Constitutive laws and rigidity transitions for dense, dry, self-propelled frictional grains (the $\mu(I,\mathcal{A})$ program)

Dense flows of dry, frictional, hard grains are described by the inertial rheology framework, in which the effective friction coefficient $\mu$ and the volume fraction $\phi$ are state functions of the inertial number $I$. When the grains are additionally self-propelled — each carrying a propulsion force $F$ along an intrinsic orientation — this framework has no established extension. Peshkov, Claudin, Clément and Andreotti (EPL 2016) introduced the activity number $\mathcal{A}=F/(Pd)$ (propulsion force versus confining pressure $P$ times grain diameter $d$) and showed, in a single two-dimensional discrete-element simulation, that $\mu$ and $\phi$ become functions of both $I$ and $\mathcal{A}$, with two rigidity transitions: an 'active solid' appearing as soon as activity is turned on, and a second 'active rigidity transition' at $\mathcal{A}_t\simeq 0.635$ (the isostatic point) beyond which the quasi-static friction vanishes and the effective rheology becomes Newtonian while the contact dynamics stays highly intermittent. The open problem is to establish this constitutive description on firm ground — validated relations $\mu(I,\mathcal{A})$ and $\phi(I,\mathcal{A})$ beyond the initiating simulations, the existence, location and nature of the rigidity transitions, and the characterization of the intermittent active fluid and of how contact-network topology, particle orientations and intermittency couple to the macroscopic flow laws. No established extension of the $\mu(I)$ rheology to self-propelled frictional grains under imposed pressure and shear currently exists: adjacent programs treat soft or Brownian active particles under shear, single-particle dry-friction dynamics, or unsheared dense active matter.

Background

Dense granular flows of passive grains are well described by the '$\mu(I)$ rheology' (GDR MiDi 2004; Jop, Forterre and Pouliquen 2006). A grain assembly of diameter $d$ and density $\rho$ is sheared at rate $\dot\gamma$ under confining pressure $P$; the ratio of inertial to pressure forces defines the inertial number $I=d\sqrt{\rho}\,\dot\gamma/\sqrt{P}$, and experiments and simulations show that the macroscopic friction coefficient $\mu=\tau/P$ ($\tau$ the shear stress) and the volume fraction $\phi$ are, to good approximation, functions of $I$ alone — a 'liquid-like' constitutive law in which $\phi(I)$ plays the role of a dilatancy relation. This framework has no intrinsic length or force scale beyond $d$, $\rho$ and $P$, so it cannot describe grains that propel themselves.

Active matter offers the missing ingredient: particles that convert stored energy into persistent directed motion, as in flocking models (Vicsek-type alignment), active Brownian colloids, or swimming microorganisms. That literature, however, predominantly addresses ordering transitions, motility-induced clustering and hydrodynamic suspension effects, not mechanical constitutive relations of frictional grains under imposed pressure and shear. The two threads meet in 'active dry granular matter': assemblies of macroscopic self-propelled particles (realizable as vibrated polar disks or robots) whose interactions are hard, frictional contacts.

Peshkov, Claudin, Clément and Andreotti ('Active dry granular flows: rheology and rigidity transitions', EPL 116, 14001 (2016)) initiated this program with two-dimensional discrete-element simulations (DEM) of frictional disks, each subject to a propulsion force of fixed magnitude $F$ along a slowly relaxing orientation vector, under imposed confining pressure $P$ and shear rate $\dot\gamma$. Comparing propulsion to confinement defines the activity number $\mathcal{A}=F/(Pd)$. Their central results: (i) the rheology becomes $\mu(I,\mathcal{A})$ and $\phi(I,\mathcal{A})$, with the quasi-static friction $\mu_c$ and critical volume fraction $\phi_c$ decreasing with $\mathcal{A}$; (ii) in the Maxwell rigidity sense — a packing is solid when its coordination $Z$ exceeds the isostatic count, measured in 2D frictional packings by the distance to isostaticity $\delta Z=3+\zeta-Z$ with $\zeta$ the fraction of sliding contacts — two liquid/solid transitions occur: any nonzero activity produces an 'active solid' with above-isostatic contacts, and at $\mathcal{A}_t\simeq 0.635$, where isostaticity is regained, the quasi-static friction vanishes and the effective rheology becomes Newtonian; (iii) for $\mathcal{A}>\mathcal{A}_t$ the 'active fluid' is highly intermittent, its contact network alternating between solid and liquid rigidity states in space and time.

