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Memory of jamming - multiscale models for soft and granular matter

Nishant Kumar, Stefan Luding

arXiv:1407.6167v5cond-mat.softphysics.data-an

TL;DR

Jamming density can depend on preparation and deformation history, yet quantitative characterization of its changes remains a challenge. The paper develops a history-dependent constitutive model and a meso-scale explanation, finding distinct slow isotropic and fast shear responses of packing efficiency.

  • Problem

    Jamming density varies with preparation history, but quantitatively characterizing how it changes with deformation history remains an open challenge.

    Source: §1 · p. 2

  • Method

    The paper combines a history-dependent macroscopic constitutive model with a stochastic meso-scale account of packing changes under isotropic and shear deformation.

    Source: §7 · p. 17

  • Results

    Packing efficiency responds very slowly to isotropic perturbations and exponentially fast to finite shear deformation.

    Source: §7 · p. 17

  • Takeaways & Limitations

    The study proposes extracting history-dependent jamming density indirectly from measurable quantities and compares alternative state variables.

    Source: §6 · p. 14

    The model omits granular-temperature fluctuations and fluidity fields that introduce an additional relaxation time-scale.

    Source: §5.1.2 · p. 13

Abstract

from arXiv · show

Soft, disordered, micro-structured materials are ubiquitous in nature and industry, and are different from ordinary fluids or solids, with unusual, interesting static and flow properties. The transition from fluid to solid -at the so-called jamming density- features a multitude of complex mechanisms, but there is no unified theoretical framework that explains them all. In this study, a simple yet quantitative and predictive model is presented, which allows for a variable, changing jamming density, encompassing the memory of the deformation history and explaining a multitude of phenomena at and around jamming. The jamming density, now introduced as a new state-variable, changes due to the deformation history and relates the system's macroscopic response to its microstructure. The packing efficiency can increase logarithmically slow under gentle repeated (isotropic) compression, leading to an increase of the jamming density. In contrast, shear deformations cause anisotropy, changing the packing efficiency exponentially fast with either dilatancy or compactancy. The memory of the system near jamming can be explained by a microstatistical model that involves a multiscale, fractal energy landscape and links the microscopic particle picture to the macroscopic continuum description, providing a unified explanation for the qualitatively different flow-behavior for different deformation modes. To complement our work, a recipe to extract the history-dependent jamming density from experimentally accessible data is proposed, and alternative state-variables are compared. The proposed simple macroscopic constitutive model is calibrated with the memory of microstructure. Such approach can help understanding predicting and mitigating failure of structures or geophysical hazards, and will bring forward industrial process design/optimization, and help solving scientific challenges in fundamental research.

1 Introduction

Jamming density varies with preparation and deformation history, but its quantitative characterization remains an open challenge. This study combines isotropic and shear deformation to model that history dependence and connect changes in packing structure to macroscopic behavior.

1 Introduction

  • Jamming density can vary with preparation protocol, and quantitatively characterizing its dependence on deformation history remains an open challenge.Prior work also reported that jamming density can scale macroscopic properties and that shear can jam unjammed frictional systems.Source: §1 · p. 2

1.1 Application examples

  • Predicting pre-failure deformation and flow onset matters for geotechnical applications, including ground movement, infrastructure settlement, and seismic response.Material parameters influence computed deformations before failure, while elastic properties inform assessments of liquefaction and anisotropic strata.Source: §1.1 · p. 2

1.2 Approach of this study

  • The study probes isotropic compression and volume-conserving pure shear, treating them as modes that can be combined to represent other deformations.The compression protocol is assumed to be truly isotropic in strain; treating tapping as isotropic is an acknowledged oversimplification.Source: §1.2 · p. 2
  • A multiscale, fractal energy landscape links deformation-driven changes in configuration populations to the history-dependent jamming density.The proposed continuum model adds one isotropic state variable, while structural anisotropy alone is stated to be insufficient.Source: §1.2 · p. 3

1.3 Overview

  • The paper proceeds from particle- and contact-scale observations to a qualitative meso-scale model, a predictive macroscale model, and an experimental validation procedure.The macroscale model is compared with microscale simulations before the validation procedure is discussed.Source: §1.3 · p. 3

2 Simulation method

The study uses discrete-element simulations of frictionless spheres under isotropic compression and pure shear. It derives macroscopic stress and fabric measures to characterize the packing state and its structure.

