Force fluctuations in sheared granular disks

Abstract

The internal structure established within granular materials, often observed as force chains, is dominant in controlling bulk mechanical properties. We designed a two-dimensional Hele-Shaw cell to contain photoelastic disks, and two servos were used on the top and right boundaries individually. We experimentally monitored the fluctuations in force on the top plate while slowing the shearing of the well-confined disks and keeping the right boundary at a contact-confined force of 0.2 kN. The particle rearrangements were found to correspond to bulk force drops and were observed in a localized zone with a length of approximately 5 particle diameters. These results help reveal the structure and mechanics of granular materials, and further investigations are ongoing.

Keywords

granular matter / force chain / stick and slip

Introduction

Because of the discreteness of particles, granular materials have a strong tendency to develop localized rearrangements when subjected to shearing [1]. Many careful experiments have observed the dynamic characteristics of slowly driven granular systems, such as the evident macroscopic force fluctuation known as stick-slip. The statistics of events in sheared tapioca grains in an annular cell were observed as a power law distribution of responses. There have been many theoretical studies of the Gutenberg-Richter law for the power law rank-order distribution of earthquake sizes N(≥m), which obey Log10(N(≥m)) ~ a–bm, where the seismic magnitude m is a logarithmic measure of event size. In contrast to these many theoretical and numerical studies, only a few well-controlled laboratory experiments of sheared systems have generated event size distributions [2]. In a recent study of triaxial compressions, the distribution of deviatoric stress triggered by the stick-slip instabilities was determined [3]. The event size m was taken as the absolute value of the derivatives of deviatoric stress with respect to axial strain. Before the peak strength state, there are many small-sized events that may correspond to small earthquakes, whereas after the peak strength state, there are many large-sized events that may correspond to large earthquakes. Recent results suggest that such rearrangements tend to occur at soft spots, which are structurally distinct [4]. Soft spots might control the localized rearrangements as well as the dynamical responses of stress and macroscopic deformation. Granular materials display more abundant dissipation phenomena than ordinary materials. A brief energy flow path with irreversible processes is illustrated, in which the concept of granular temperature, initially proposed for dilute systems, is extended to dense systems to quantify disordered force chain configurations [5].

In this work, we briefly report results from force fluctuations in sheared 6 400 bidisperse disks. The particle rearrangements are observed and corresponded to bulk fluctuations process of force.

Experimental setup and results

On the basis of previous photoelastic tests [6], our experiments were conducted in a Hele-Shaw cell, as illustrated in Fig. 1. The cell consists of two parallel glass plates with a height of 840 mm and a width of 835 mm, separated by a depth of 3.4 mm, as illustrated in Fig. 1. The system consists of a bidisperse mixture of 3400 large (diameter d = 15.0±0.2 mm) and 3000 small (diameter d = 10.0±0.25 mm) polymer disks with a thickness of 3.2 mm. The disks are cut from a 3-mm-thick sheet of Polystyrene PS Sheet (ST313300) obtained from Goodfellow Cambridge Limited, England. The material density is 1 050 kg·m^‒3, Young's modulus is 2.3 GPa, and Poisson’s ratio is 0.35. The coefficient of static friction, μ_s, was measured by placing the sled on an inclined steel sheet with the same material glued on it and noting the angle at which acceleration begins. Several values of mass were added to the sled to calculate μ_s. In this work, μ_s was measured to be 0.417±0.053, independent of the applied load. All of the disks are randomly arranged in a mono-layer between two glass plates in the cell.

To track the movements of each disk, i.e., translational and rotational displacements, one face of each disk was painted black to determine particle size and center position. Another face was painted white and marked with a short black line across its center to measure the rotation angles. The subsequent particle movements can be easily observed due to the sharp contrast between the dark color and bright white background. To capture the particle images, we used a Nikon D200 camera mounted with a long-focus lens that provides high-quality pictures with 12 million pixels. The positions and rotation angles were obtained for individual disks. The material was initially compacted into a dense, random, isotropic arrangement, with an initial packing fraction ϕ = 0.86.

The right boundary and upper boundary are movable to exert servo (either force or velocity) loading onto the assembly. In this study, the assembly was first isostatically compressed. The right boundary and the upper boundary moved inward at the same speed of 1.0 mm/min. The forces on both boundaries simultaneously increased. The force on the right plate was slightly larger than that on the upper plate because of the disks’ weight acting on the right plate. A smooth increase of force on both boundaries was observed at the initial compression stage. This confining process was stopped until the forces on both boundaries were approximately 0.2 kN. Afterwards, to keep the assembly in a quasi-static condition, a constant vertical loading speed of 1.0 mm/min was imposed on the upper boundary. A confining force of 0.2 kN was maintained on the right boundary. As shown in Fig. 2, the force on the upper boundary exhibited more abrupt changes during progressive compression, which is a signature of a friction-induced hysteretic response as the system experiences repeated partial jamming and unjamming. Moreover, the force increased relatively slowly.

