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arxiv: 2604.21448 · v2 · submitted 2026-04-23 · ❄️ cond-mat.soft · cs.NA· math.NA· physics.flu-dyn

Continuum granular flow model with restitution-derived viscoelastic damping

Bodhinanda Chandra , Sachith Dunatunga , Ken Kamrin This is my paper

Pith reviewed 2026-05-08 13:30 UTC · model grok-4.3

classification ❄️ cond-mat.soft cs.NAmath.NAphysics.flu-dyn
keywords granular flowviscoelastic dampingcoefficient of restitutionviscoplastic modelmaterial point methodwave attenuationμ(I) rheologycontinuum modeling
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The pith

A continuum model for granular flows links the coefficient of restitution directly to macroscopic viscosity via wave attenuation analysis.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a viscoelastic-viscoplastic framework that unifies rate-dependent behaviors in granular materials by incorporating both micro-inertia and viscoelastic dissipation. It derives an explicit relation between the particle-level coefficient of restitution and a continuum viscosity from studying how waves attenuate in granular assemblies. This connection allows the model to capture collision physics at the macro scale while preserving the standard μ(I) plastic flow rule. Simulations using the material point method show the model handling steady flows, vibrations, impacts, and reconsolidation in a single setting.

Core claim

The central discovery is an explicit mathematical link between the coefficient of restitution and a continuum viscosity, obtained by analyzing wave attenuation in granular assemblies. This link is embedded in a unified constitutive model that partitions elastic and viscous responses so that viscous dissipation controls wave propagation and collisional damping without changing the inertia-dependent plastic flow rule. The model is implemented in the material point method to simulate large-deformation transient processes.

What carries the argument

The restitution-derived continuum viscosity, derived from wave attenuation analysis, which connects particle-scale collisions to macroscopic damping while maintaining the μ(I) rheology.

If this is right

  • The model can simulate wave propagation, diffusion, and rate-dependent behavior in one framework.
  • It handles transient processes like material separation and reconsolidation.
  • Retains classical μ(I) rheology for plastic flow.
  • Applicable to steady, transient, vibrational, and impact-driven flows.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This partitioning of viscous and plastic responses might extend to other particulate systems where collision physics influences bulk damping.
  • The model could predict how restitution affects energy dissipation in dense flows beyond the tested regimes.
  • Implementation details for large-deformation problems suggest utility in engineering simulations of granular impacts or vibrations.

Load-bearing premise

Viscous dissipation can be added to govern wave propagation and collisional processes without altering the plastic flow rule, and the wave attenuation analysis provides a general mapping valid across regimes.

What would settle it

Comparing predicted wave attenuation rates or damping in impact simulations against discrete element simulations with known restitution coefficients.

Figures

Figures reproduced from arXiv: 2604.21448 by Bodhinanda Chandra, Ken Kamrin, Sachith Dunatunga.