2 Simulation method

  • Discrete Element Method simulations model 9,261 soft, frictionless spheres in a three-dimensional periodic box with linear visco-elastic contacts.The assemblies have uniform polydispersity and include contact and background damping.Source: §2 · p. 3

2.1 Preparation procedure and main experiments

  • Preparations cycle relaxed assemblies through isotropic compression and decompression, then apply volume-conserving plane-strain shear to selected configurations.The isotropic cycle is repeated across 100 cycles; shear is applied for four cycles at a quasi-static strain rate.Source: §2.1 · pp. 3–4
  • The shear protocol is chosen as a fundamental deviatoric mode, with extrapolation to more realistic materials left uncertain.The authors note that wall-motion methods that remain homogeneous for the modeled frictionless spheres need not do so with friction or cohesion.Source: §2.1 · p. 3, §2.2 · p. 4

2.2 Macroscopic (tensorial) quantities

  • Macroscopic quantities include contact-based static stress and fabric, which characterize stress and the geometry of the particle-contact network.The stress calculation neglects kinetic fluctuation energy; pressure is the isotropic stress component, and the fabric tensor describes packing structure.Source: §2.2 · p. 4
  • Deviatoric tensor measures retain a sign for shear direction, and pressure and shear stress are nondimensionalized using particle size and stiffness.The stated scale is 2⟨r⟩/k.Source: §2.2 · p. 4

3 Micromechanical results

The results show that jamming density depends on deformation history: isotropic over-compression raises it, while shear can lower it toward a critical limit. A history-dependent phase diagram and constitutive model connect these changes to force-network structure and predict shear-jamming behavior.

3.1.1 Identification of the jamming density

  • Unloading pressure fits identify φJ by setting logarithmic volumetric strain −εv = log(φ/φJ), where pressure approaches zero at φJ.The fit uses p0, γp, and φJ; scaled pressure is proportional to dimensionless deformation and converges to zero as φ approaches φJ.Source: §3.1.1 · p. 4

3.1.2 Isotropic cyclic over-compression

  • Repeated isotropic over-compression raises φJ, with successive-cycle growth slowing and following a stretched-exponential relation.The lower limit is φc = 0.6567; the extrapolated steady-state limit depends on maximum over-compression, and a relation predicts that limit from the first-cycle jamming density.Source: §3.1.2 · p. 5

3.2.1 Shear jamming below φJ(H)

  • Under shear, strong-force networks progress from unjammed through fragile to shear-jammed states as they percolate in more directions.For the example at φ = 0.6584, all three directions percolate when fNR > 0.82 ± 0.01; shear jamming is also marked by C* crossing its isostatic limit of 6.Source: §3.2.1 · pp. 6–7, Figure 3 · p. 7
  • At shear reversal, strong-force networks break and the system may unjam temporarily, before a network rapidly reforms in the perpendicular direction.Systems with volume fraction higher and farther from φc can resist reversal.Source: §3.2.1 · p. 8

3.2.2 Relaxation effects on shear jammed states

  • Relaxation leaves isotropic fabric nearly unchanged while pressure decreases; extrapolated relaxed pressure gives φc = 0.6567 ± 0.0005.The relaxed shear-stress extrapolation gives a similar critical density, consistent with estimates from coordination number or energy.Source: §3.2.2 · p. 8

3.3 Jamming phase diagram with history H

  • The phase diagram predicts shear-jamming strain from the initial history-dependent jamming density, with different histories converging toward φc ≈ 0.6567.Scaled curves collapse onto a master curve with α = 1.37 ± 0.01 and shear-strain scale ε0_d = 0.102 ± 0.001; the model predicts shear jamming vanishes without prior tapping, tempering, or over-compression.Source: §3.3 · pp. 8–9

4 Meso-scale stochastic slow dynamics model

A multiscale, fractal energy-landscape picture links deformation history to changes in jamming density and macroscopic response. Isotropic perturbations promote slow relaxation toward denser configurations, whereas shear drives faster rearrangements and can lower the jamming density.

