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Edited by: Daniel Bonamy, Commissariat à l'Énergie Atomique et aux Énergies Alternatives, France

Reviewed by: Eric Josef Ribeiro Parteli, University of Erlangen-Nuremberg, Germany; Ferenc Kun, University of Debrecen, Hungary

*Correspondence: Benjy Marks, Condensed Matter Physics, Department of Physics, University of Oslo, Sem Sælands Vei 24, Fysikkbygningen, Oslo 0371, Norway

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In both nature and engineering, loosely packed granular materials are often compacted inside confined geometries. Here, we explore such behavior in a quasi-two dimensional geometry, where parallel rigid walls provide the confinement. We use the discrete element method to investigate the stress distribution developed within the granular packing as a result of compaction due to the displacement of a rigid piston. We observe that the stress within the packing increases exponentially with the length of accumulated grains, and show an extension to current analytic models which fits the measured stress. The micromechanical behavior is studied for a range of system parameters, and the limitations of existing analytic models are described. In particular, we show the smallest sized systems which can be treated using existing models. Additionally, the effects of increasing piston rate, and variations of the initial packing fraction, are described.

When granular materials are placed in confined geometries, we often observe a significant portion of the stress being redirected toward the confining boundaries. This phenomenon has been systematically studied for many systems [_{r} is the redirected stress due to some applied normal stress σ_{n}. Here we investigate the development of stresses within a granular packing, confined vertically between two horizontal plates, with no walls in the remaining directions, subjected to a rigid piston impacting it from one side. As the piston moves, granular material is compacted near the piston, and with increasing displacement of the piston, the size of the packing increases. Such an accumulation process is known to occur in the petroleum industry, where sand is liberated from the host rock during extraction, altering the underground morphology of cracks [

There are a number of interesting patterns which form when a granular material is displaced by a flexible interface in such a geometry [

We are interested in systems which are highly confined. In common experiments with granular material inside Hele-Shaw cells, there are in general fewer than 20 grain diameters between the two Hele-Shaw plates, typically down to around 5 grain diameters [

Existing analytic models for the micromechanics of such a system generally reduce the problem to one spatial dimension (

Firstly, in Section 2 describe the numerical model that has been used to simulate this system. In Section 3, we establish continuum properties which correspond to the analytic formulation, and show comparisons between the two. In particular, the limitations of current analytic models are identified. Finally, a parameter set is proposed that best fits the analytic theories for a wide range of system variables.

This paper is an investigation into the micromechanics of a system which is highly constrained by external boundaries. For this reason, it is ideal to use a particle based method to model the behavior, as the total number of grains in the system is small. Toward this end, we use a conventional soft sphere discrete particle approach, implemented in the open source code MercuryDPM (

_{0}. Particles are colored by size, darker colors representing smaller particles.

We work in a system of non-dimensionalized units with the following properties; length and mass have been non-dimensionalized by the length _{m}_{m} of the largest particle in the system, respectively, where the prime indicates that the quantity has dimension. The particle diameters, ^{3}ρ_{p}/3 = 1, or ρ_{p} = 6/π. Time is non-dimensionalized by the time taken for the largest particle to fall from rest its own radius under the action of gravity, so that a unit time is _{m}g′/d′^{2}_{m}.

Particles are filled into the available space by assigning them to positions on a regular hexagonally close packed lattice, dimensioned such that particles of diameter 1 would be in contact. In all cases we use particles distributed uniformly in the range 0.5 ≤ _{0} defined in Equation (3), is approximately constant throughout the cell. From

As shown in Figure _{r} = 1. Otherwise, the interaction properties are the same as between two particles, except that the walls and piston are of infinite mass. We therefore have a well defined macroscopic sliding friction of μ = 0.4 that does not depend on the rate of loading.

_{t}_{r}_{t} and γ_{r} and friction coefficients μ_{t} and μ_{r}. Full details are given in Luding [

Normal | 1,00000 | 1000 | – |

Tangential | 80,000 | 0 | 0.4 |

Rolling | 80,000 | 0 | 10^{−3} |

In the following Section we will firstly detail the important macroscopic quantities measured from a single simulation. We will then investigate the effect of three controlling parameters on the evolution of the system: the Hele-Shaw spacing _{0} and the velocity of the piston, _{p}gD

As depicted in Figure _{s}/V_{t}_{i}_{i}_{m}, as an average over the solid fraction close to the wall at some time when the transition zone is far from the piston as
_{m}, and note that in the absence of volumetric expansion or dilation of the granular material, ϕ is directly proportional to the height of the packing between the Hele-Shaw walls. The undisturbed zone is that which is maintained at the initial packing fraction ϕ_{0}, which is defined at time

To delineate the plug, transition and undisturbed zones, at each time _{0} + 0.1 to ϕ = 0.9, such that ϕ_{0}. The coordination number,