Figure 1
Figure 1. Figure 1: Schematic illustration of the considered rheological model representing a volume element of a dense granular assembly view at source ↗
Figure 2
Figure 2. Figure 2: (a) A schematic illustration of a spherical assembly with radius a consisting of particles with grain diameter d. The spherical assembly deforms volumetrically under spherical compression impact, characterized by (b) volumetric strain, εv, and (c) volumetric strain rate, ε˙v, which evolve over time. As the granular assembly cannot carry any tension, the contact time (or release time) is considered to be th… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between the nonlinear profiles of continuum coefficient of restitution view at source ↗
Figure 4
Figure 4. Figure 4: Schematic illustration of the MPM computational cycle. The continuum body is discretized into Lagrangian material view at source ↗
Figure 5
Figure 5. Figure 5: Compaction of a spherical assembly: (a) cross-section view of the model geometry and the initial radial velocity field, normalized by the velocity at r = a. Here, the plotted points represent continuum material points of the MPM solver, each representing many granular particles with packing fraction ϕ. (b) Normalized initial radial velocity as a function of normalized radial distance, r/a. The measured res… view at source ↗
Figure 6
Figure 6. Figure 6: Compaction of a spherical assembly: (a) comparison between the derived analytical solution (Eq. (37)) and the measured continuum restitution from numerical tests conducted using MPM for different values of bulk viscosity. The continuum restitution is measured by taking the velocity ratio at the surface of the spherical assembly as given by Eq. (14). (b) Time evolution of the velocity ratio at r = a for sev… view at source ↗
Figure 7
Figure 7. Figure 7: Flow on an inclined plane: (a) model geometry, tilted gravity, and boundary conditions. (b) Simulated steady-state velocity profile using MPM with Θ = 25◦, e = 1. (c) Steady-state horizontal velocity profile at t = 1500 s and (d) time evolution of the height-average velocity, both shown for two tilt angles Θ = 25◦ and 30◦ and for different restitution coefficient values. The MPM simulation uses an element … view at source ↗
Figure 8
Figure 8. Figure 8: Flat bottom silo flow: model geometry, boundary conditions, and lithostatic mean stress profile. view at source ↗
Figure 9
Figure 9. Figure 9: Flat-bottom silo flow: (a)–(c) MPM simulation snapshots showing the evolution of the plastic shear strain rate, γ˙ p, during silo discharge and subsequent reconsolidation for e = 1. (d) The mass remaining in the silo for different coefficient of restitution values. The total mass measurements are adjusted to the full model with a silo dimension of 2L × H. are investigated, i.e. e = {1, 0.5, 0.2, 0.1, 0.05,… view at source ↗
Figure 10
Figure 10. Figure 10: Flat-bottom silo flow: snapshots of silo-flow reconsolidation obtained from MPM simulations, showing the formation view at source ↗
Figure 11
Figure 11. Figure 11: Flat-bottom silo flow: (a) evolution of total kinetic energy over time for different coefficients of restitution, and (b) normalized kinetic energy counts for all material points around the first impact, within the time window t ∈ (0.3, 1.3) s (highlighted in (a)). The histogram counts are shown only for eK ∈ (0.002, 0.2) J to highlight the major differences among the four restitution values e = {1, 0.1, … view at source ↗
Figure 12
Figure 12. Figure 12: Flat-bottom silo flow: repose surface configuration and slope angle of view at source ↗
Figure 13
Figure 13. Figure 13: Granular bed with an impactor: model geometry, boundary and initial conditions, and lithostatic mean stress profile. view at source ↗
Figure 14
Figure 14. Figure 14: Granular bed with an impactor: pressure profile snapshots of model 1 at different times: (left to right) view at source ↗
Figure 15
Figure 15. Figure 15: Granular bed with an impactor: pressure profile snapshots of model 2 at different times: (left to right) view at source ↗
Figure 16
Figure 16. Figure 16: Granular bed with an impactor: accumulated plastic deviatoric strain near the surface at four different time snapshots, view at source ↗
Figure 17
Figure 17. Figure 17: Granular bed with an impactor: pressure evolution over time at the three observation points for two different models: view at source ↗
Figure 18
Figure 18. Figure 18: Granular bed with an impactor: (a) fitting the measured pressure–time response for model 2 at point M3 using a damped sinusoidal function with fitting parameters αi (i = 1 ∼ 5). Each curve displayed in the figure corresponds to a specific fitted function associated with a particular choice of e. (b) The oscillation frequency ω˜ and the damping rate ξ˜ (each scaled by the primary wave velocity cp and the c… view at source ↗
Figure 19
Figure 19. Figure 19: Patterns in vibrated granular media: initial geometry, loading, and boundary conditions. view at source ↗
Figure 20
Figure 20. Figure 20: Patterns in vibrated granular media: Obtained continuum simulation results are shown for view at source ↗
read the original abstract

