Discrete-element simulations by Mouraille and Luding (2008) showed that in a dense granular packing built from a face-centered-cubic (FCC) crystal of spheres, even a weak polydispersity (particle-size variation comparable to the static contact deformation) makes the transmission spectrum discontinuous: a low-frequency band propagates while higher frequencies are strongly attenuated. The authors left open how these transmission bands, band gaps, and attenuation lengths depend systematically on confining pressure, polydispersity width, and structural disorder; what the frequency dependence and mechanisms of compressional-to-shear (P-to-S) and shear-to-compressional (S-to-P) mode conversion are; and whether rotational-mode branches (particle-spin waves) are visible in polydisperse frictional packings. The same source line frames the micro-to-macro description as itself open - Cosserat-type continua incorporating micro-rotations have "still a long way to go", leaving the extrapolation to the macro-scale as "the open issue" (Mouraille 2009, Sec. 1.2) - posing the further question of how to derive micro-based continuum constitutive relations that reproduce the dynamic, frequency-dependent response. Later work resolved pieces of this agenda - rotational sound in two-dimensional disordered frictional packings, attenuation mechanisms in disordered disk assemblies, stress dependence of wave speeds and dispersion, and micropolar/micromorphic continua for ordered crystals - but the frequency-resolved mode-conversion question and the joint pressure-polydispersity-disorder map for three-dimensional frictional packings remain unaddressed, and a micro-based dynamic continuum for fully disordered packings is still explicitly future work.

Background

A dense granular packing is an assembly of macroscopic particles that interact only through repulsive contact forces. Under a confining pressure the contacts deform elastically; for spherical grains the Hertz law makes the normal contact stiffness grow with the static contact force, so the effective elastic moduli and sound speeds of the packing increase with pressure (approximately as p1/6p^{1/6} for a random Hertzian assembly under pressure pp). Wave propagation in such media is probed experimentally by ultrasonics and in simulation by the discrete element method (DEM), which integrates the equations of motion of each particle under its contact forces. Two coherent bulk modes exist in an elastic continuum: P-waves (compressional, displacement parallel to propagation) and S-waves (shear, displacement transverse to propagation). Because grains can also rotate, ordered granular assemblies additionally support rotational-mode branches, analogous to optical phonon branches in crystals, in which the particle spins oscillate; these are captured phenomenologically by Cosserat (micropolar) continuum theories, which augment the classical elasticity of a granular medium with a couple-stress field conjugate to particle rotations. Mode conversion is the transfer of energy between P- and S-wave polarization during propagation or scattering. In an ordered (crystalline) packing, plane waves have well-defined dispersion relations and band gaps (frequency intervals in which no propagating mode exists), exactly as in a phononic crystal; in a disordered packing the same concepts survive statistically, through transmission spectra and attenuation lengths, the distance over which a coherent wave amplitude decays. Mouraille and Luding (2008) simulated P- and S-wave propagation in three-dimensional packings consisting of FCC lattices of spheres perturbed by a narrow size polydispersity (a continuous distribution of particle sizes, here weak), with both linear and Hertz-nonlinear contact laws. Their central observation was dimensional: once the size variation becomes comparable to the typical static contact deformation (overlap), the transmission spectrum turns discontinuous - lower frequencies transmit, higher frequencies are cut off - and this low-pass filtering is intrinsic, persisting with Hertz contacts, friction, smaller amplitudes, and larger systems. Related earlier experiments had shown strong multiple scattering of elastic waves in dense granular media (Jia 2004; Liu and Nagel 1992 measured non-Hookean, pressure-dependent sound speeds in compressed sand), and subsequent work on ordered granular crystals established their band structure, including rotational branches confirmed experimentally (Merkel, Tournat and Gusev 2010, 2011) and reproduced by a micropolar continuum (Merkel and Luding 2017). What no existing result provides is the systematic bridge from the crystalline limit to genuinely disordered, polydisperse, frictional packings: how confinement, size distribution, and disorder jointly control wave transmission, how polarization conversion behaves in frequency, and which continuum description reproduces it.

Problem Statement

Establish a quantitative, micro-mechanically grounded account of elastic wave transmission in dense polydisperse granular packings, spanning the range from weakly perturbed crystalline (FCC) packings to fully disordered configurations, by answering four linked questions that share one objective - predicting wave transport from contact-scale mechanics. (1) Map systematically how the transmission bands, band gaps, and attenuation lengths of P- and S-waves depend on confining pressure (static contact deformation), on the width of the polydispersity distribution up to full polydispersity, and on disorder amplitude (structural disorder beyond weak lattice perturbation), including packings that are fully disordered rather than weakly perturbed FCC crystals. (2) Resolve the frequency dependence and physical mechanisms of mode conversion, i.e. the transfer of energy between compressional (P) and shear (S) polarizations during propagation through such packings, via detailed spectral analysis of the mode-resolved wave field. (3) Determine the visibility and dispersion of rotational-mode branches (collective particle-rotation waves) in polydisperse, frictional packings, including whether and where they decouple from or hybridize with the acoustic branches. (4) Derive micro-based continuum constitutive relations (of Cosserat/micropolar or more general micromorphic type) that encode the contact-scale physics and reproduce the observed dynamic, frequency-dependent response, including dispersion and attenuation, of the disordered packings. The four deliverables are scientifically linked: the parameter map (1) and the mode-resolved mechanisms (2)-(3) are the phenomena any valid continuum closure (4) must reproduce, so together they define one coherent objective rather than four independent problems.

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