_{0} = 0.5 and _{xx}_{yy}_{zz}_{xz}_{zz}_{y}_{yy}_{xx}_{z}_{zz}_{p}ν_{xx}

Coarse graining techniques in general cause measured fields to converge toward zero near boundaries [^{p}_{n}

An analytic expression to describe this stress evolution was first derived in Knudsen et al. [_{z}_{zz}_{xx}_{zz}_{zz}_{p}ν_{m}g_{xy}_{zz}

By further assuming that the stress at _{T} ≡ σ_{xx}

Previously, the threshold stress σ_{T} has been modeled as either estimated from experimental data [_{T} as a ϕ_{0} dependent quantity by considering limit equilibrium of a wedge of material being displaced into the undisturbed zone, as shown in Figure _{x}_{b}_{i}_{x}_{T}, _{b}_{0} = μ^{2}ρ_{p}ν_{m}g_{i}_{1} cos θ = μ^{2}ρ_{p}_{m}g^{2}_{0} cos θ/(2 tan θ) per unit length in the

_{x}_{b}_{0} of the green region, and the _{i}_{1}, of the blue region.

This assumption of the failure surface introduces no new parameters into the model, and as will be shown in the following, closely predicts the measured value of σ_{T} for a large range of system parameters. In the limit where ϕ_{0} → 0, this definition reduces to that used in Knudsen et al. [

A best fit estimate is used to find _{z}_{m}, θ and ϕ_{0}, which adequately captures the behavior of the system past _{xx}

The measured value of apparent friction μ = σ_{xz}_{zz}_{y}_{yy}_{xx}_{z}_{zz}_{xx}

An underlying assumption of the Janssen stress redistribution is that when averaging over the width of the system (here in the

As motivated in Equation (6), the gap spacing _{0} = 0.5 ± 0.05 and _{0} ≈ 0.66), as depicted for four values of _{T} are averaged temporally over values in the range _{m}, and slope angle, θ, with increasing gap spacing, as the effect of the boundaries on the system decreases.

_{0} ≈ 0.5,

_{m}. _{z}^{p}_{n}_{z}_{y}_{T}. Dots represent the mean value of the coarse grained continuum field σ_{xx}_{m}, θ and ϕ_{0}.

For each simulation, the measured normal stress at the piston, σ^{p}_{n}_{z}_{z}_{y}_{z}_{y}_{y}_{z}_{z}_{z}_{T} also from _{m}, θ and ϕ_{0}. In all cases, we find the measured and fitted values of _{z}_{T} agree only with _{T} depends strongly on θ, and we have as yet no means for predicting this quantity. The dependence of σ_{T} on θ is in contrast to studies on fold and thrust belts [

Existing models [_{0}. To test this assumption, we here vary ϕ_{0} from 0.1 to 0.6, while maintaining _{z}_{0}, and that these values are lower than the measured values of _{y}_{0}, slightly under-predicts the threshold stress at ϕ_{0} = 0.1. Nevertheless, both the measured and predicted values of the threshold stress in Figure _{T} = 10.7 was found to reproduce the observed pattern formation behavior in the quasi-static limit, at _{0} ≤ 0.5.

_{0}_{m}. _{z}^{p}_{n}_{z}_{y}_{T}. Dots represent the mean value of the coarse grained continuum field σ_{xx}_{m}, θ and ϕ_{0}.

Finally, we wish to comment on inertial effects in such a system. Toward this end we systematically vary the inertial number ^{−2} to 10 while maintaining _{0} = 0.5 ± 0.05. As shown in Figure _{y}_{z}_{T}, also diverges above _{T} in the best fit estimation of _{z}

_{m}. _{z}^{p}_{n}_{z}_{y}_{T}. Dots represent the mean value of the coarse grained continuum field σ_{xx}_{m}, θ and ϕ_{0}.

We have here described a large number of simulations of granular materials which have been compacted in a confined geometry. For all cases, we observed that the stress distribution within the packing is well approximated by previous models, once a more rigorous definition of the threshold stress is used. This is true for a wide range of gap spacings, initial filling fractions and piston rates.

In this study we have used a rough boundary condition, where macroscopic friction at the piston and walls is equal to the inter-particle friction. However, in many systems we expect the roughness at the boundaries to be lower than that between particles. It is unclear how this difference will affect either the accumulation of material near the piston head, or the stress distribution within the packing.

Below a gap spacing of 3 particle diameters, the stress distribution is not well represented by this model. We conclude that _{z}_{y}

The slope angle, θ, has been measured for different system parameters to lie in the range of 2° – 18°.

With regards to the two Janssen parameters, we can clearly distinguish the values of _{y}_{z}

BM conducted the simulations, post-processing and authored the paper. BS conceived the idea for the simulations. BS, GD, JE, and KM assisted with the interpretation and analysis of the simulations, and editing of the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors would like to acknowledge grants 213462/F20 and 200041/S60 from the NFR.