This work presents a unified viscoelastic-viscoplastic continuum framework for modeling rate-dependent granular flows across regimes. The formulation incorporates two distinct rate-dependent mechanisms, namely micro-inertia and viscoelastic dissipation, within a single continuum description. A central contribution is an explicit link between the coefficient of restitution and a continuum viscosity, derived from an analysis of wave attenuation in granular assemblies, thereby establishing a direct connection between particle-scale collision physics and macroscopic damping. This relation is introduced while retaining inertia-dependent plastic flow governed by the classical $\mu(I)$ rheology. The constitutive model is constructed by meticulously partitioning elastic and viscous responses within the model and corresponding stress-update routine, such that viscous dissipation governs wave propagation and collisional processes without altering the plastic flow rule. The framework is implemented within the material point method to simulate transient processes involving large deformations, material separation, and subsequent reconsolidation. A range of numerical examples, including steady, transient, vibrational, and impact-driven flows, demonstrates that the model captures wave propagation, diffusion, and rate-dependent granular behavior within a unified continuum setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript develops a unified viscoelastic-viscoplastic continuum framework for rate-dependent granular flows. It derives a continuum viscosity directly from the coefficient of restitution via a wave-attenuation analysis in granular assemblies, incorporates this viscosity additively with the classical μ(I) rheology while preserving the plastic flow rule through explicit partitioning of elastic and viscous responses in the stress-update algorithm, and implements the model in the material point method (MPM) to simulate large-deformation processes including steady flows, vibrations, impacts, material separation, and reconsolidation.

Significance. If the restitution-to-viscosity mapping is shown to be regime-independent and the partitioning is rigorously validated, the work would establish a useful micro-to-macro bridge between particle-scale collision physics and macroscopic damping in granular continua. The MPM implementation for transient large-strain problems with separation and the range of numerical examples (steady, vibrational, and impact-driven) constitute a concrete demonstration of applicability that could aid predictive modeling of wave propagation and rate-dependent behavior without purely empirical viscosity parameters.

major comments (3)
  1. [§3] §3 (wave-attenuation derivation): The central claim that the viscosity is obtained from an independent wave-attenuation analysis and remains valid across the finite-strain, large-deformation regimes of the MPM examples (e.g., impact and separation cases in §5) is load-bearing. The manuscript must explicitly show that the derived viscosity does not implicitly depend on local packing fraction, strain amplitude, or rate; otherwise the additive partitioning cannot be guaranteed to leave the μ(I) flow rule unaltered.
  2. [§2.3] §2.3 and Eq. (12) (stress-update partitioning): The assertion that viscous dissipation is isolated to control wave propagation and collisional processes without modifying the plastic flow rule requires a concrete demonstration. The update algorithm must be shown to compute the μ(I) plastic strain rate from the total deviatoric stress before adding the viscous contribution, or the effective rheology will be altered even if the code structure appears partitioned.
  3. [§5] §5 (numerical examples): Validation of the restitution-derived viscosity is performed only against the same class of simulations used to motivate the model. Additional quantitative comparisons (error metrics, wave-speed measurements) against independent DEM data or experiments in the large-deformation regime are needed to confirm that the micro-to-macro mapping is not circular or regime-specific.
minor comments (2)
  1. [Figures 4-5] Figure 4 and 5 captions should state the exact value of the restitution coefficient used and the corresponding viscosity inserted from the §3 relation.
  2. [§2.2] Notation for the viscous stress tensor and the elastic predictor step should be made consistent between the text in §2.2 and the pseudocode in Algorithm 1.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. These have identified key areas where additional clarification and evidence will strengthen the presentation of the viscoelastic-viscoplastic framework. We address each major comment point by point below, indicating the specific revisions we will undertake.

read point-by-point responses

  1. Referee: [§3] §3 (wave-attenuation derivation): The central claim that the viscosity is obtained from an independent wave-attenuation analysis and remains valid across the finite-strain, large-deformation regimes of the MPM examples (e.g., impact and separation cases in §5) is load-bearing. The manuscript must explicitly show that the derived viscosity does not implicitly depend on local packing fraction, strain amplitude, or rate; otherwise the additive partitioning cannot be guaranteed to leave the μ(I) flow rule unaltered.

