A 2026 study of shock transmission through a two-dimensional granular layer bonded to an elastic plate showed that the wavenumber-domain spectrum of the transmitted wavefront evolves like the solution of an anisotropic heat equation — a hyperbolic-to-parabolic transition induced by a compact, strongly nonlinear scattering region — but the diffusion coefficients were chosen arbitrarily, and the analogy was validated only qualitatively, for one ordered packing, one shock amplitude, and early times before boundary reflections return. The open problem is to convert this qualitative analogy into a predictive, validated reduced description: derive effective wavenumber-domain diffusion coefficients (or an explicitly stated, demonstrably better transport-type model) from the granular microphysics, and establish the mapping's domain of validity across packing order and disorder, input-energy regimes spanning persistent contact and granule–plate separation, scatterer-region size, and long-time dynamics including boundary reflections and plate finite-size modes, with the modal diagnostics shown robust to the wavenumber-filtering and discretization parameters.

Background

Granular crystals are ordered assemblies of macroscopic elastic grains whose neighbors interact through Hertzian contacts — repulsive normal forces proportional to the contact overlap raised to the power 3/2 — so that an uncompressed assembly has no linear stiffness to perturb about. Nesterenko (1994) showed that such media behave as "sonic vacua" supporting strongly nonlinear solitary waves whose speed depends on wave amplitude rather than on a material sound speed, and Yang et al. (2011) studied how these waves interact with ordinary linear elastic media. A closely related line of work concerns hybrid discrete–continuum systems: Wang, Zhang and Vakakis (2021) simulated shock transmission through a two-dimensional hybrid interface comprising an ordered hexagonal layer of steel granules — with Hertzian normal contacts, frictional tangential contacts, grain rotations, and unilateral contacts that can separate and recollide — bonded at five contact points to a linear elastic plate modeled under plane stress. Compared with an isoenergetic shock entering a monolithic plate–plate interface, the granular layer drastically delays and diminishes energy transmission into the plate (about 41% versus about 88% peak transmitted energy in the studied configuration) and makes the response passively tunable to the input amplitude (Wang, Tawfick and Vakakis 2021, 2022). Tempelman, Vakakis and Matlack (2023) had earlier quantified mode-resolved spectral scattering for one-dimensional phononic lattices with vibro-impact nonlinearities.

Tempelman, Wang and Vakakis (2026) supplied the corresponding spectral picture for the two-dimensional granular–elastic interface by importing acoustic wavenumber imaging, a postprocessing technique from non-destructive evaluation: each snapshot of the plate velocity field is spatially Fourier transformed, multiplied by a ring-shaped Log–Gabor filter (a band-pass filter selecting a narrow annulus of the wavenumber plane centered at a chosen wavenumber kck_c, with bandwidth set by a spread parameter), and transformed back, isolating the energy carried in each wavenumber band. Two scalar diagnostics track the mode-resolved evolution: the spectral variance σk(τ)\sigma_k(\tau), the second moment of the wavenumber–energy distribution, and the spectral entropy Hk(τ)H_k(\tau), the Shannon entropy of the normalized wavenumber–energy distribution; in the source's diffusion comparison these are computed through a discrete-cosine/discrete-sine (DCT/DST) modal representation matched to the heat-solution basis, a chain distinct from the Log–Gabor wavenumber-band imaging. Through a monolithic interface both remain nearly constant — the shock stays a coherent "block of energy" — whereas the primary wavefront transmitted through the granular layer (the packet launched by the shock before any boundary reflections return) breaks up and its spectrum collapses toward low wavenumbers.

That collapse is reminiscent of a classical fact: linear multiple scattering by large ensembles of scatterers produces diffusive energy transport in the short-wavelength limit, formalized by radiative transfer theory (a Boltzmann-type balance for wave energy density in phase space) and its diffusion approximation, with applications to coda waves in dense granular media (Jia 2004) and to classical-wave multiple scattering generally (van Rossum and Nieuwenhuizen 1999); disordered granular chains similarly act as frequency filters (Lawney and Luding 2014). Tempelman et al. show that a qualitatively similar transition — from hyperbolic (wave-like, finite propagation speed) to parabolic (diffusion-like, smoothing) evolution — arises with no linear-scattering ensemble at all: a compact, strongly nonlinear, ordered granular region only a few grain diameters across suffices. Quantitatively, they compare the simulated metrics with the exact separation-of-variables solution of an anisotropic heat equation ψ′=αxψxx+αyψyy\psi'=\alpha_x\psi_{xx}+\alpha_y\psi_{yy} on the plate domain with matching boundary conditions, and find that the normalized spectral variance and entropy track the diffusion decay (roughly 1/τ21/\tau^2 in a normalized time τ∈[0,1]\tau\in[0,1]).

The comparison is, however, deliberately qualitative. The anisotropy ratio αx=2αy\alpha_x=2\alpha_y and the coefficient scale were chosen arbitrarily "to better approximate" the observed spreading, and only normalized decay trends were matched. The demonstration covers one ordered compact packing, one shock amplitude, the early-time window before boundary reflections and granule–plate separation re-enter the dynamics, and one setting of the filter-bank and discretization parameters. The open question is what, if anything, replaces the arbitrary coefficients: a predictive mapping from the granular microphysics to the effective spectral transport.

Problem Statement

Establish a quantitative, predictive reduced description of the mode-resolved spectral evolution of a wavefront transmitted from a strongly nonlinear granular layer into a coupled elastic plate, replacing the qualitative diffusion analogy of Tempelman, Wang and Vakakis (2026) with a rigorous mapping. The problem has four jointly supporting parts: (i) derive — analytically, by systematic simulation-based homogenization, or through an equivalent transport-type closure — effective wavenumber-domain diffusion coefficients αx\alpha_x and αy\alpha_y, or an explicitly stated alternative parabolic or transport model that demonstrably fits better, as functions of the granular microphysics (grain material and radius, packing geometry and degree of disorder, Hertzian and frictional contact laws including grain rotations and separations, and incident shock amplitude); (ii) determine the domain of validity of this mapping across packing orders, sizes, and degrees of disorder, across input energies spanning the persistent-contact and separation/collision (non-contact) regimes, and across scatterer-region sizes up to spatially extended regions where radiative-transport descriptions might become valid; (iii) extend the comparison beyond early times to long-time spectral statistics that include boundary reflections, plate finite-size modes, and granule–plate decoupling, establishing whether a unified hyperbolic-to-parabolic reduced model can account for these effects; and (iv) quantify the sensitivity of the modal diagnostics — the spectral variance and entropy computed through the modal (DCT/DST-type) wavenumber representation, together with the Log–Gabor-type wavenumber-band decompositions used to image the transmitted wavefront — to filter bandwidth, modal-basis truncation, spatial sampling, and numerical resolution, so that inferred diffusive behavior is demonstrably not a postprocessing artifact. These parts serve one overarching objective: converting the demonstrated qualitative hyperbolic-to-parabolic transition into a predictive, validated model of spectral energy transport across granular–elastic interfaces, with part (iv) supplying the methodological robustness on which the claims of parts (i)–(iii) rest. A complete answer addresses all four parts: it states the mapping (or its impossibility in specified regimes, which resolves only that part), delimits its validity domain with evidence, extends the comparison to long-time spectral statistics, quantifies the sensitivity of the modal diagnostics, and quantifies the agreement between predicted and observed spectral evolution.

References

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