4 Meso-scale stochastic slow dynamics model

  • Over-compression and tapping drive slow reorganization toward more efficient packing, while repeated rearrangements become less likely as easy configurations are exhausted.The model represents the packing as random walkers in a hierarchical landscape; larger perturbations can overcome larger barriers.Source: §4 · pp. 9, 11
  • Shear pushes parts of the ensemble uphill in the landscape, producing a net drift toward lower jamming density when the material was previously relaxed or forced toward local minima.Under planar simple shear, one eigen-direction is extensive and another compressive.Source: §4 · p. 11
  • Ongoing over-compression slowly raises coordination number, pressure, and jamming density, whereas decompression lowers pressure until the system unjams.At constant volume, ongoing tapping or finite temperature slowly lowers coordination number and pressure; at fixed confining pressure, volume decreases.Source: §4 · p. 11
  • Ongoing shear lowers jamming density and raises coordination number, pressure, and shear stress until a steady state is reached, on a faster timescale than relaxation.At large strain, rearrangement randomizes or rejuvenates the system toward its quenched state near φc.Source: §4 · p. 11, Figure 7 · p. 11
  • Concurrent temperature-driven relaxation and continuous shear can balance opposing jamming-density trends, leaving the jamming density constant in a steady state.The schematics depict stress-density trajectories under isotropic compression and decompression, and under shear, reversal, and ongoing shear.Source: §4 · pp. 11–12, Figure 8 · p. 12, Figure 9 · p. 12

5 Macroscopic constitutive model

The constitutive model makes the jamming density φJ(H) history-dependent, using it to connect deformation history with isotropic and shear responses. It applies separate evolution rules to isotropic compression and shear, including a reversal rule, and predicts cyclic behavior.

5.1.1 During cyclic isotropic deformation

  • For cyclic isotropic deformation, history-dependent φJ(H) enters the pressure, corrected coordination, and non-rattler relations, with φJ updated during compression but held constant during unloading.The model uses the volumetric strain −εv = log(φ/φJ(H)); the pressure correction γp is described as small.Source: §5 · p. 12, §5.1.1 · pp. 12–13

5 Macroscopic constitutive model

  • The model’s central change from the cited isotropic-unloading relations is to make φJ a variable that depends on deformation history.The constitutive equations use Eq. (4) for isotropic and Eq. (8) for shear deformation.Source: §5 · p. 12

5.1.2 Cyclic (pure) shear deformation

  • Cyclic pure shear uses simplified evolution relations for shear stress ratio τ/p and deviatoric fabric Fd, with their saturation values and growth rates depending on volume fraction.The model form used here omits kinetic-energy fluctuations, or granular temperature Tg, and fluidity fields that would introduce another relaxation timescale.Source: §5.1.2 · p. 13, Table 2 · p. 13

5.1.3 Behavior of the jamming density at strain reversal

  • Below φcr, shear reversal unjams states and requires additional strain to re-jam them; the scaled reversal strain collapses onto a master curve, with α = 1.37 ± 0.01 and ε0,R d = 0.102.The proposed rule sets φJ to φcr on reversal for states at or below φcr.Source: §5.1.3 · pp. 13–14, Figure 10 · p. 14

5.2 Prediction: minimal model

  • The model predicts cyclic shear stress ratio τ/p and deviatoric fabric Fd behavior, quantitatively after first loading and qualitatively close to cyclic granular-assembly behavior.The isotropic quantities are also well predicted while only φJ varies with shear strain; the authors note some arbitrariness in its sudden changes at reversal.Source: §5.2 · p. 14

6 Towards experimental validation

The paper proposes indirect ways to infer the history-dependent jamming density from measurable quantities, since it is difficult to measure directly. For isotropic over-compression and volume-conserving shear, the inferred values show agreement with the model under the stated conditions.

