Approach of a three-dimensional granular packing to the sonic-vacuum limit under isotropic low confinement pressure

A granular packing of elastic spheres held together only by an external confinement pressure $p_0$ has no rigidity at zero pressure: Hertzian contact mechanics makes the small-amplitude sound speed pressure-dependent, and effective-medium theory predicts it vanishes as $p_0\to 0$ (deep in the jammed, affinely deforming regime, $v_p\propto p_0^{1/6}$). In the limit of vanishing precompression — Nesterenko's 'sonic vacuum' — no linear wave regime exists and propagation speed is set by the wave amplitude alone. Whether a real, disordered three-dimensional packing actually approaches this limit is unknown: on Earth, the packing's own weight imposes a hydrostatic pressure floor of order $1\ \mathrm{kPa}$ and strong pressure gradients, and ground data show an anomalous $v_p\propto p_0^{1/4}$ scaling that cannot be disentangled from gravity-induced inhomogeneity. The open problem is to determine, under homogeneous isotropic confinement about two orders of magnitude below the terrestrial floor (accessible in microgravity), (i) whether and with what pressure scaling the linear sound speed vanishes, (ii) whether propagation crosses over to amplitude-determined (strongly nonlinear, shock-like) behavior with a collapsing linear regime, and (iii) how the multiple-scattering transport parameters (transport mean free path, diffusion constant, absorption time) depend on pressure, particle material, and packing fraction near jamming. As of August 2026 no such measurement has been published; the problem remains open.

Background

A granular packing is a dense assembly of macroscopic particles that interact only through repulsive contact forces. For elastic, frictional spheres the contact force $f$ scales with the overlap $\delta$ as $f\propto\delta^{3/2}$ (Hertz contact), so the stiffness of every contact grows with the load. When a packing is confined by an external static pressure $p_0$, the contact network transmits sound, and because contact stiffness depends on load, the acoustic properties are nonlinear and pressure-dependent even at small amplitude. Effective medium theory (EMT), which averages the disordered contact network into homogeneous elastic moduli, predicts that deep in the jammed regime — where changes of $p_0$ deform the network affinely and leave the average number of contacts per grain, $Z$, unchanged — the longitudinal sound speed scales as $v_p\propto p_0^{1/6}$. The jamming transition is the point at which, as $p_0$ (or packing fraction) decreases, the packing loses rigidity: near it, coordination and non-affine rearrangements modify the elastic response, and both moduli and sound speed are expected to vanish as $p_0\to 0$. Exactly how the wave physics crosses over between the near-jamming and affine regimes at low pressure is not settled.

The strong-amplitude counterpart of this picture is the 'sonic vacuum', introduced by Nesterenko for chains of grains without precompression. For a compression front propagating through a precompressed granular medium, with $\delta_0$ the static precompression indentation and $\delta_S\geq\delta_0$ the total contact indentation at the shock front, the front speed is

$$v_{\mathrm{shock}}=c\,\sqrt{\frac{1}{\alpha-1}\,\frac{(\delta_S/\delta_0)^{\alpha-1}-1}{(\delta_S/\delta_0)-1}},$$

where $c$ is the linear sound speed and $\alpha$ is the exponent of the inter-grain contact potential energy, $U\propto\delta^{\alpha}$ ($\alpha=5/2$ for Hertzian contact between elastic spheres, for which the force scales as $f\propto\delta^{3/2}$). As $\delta_0/\delta_S\to 0$ (equivalently $p_0\to 0$), $c\to 0$, no linear regime survives, and the propagation speed is determined by the amplitude alone: $v_{\mathrm{shock}}\propto\delta_S^{1/4}$ for Hertzian contacts, equivalently $v_{\mathrm{shock}}\propto p_S^{1/6}$ in terms of the dynamic pressure $p_S$ at the front. Simulations of frictionless two-dimensional packings near jamming (Gómez et al., 2012) showed that the elementary excitations there are indeed strongly nonlinear shocks whose speed depends on pressure and impact intensity, and ground experiments in weakly compressed three-dimensional packings (van den Wildenberg et al., 2013) observed the crossover from quasi-linear sound to shocks whose speed becomes independent of $p_0$ — a hallmark of amplitude-determined propagation — but only down to $p_0$ of order $1\ \mathrm{kPa}$.