These results rest on a single 2D, one-parameter-family DEM study, albeit of polydisperse disks whose diameters are drawn uniformly between $0.8d$ and $1.2d$ to prevent crystallization under shear. The authors themselves flag what remains: the sensitivity of the critical values $\mathcal{A}_t$ and $\phi_t$ (the critical volume fraction at the active rigidity transition, $\phi_t=\phi_c(\mathcal{A}_t)\simeq 0.774$) to system size and particle properties 'is to be systematically investigated', and the new type of transition and the nature of the active states 'would be interesting to understand in depth'. Subsequent literature has built closely related constitutive descriptions only in other limits — soft, frictionless, Brownian active particles under shear (Wiese, Kroy and Levis 2023; Shee, Bandyopadhyay and Yue 2025) and single self-propelled particles with dry Coulomb friction (Antonov et al. 2024) — so the dry, dense, frictional, many-body regime of the original question is still without an established constitutive law, an experimentally confirmed transition picture, or a characterization of the intermittent active fluid.

Problem Statement

Establish the macroscopic constitutive description of a dense, dry assembly of frictional, self-propelled hard grains under simple shear, in the regime where both inertial and activity effects matter. Concretely: (1) determine and validate, beyond the initiating 2D discrete-element study, the constitutive relations $\mu(I,\mathcal{A})$ and $\phi(I,\mathcal{A})$ — including their quasi-static limits $\mu_c(\mathcal{A})$, $\phi_c(\mathcal{A})$ and the normal-stress/dilatancy behavior — where $I=d\sqrt{\rho}\,\dot\gamma/\sqrt{P}$ is the inertial number and $\mathcal{A}=F/(Pd)$ the activity number comparing the propulsion force $F$ to the confining pressure $P$ (with $d$ the grain diameter, $\rho$ the grain material density, and $\dot\gamma$ the imposed shear rate), and establish over what parameter range (dimension, degree of polydispersity, inter-particle friction, propulsion rule and persistence, system size) these state functions exist at all; (2) determine the existence, location and nature of the rigidity (liquid/solid) transitions — the onset of the 'active solid' at infinitesimal activity and the 'active rigidity transition' near the isostatic condition at $\mathcal{A}_t$, including whether $\mu_c$ genuinely vanishes there and whether the transition is a true phase transition or a crossover; and (3) characterize the intermittent 'active fluid' regime above $\mathcal{A}_t$ and quantify how microscopic organization — contact-network topology relative to isostaticity, particle-orientation statistics, and spatiotemporal intermittency — couples to and determines the macroscopic flow laws. Answers based on experiment, theory, or independently validated simulation are all admissible; a demonstration that no single-valued constitutive state function $\mu(I,\mathcal{A})$, $\phi(I,\mathcal{A})$ exists (e.g. strong history dependence) would equally answer part (1) only, provided it is established with the same generality, leaving parts (2) and (3) to be addressed independently.

References

1. Anton Peshkov, Philippe Claudin, Éric Clément, Bruno Andreotti, 'Active dry granular flows: rheology and rigidity transitions', EPL 116, 14001 (2016). DOI: 10.1209/0295-5075/116/14001, https://doi.org/10.1209/0295-5075/116/14001 (also arXiv:1607.02648, https://arxiv.org/abs/1607.02648)
2. GDR MiDi (collective author), 'On dense granular flows', European Physical Journal E 14 (2004). DOI: 10.1140/epje/i2003-10153-0, https://doi.org/10.1140/epje/i2003-10153-0
3. Pierre Jop, Yoël Forterre, Olivier Pouliquen, 'A constitutive law for dense granular flows', Nature 441 (2006). DOI: 10.1038/nature04801, https://doi.org/10.1038/nature04801
4. Adeline Favier de Coulomb, Mehdi Bouzid, Philippe Claudin, Éric Clément, Bruno Andreotti, 'Rheology of granular flows across the transition from soft to rigid particles', Physical Review Fluids 2, 102301 (2017). DOI: 10.1103/PhysRevFluids.2.102301, https://doi.org/10.1103/PhysRevFluids.2.102301
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8. Roland Wiese, Klaus Kroy, Demian Levis, 'Fluid-Glass-Jamming Rheology of Soft Active Brownian Particles', Physical Review Letters 131, 178302 (2023). DOI: 10.1103/PhysRevLett.131.178302, https://doi.org/10.1103/PhysRevLett.131.178302
9. Amir Shee, Ritwik Bandyopadhyay, Haicen Yue, 'Tuning Steady Shear Rheology through Active Dopants', Physical Review Letters 135, 168302 (2025). DOI: 10.1103/qvr9-rvw4, https://doi.org/10.1103/qvr9-rvw4 (also arXiv:2506.05262, https://arxiv.org/abs/2506.05262)
10. Alexander P. Antonov, Lorenzo Caprini, Anton Ldov, Christian Scholz, Hartmut Löwen, 'Inertial Active Matter with Coulomb Friction', Physical Review Letters 133, 198301 (2024). DOI: 10.1103/PhysRevLett.133.198301, https://doi.org/10.1103/PhysRevLett.133.198301 (also arXiv:2404.06615, https://arxiv.org/abs/2404.06615)
11. Cunyuan Jiang, 'Absence of solid phase in dense amorphous active granular matter', arXiv:2604.14610 (2026), https://arxiv.org/abs/2604.14610