (modeled as plate or beam), and the entire granular interface (with and without dissipation effects added): (a), (c), (e) excitation is applied to granule 3 of the LSGM for weak, moderate, and strong shocks, respectively; and (b), (d), (f) excitation is applied to granule C of the SSGM granule for weak, moderate, and strong shock excitations, respectively; the inset in (f) is an enlargement of the energy of the elastic solid in this case when modeled as plate or beam.

Figure 5

Normalized energy density snapshots of the granular-elastic solid interface for the case of strong shock excitation: (a), (b) $\tau =0.2\mathrm{ms}$ with the excitation applied to granule 3 of the LSGM, and the elastic solid modeled as thin plate or Euler-Bernoulli beam, respectively; (c), (d) $\tau =0.05\mathrm{ms}$ with the excitation applied to granule C of the SSGM, and the elastic solid modeled as thin plate or Euler- Bernoulli beam, respectively; note the much faster acoustics in the latter case.

Figure 6

Selected modes of the thin plate model of the elastic solid of the granular interface of Fig. 1: (a) Undeformed FE mesh, (b) first bending mode with natural period $0.199\mathrm{ms}$, (c) fourth axial mode with natural period $0.0395\mathrm{ms}$—due to its antisymmetry it cannot contribute to the acoustics, and (d) seventh bulk mode with natural period $0.0198\mathrm{ms}$—due to its symmetry it may participate in the acoustics.

Figure 7

Normalized accumulated modal energy in the elastic solid when modeled as a thin plate in plane stress, subject to strong shock applied (a) at granule 3 of the LSGM, and (b) granule C of SSGM; in each case the accumulated total energy and modal energies of the leading 6, 100, and 1000 plate modes are depicted as functions of time.

Figure 8

Instantaneous normalized total, elastic, and kinetic energies of the elastic solid when modeled as a thin plate in plane stress subject to strong shock excitation applied to (a) granule 3 of the LSGM and (b) granule C of the SSGM.

Figure 9

Accumulated energy captured by the leading modes, as defined in (16): (a) weak $\left(F_{max}=63.67\mathrm{N}\right)$, (b) moderate $(F_{max}=318.3\mathrm{N}$), and (c) strong $(F_{max}=1591.6\mathrm{N})$ shocks.

Figure 10

Force-velocity constitutive relation of (a) the smooth Coulomb-tanh friction model and (b) the discontinuous Coulomb dry friction model.

Figure 11

Moduli of the six eigenvalues for strong shock applied to the granule C of the SSGM, at the three contact points (a) on the left boundary of the plate and (b) on the right boundary of the plate; note the discontinuous nature of the eigenvalues due to the adaptive change of the time step.