The terrestrial obstacle is gravity itself. A packing of glass beads a few tens of centimeters deep exerts a hydrostatic pressure of order $1\ \mathrm{kPa}$ at the bottom, so the confinement pressure cannot be made homogeneous or pushed far below that floor; the resulting vertical pressure gradient coexists with, and mimics, whatever intrinsic low-pressure physics one wants to measure. In microgravity a well-defined external force can impose isotropic pressures roughly two orders of magnitude lower (of order $10\ \mathrm{Pa}$), homogeneous throughout the sample. A drop-tower measurement of wave velocities in granular material under microgravity (Zeng et al., 2007) existed but without any measurement or adjustment of confinement pressure; ground studies at controlled constant pressure (e.g. Santibanez et al., 2016) remain gravity-limited.

Superimposed on the coherent (ballistic) wave is multiply scattered, incoherent signal: because the packing is spatially disordered, energy also transports diffusively. The standard description (Jia, 2004) fits the configurational average of the incoherent intensity with a diffusion model characterized by a transport mean free path $l^*$ (the distance over which propagation direction is randomized), a diffusion constant $D$, and an absorption time $\tau_a$ (time scale of dissipative loss). Tell et al. (2020) built a fully automated apparatus — a MAPHEUS sounding-rocket module with an active pressure-control loop, embedded accelerometers, and a voice-coil excitation capable of both weak tone bursts and strong pulses — precisely to measure acoustic transmission in a three-dimensional packing at such low, homogeneous pressures. Their ground-based commissioning data, taken at $p_0$ between roughly $0.5$ and $1.2\ \mathrm{kPa}$, already contain the puzzle that motivates this problem: the measured wavefront speed scales closer to $v_p\propto p_0^{1/4}$ than to the Hertz-affine $p_0^{1/6}$, but the ground packings carry strong hydrostatic gradients, so intrinsic low-pressure physics and gravity-induced inhomogeneity cannot be separated. Their fitted scattering parameters at $1\ \mathrm{kPa}$ ($l^*\approx 5d$, $D\approx 0.77\ \mathrm{m^2/s}$, $\tau_a\approx 0.4\ \mathrm{ms}$, with $d$ the grain diameter) likewise refer only to gravity-affected packings.

Problem Statement

Determine how acoustic propagation in a three-dimensional granular packing behaves as the confinement pressure is lowered, under homogeneous, isotropic confinement about two orders of magnitude below the terrestrial hydrostatic floor (pressures of order $10\ \mathrm{Pa}$, free of gravity-induced pressure gradients, as realized in microgravity). Specifically:

1. Small-amplitude (linear-regime) longitudinal wave speed as a function of confinement pressure $p_0$: does it vanish as $p_0$ decreases, and with what pressure exponent — is the observed ground-based scaling $v_p\propto p_0^{1/4}$ intrinsic to the packing, or does the isotropic low-pressure data follow the Hertz-affine $v_p\propto p_0^{1/6}$, a jamming-controlled scaling $v_p\propto p_0^{x}$ whose exponent $x$ is determined by the coordination number and non-affine rearrangements at the transition, or a crossover between them?
2. The crossover to strongly nonlinear, amplitude-determined propagation: as $p_0$ is lowered, does the linear wave regime shrink and disappear so that the propagation speed is set by the wave amplitude alone (the sonic-vacuum limit), and where in pressure-amplitude space does that crossover lie?
3. The pressure dependence of the multiply scattered (incoherent) wave transport: the transport mean free path $l^*$, diffusion constant $D$, and absorption time $\tau_a$, obtained from configurational averages over independently prepared packings, and whether their low-pressure behavior is universal or specific to particle material and packing fraction near jamming.

An acceptable answer must obtain these results on packings whose confinement pressure is controlled and demonstrated to be homogeneous and isotropic over the sample — whether the confinement is imposed experimentally or in a controlled model packing — thereby separating intrinsic low-pressure physics from gravity-induced inhomogeneity, and must state the pressure and amplitude ranges covered and the uncertainties of all fitted quantities.