By tracking the centers of particles, one can observe clear motion in this granular assembly. As shown in Fig. 3, there were rare buckling events in which particles suddenly changed their relative positions. These events were discrete and localized. More information can be obtained from the displacement field of the system. To visually illustrate the particle displacement field, we subtracted two images of the system at different times. Any stationary part of the system will appear black in such difference images because the individual pixels are identical in that region. However, if a particle moves, the image subtraction will produce an area with a crescent shape along the front boundary in the movement direction (shown in cyan) and leave a similar crescent area in the rear (shown in yellow). Hence, only front boundaries in the moving direction are indicated in the image. The curvature and area of the crescent show the direction and magnitude of a particle’s displacement, respectively; Fig. 3 shows the displacement field of a typical experiment. Their time interval was the same, i.e., 4 s. The corresponding displacement fields are shown as (a), (b) and (c). From Fig. 1, we can see that the forces smoothly increased when the system evolves before A, and the internal structure only changes slightly. For the transition from A to B, a large force drop occurs, and the transient positions of local particles adjust. The remaining particles are changing nearly uniformly. This process can be separated into stick or slip phases. In the stick regime, the system evolves as it adapts to the increasing stress, but no macroscopic motion is observed. In the slip regime, the evolution is composed of small slips, or rearrangement of particles, as shown in Fig. 3.

The standard deviation of the particle displacement is denoted by s and is defined as follows:

(1)s=\sqrt{{\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle \sum _{i=\mathrm{1}}^N{\left({\stackrel{\rightharpoonup }{r}}_i-\langle \stackrel{\rightharpoonup }{r}\rangle \right)}^{\mathrm{2}}}},

where {\stackrel{\rightharpoonup }{r}}_i is the observed displacement of particle i, \langle \stackrel{\rightharpoonup }{r}\rangleis the mean value of these observations, and the denominator N is the number of particles.

Obvious particle rearrangements would lead to a larger s. From Figs. 2, 3 and 4, we can also see that larger s would result in larger drops in force. For the stage during which force increases, the standard deviation of the particle displacement is much smaller. These findings imply that uniform deformation occurs as force increases, while localized rearrangement leads to force drops.

Outlook

We have discussed particle rearrangements in a sheared granular assembly. The larger force drop was found to correspond to evident particle rearrangements in a local zone with a typical length of 5 particle diameters. These results are preliminary, and the following is our consideration of the structural analysis.

It is of great importance to determine the mesostructure based on overall macroscopic characteristics and to establish corresponding macroscopic constitutive relations. Several homogenization approaches have been presented to achieve these goals, such as Rowe’s stress dilatancy relation for the regular packing of mono-dispersed particles in 2D. At present, many methods are restricted to very simple mesoscopic geometries (or media with periodic microstructures) and simple material models, mostly at small strains. Rigorous treatment of non-uniformly distributed strains requires tools that have not yet been fully developed. In continuum theories for granular materials, such as Granular Solid Hydrodynamics, many physical quantities need to be determined, either by careful observation of experimental data-an exercise in trial and error-or more systematically through discrete element simulations and advanced techniques, such as confocal microscopy, CT and MRI, and mesoscopic considerations. We believe that studies of both meso- and macro-scales have advantages, and their relationships can be strengthened.

References

[1] Bi Z, Sun Q, Jin F, Zhang M. Numerical study on energy transformation in granular matter under biaxial compression. Granular Matter, 2011, 13(4): 503–510

[2] Pica Ciamarra M, Lippiello E, de Arcangelis L, Godano C. Statistics of slipping event sizes in granular seismic fault models. Europhysics Letters, 2011, 95(5): 54002

[3] Sun Q, Song S, Jin F, Bi Z. Elastic energy and relaxation in triaxial compressions. Granular Matter, 2011, 13(6): 743–750

[4] Manning M L, Liu A J. Vibrational modes identify soft spots in a sheared disordered packing. Physical Review Letters, 2011, 107(10): 108302

[5] Sun Q, Song S, Jin F, Jiang Y. Entropy productions in granular materials. Theoretical and Applied Mechanics Letters, 2012, 2(2): 021002

[6] Liu J, Sun Q, Jin F. Visualization of force networks in 2D dense granular materials. Frontiers of Architecture and Civil Engineering in China, 2010, 4(1): 109–115

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