6 Towards experimental validation

  • Indirect jamming-density estimates from coordination number and pressure are consistent, providing a way to obtain φJ(H) from experimentally accessible quantities.The authors describe direct access as limited to the unloading limit p → 0, where the measurement itself may change φJ(H).Source: §6 · pp. 14–15
  • During isotropic over-compression, estimates from C∗ and pressure indicate that φJ rises rapidly and then saturates, including an increase during unloading.The inferred values are reproducible and follow the same first-cycle master curve across maximum compression levels; subsequent deformation depends on the previous maximum density.Source: §6 · pp. 15–16, Figure 12 · p. 16

For shear deformation

  • For volume-conserving shear at φ = 0.66 with initial φJ = 0.6652, indirect estimates agree with Eq. (8) after shear jamming, when φJ(H) < φ.The pressure-based estimate fits best, lying between the other estimates and varying more smoothly.Source: For shear deformation · p. 16, Figure 13 · p. 16, §6 · p. 16

7 Summary, Discussion and Outlook

The model treats history-dependent jamming density as a state variable linking deformation history to macroscopic behavior, with isotropic and shear deformation changing it differently. It connects this response to microstructure and discusses the model’s scope, measurement, and open validation needs.

7 Summary, Discussion and Outlook

  • The macroscopic model quantitatively matches the considered simulations using isotropic and anisotropic microstructure as state variables, with a multiscale stochastic model explaining their different deformation responses.The jamming density responds slowly to isotropic deformation and exponentially fast to finite shear.Source: §7 · p. 17

7.1 Some questions answered

  • History-dependent jamming density φJ(H) controls jamming, unjamming, and shear jamming, which can all occur in 3D frictionless systems through microstructural reorganization.The model also identifies a unique lower critical density φc in the stated zero-pressure limits.Source: §7.1 · p. 17

7.2 Lower limit of jamming

  • The lower limit φc is difficult to access because shear also causes competing relaxation, but it can be estimated by extrapolating relaxed steady-state pressure to zero.Special deformation modes or careful preparation may produce jammed states below φc; the authors could not clearly identify this mechanism in their frictionless simulations.Source: §7.2 · p. 17

7.3 Shear jamming as consequence of a varying φJ(H)

  • Shear jamming follows when shear lowers φJ(H) below the system density, while the direction of volume change depends on shear relative to prior preparation and relaxation.Slower shear can produce dilatancy as φJ falls; relatively fast, violent shear can instead increase φJ and produce compactancy.Source: §7.3 · p. 17

7.4 Rheology

  • The model’s isotropic and deviatoric responses appear mostly decoupled in the data, but their superposition cannot be assumed for more realistic granular and powder systems.The study’s model is based on frictionless particle simulations.Source: §7.4 · pp. 17–18
  • The model links compactancy and dilatancy at constant confining stress, and pressure changes at constant volume, to increases or decreases in jamming density.It also describes shear-thinning or shear-thickening through jamming-density changes; including granular temperature is proposed to capture additional rate-dependent effects.Source: §7.4 · p. 18

7.5 Towards experimental validation

  • History-dependent φJ(H) can be inferred from measurable quantities such as pressure, coordination number, or non-rattler fraction, and the indirect measurements show it rising during isotropic deformation and falling during shear.The state variable need not be jamming density itself; another isotropic quantity can serve if it characterizes packing efficiency.Source: §7.5 · p. 18
  • The authors found no considerable crystallization or phase separation, but note that their observations do not rule out crystallization that is too weak to detect clearly.They attribute the limited ordering to particle-size polydispersity and leave the microstructural origin of increased packing efficiency for future research.Source: §7.5 · p. 18
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