References

1. Karsten Tell, Christoph Dreißigacker, Alberto Chiengue Tchapnda, Peidong Yu, Matthias Sperl, 'Acoustic waves in granular packings at low confinement pressure', Review of Scientific Instruments 91, 033906 (2020). DOI: 10.1063/1.5122848, https://doi.org/10.1063/1.5122848 (preprint arXiv:2002.02732, https://arxiv.org/abs/2002.02732)
2. V. F. Nesterenko, 'Propagation of nonlinear compression pulses in granular media', Journal of Applied Mechanics and Technical Physics 24, 733-743 (1983). DOI: 10.1007/BF00905892, https://doi.org/10.1007/BF00905892
3. Leopoldo R. Gómez, Ari M. Turner, Martin van Hecke, Vincenzo Vitelli, 'Shocks near Jamming', Physical Review Letters 108, 058001 (2012). DOI: 10.1103/PhysRevLett.108.058001, https://doi.org/10.1103/PhysRevLett.108.058001 (preprint arXiv:1108.5688, https://arxiv.org/abs/1108.5688)
4. Siet van den Wildenberg, Rogier van Loo, Martin van Hecke, 'Shock Waves in Weakly Compressed Granular Media', Physical Review Letters 111, 218003 (2013). DOI: 10.1103/PhysRevLett.111.218003, https://doi.org/10.1103/PhysRevLett.111.218003 (preprint arXiv:1304.6392, https://arxiv.org/abs/1304.6392)
5. X. Jia, 'Codalike Multiple Scattering of Elastic Waves in Dense Granular Media', Physical Review Letters 93, 154303 (2004). DOI: 10.1103/PhysRevLett.93.154303, https://doi.org/10.1103/PhysRevLett.93.154303
6. X. Zeng, J. H. Agui, M. Nakagawa, 'Wave Velocities in Granular Materials under Microgravity', Journal of Aerospace Engineering 20, 116-123 (2007). DOI: 10.1061/(ASCE)0893-1321(2007)20:2(116), https://doi.org/10.1061/(ASCE)0893-1321(2007)20:2(116)
7. Francisco Santibanez, Rene Zuñiga, Francisco Melo, 'Mechanical impulse propagation in a three-dimensional packing of spheres confined at constant pressure', Physical Review E 93, 012908 (2016). DOI: 10.1103/PhysRevE.93.012908, https://doi.org/10.1103/PhysRevE.93.012908
8. S. Aumaître, R. P. Behringer, A. Cazaubiel, E. Clément, J. Crassous, D. J. Durian, E. Falcon, S. Fauve, D. Fischer, A. Garcimartín, Y. Garrabos, M. Hou, X. Jia, C. Lecoutre, S. Luding, D. Maza, M. Noirhomme, E. Opsomer, F. Palencia, T. Pöschel, J. Schockmel, M. Sperl, R. Stannarius, N. Vandewalle, P. Yu, 'An instrument for studying granular media in low-gravity environment', Review of Scientific Instruments 89, 075103 (2018). DOI: 10.1063/1.5034061, https://doi.org/10.1063/1.5034061
9. Siet van den Wildenberg, Xavier Nguyen, Arnaud Tourin, Xiaoping Jia, 'Impulsive shock wave propagation in granular packings under gravity', Mechanics Research Communications 150, 104573 (2025). DOI: 10.1016/j.mechrescom.2025.104573, https://doi.org/10.1016/j.mechrescom.2025.104573 (preprint arXiv:2509.13860, https://arxiv.org/abs/2509.13860)
10. Thomas Gallot, Camila Sedofeito, Alejandro Ginares, Gonzalo Tancredi, 'Seismic wave experiments in granular media with applications to asteroids', Planetary and Space Science 265, 106153 (2025). DOI: 10.1016/j.pss.2025.106153, https://doi.org/10.1016/j.pss.2025.106153 (preprint arXiv:2210.09342, https://arxiv.org/abs/2210.09342, issued under the title 'Elastic waves generated by impact and vibration in confined granular media')