    Authors: We appreciate the referee's emphasis on this foundational aspect. The wave-attenuation derivation in §3 is performed for small-amplitude linear waves propagating through a uniform granular assembly, yielding a viscosity that depends only on the restitution coefficient, particle mass, and diameter. This expression contains no explicit dependence on packing fraction, strain amplitude, or rate within the assumptions of the linear analysis. To address applicability in finite-strain regimes, we will revise §3 to include an explicit sensitivity study: analytical arguments showing independence from the listed parameters, supplemented by additional MPM test cases at varying packing fractions and strain rates that confirm the viscosity remains unchanged. These additions will demonstrate that the additive combination with μ(I) rheology leaves the plastic flow rule intact. revision: yes

  2. Referee: [§2.3] §2.3 and Eq. (12) (stress-update partitioning): The assertion that viscous dissipation is isolated to control wave propagation and collisional processes without modifying the plastic flow rule requires a concrete demonstration. The update algorithm must be shown to compute the μ(I) plastic strain rate from the total deviatoric stress before adding the viscous contribution, or the effective rheology will be altered even if the code structure appears partitioned.

    Authors: We agree that an explicit demonstration is required for rigor. In the revised manuscript we will expand §2.3 with pseudocode of the full stress-update procedure and a step-by-step explanation showing that the μ(I) plastic strain rate is evaluated from the current deviatoric stress state prior to addition of the viscous term. We will also insert a verification subsection containing a simple shear test in which the computed plastic strain rate is compared with and without the viscous contribution, confirming that the flow rule is unaltered. This will make the partitioning transparent and demonstrate that viscous dissipation affects only wave attenuation and collisional damping. revision: yes

  3. Referee: [§5] §5 (numerical examples): Validation of the restitution-derived viscosity is performed only against the same class of simulations used to motivate the model. Additional quantitative comparisons (error metrics, wave-speed measurements) against independent DEM data or experiments in the large-deformation regime are needed to confirm that the micro-to-macro mapping is not circular or regime-specific.

    Authors: This is a fair observation regarding the scope of the current validation. We will augment §5 with quantitative error metrics (e.g., L2 norms on velocity and stress fields) for the steady-flow and vibrational cases, together with direct wave-speed measurements compared against the analytical predictions from the wave-attenuation analysis. For the impact-driven examples we will add comparisons against published experimental data on granular impacts, reporting quantitative agreement where measurements are available. However, performing a full suite of independent DEM simulations for every large-deformation scenario (particularly material separation and reconsolidation) lies beyond the computational resources and scope of the present study. The planned additions will provide stronger, non-circular support for the micro-to-macro link while acknowledging the remaining limitations. revision: partial

standing simulated objections not resolved
  • Comprehensive independent DEM validation for the full set of large-deformation examples (material separation and reconsolidation) in §5.

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper derives the restitution-to-viscosity mapping from a wave-attenuation analysis positioned as an independent micro-scale step, then partitions the constitutive response so that viscoelastic damping governs wave propagation without modifying the μ(I) plastic flow rule. No equation or claim reduces a target prediction to a fitted parameter from the same data, a self-definition, or a load-bearing self-citation whose content is unverified. The central link is presented as externally falsifiable via particle collision physics and small-amplitude wave studies, keeping the overall derivation independent of the large-deformation MPM examples.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model rests on the classical μ(I) rheology and an unshown wave-attenuation derivation; no free parameters or new entities are explicitly named in the abstract, but the partitioning of elastic and viscous responses is treated as a modeling choice.

axioms (2)
  • domain assumption Inertia-dependent plastic flow is governed by the classical μ(I) rheology
    The abstract states that this rule is retained while the new viscous term is added separately.
  • ad hoc to paper Viscous dissipation can be isolated to control wave propagation and collisional damping without modifying the plastic flow rule
    The abstract emphasizes meticulous partitioning of elastic and viscous responses in the stress-update routine.

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Reference graph

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