1 Introduction

Granular media has near omnipresence in nature in the form of soils, sands, snow, rocks and more, and is the second most processed substance in industry after water. It has been studied from many viewpoints, perhaps due to the wide variety of behaviours and phenomena it exhibits under different conditions. Some examples of these studies examine regimes which transition from solid-like to fluid-like behaviour [1,2,3], compaction [4,5,6], ‘granular-gas’ behaviour [7,8,9], acoustic wave transmission [10,11,12], and convection rolls and particle movement in vibrating granular systems [13,14,15].

The prevalence of vibrated granular matter in industry coupled with its ability to exhibit the above phenomena has resulted in increasing interest in the subject over the last few decades. It is well established in the literature that the continuum-scale behaviour of a granular system is dominated by the grain-scale interactions between particles [16,17,18], which is in turn influenced by many factors - such as grain size and shape, boundary geometry and material, and vibration direction and acceleration, to name a few.

The most basic grain scale interaction is the normal contact force between neighbouring particles. ‘Pathways’ of normal forces between neighbouring particles form structures known as force chains which support the majority of load on the material. Even in highly ordered homogeneous granular materials, these structures are often unintuitive in their location and geometry and much research has been dedicated to determining their shape and strength characteristics, for example, [19,20,21,22,23].

Forces in contacts are generated from grain-scale interactions such as sliding, rolling, and impacts. Subject to external excitation, such as particle strikes or moving boundaries, waves propagate through granular media via a combination of these particle-particle interactions. There is a large area of literature directed towards understanding the nature of granular wave propagation at this scale and energy transfer in granular matter, for example, [24,25,26,27].

If a wave carries enough energy through a granular medium not bounded on all sides, particles within the system experience spatial displacement. The result of net particle displacement within a granular body is visible flow behaviour, of which there are three general categories exhibited by vibrating granular media, often observed simultaneously in different regions of the bulk material [28] [29].

  • ‘Jammed,’ where the particle contact network remains undisturbed and particles do not move relative to the vibrating boundaries.

  • ‘Glassy,’ in which each particle within the bulk medium moves with an amplitude smaller than its diameter but is trapped by its neighbouring particles.

  • ‘Fluid,’ in which the particles enjoy a mean free path larger than their diameter, in turn giving rise to fluid-like behaviour.

In studies where the medium is subjected to harmonic vibrations, the stroke, x, is given by Eq. 1, where f and A are the frequency and amplitude of vibration, respectively, and t is time. From this, the maximum vibration velocity, or velocity amplitude, v, is easily seen and is given by Eq. 2. The parameter often used to indicate the level of driving sinusoidal vibrations is non-dimensional acceleration, \(\Gamma\), denoted by Eq. 3, where g is gravitational acceleration.

$$\begin{aligned} x(t) = A\mathrm{{sin}}(2 \pi f t) \end{aligned}$$
(1)
$$\begin{aligned} v = 2 \pi f A \end{aligned}$$
(2)
$$\begin{aligned} \Gamma = \frac{(2 \pi f)^2 A}{g} \end{aligned}$$
(3)

Generally, as \(\Gamma\) increases, so the granular temperature - mean fluctuating velocity per particle - likewise increases. The concept of granular temperature has existed since the late 1970 s [30], where it emerged from attempts to represent granular flow mathematically with continuum-mechanical and statistical approaches. The links between \(\Gamma\), granular temperature, and particle movement are explored in many subsequent experimental efforts to determine, for example, velocity distributions and other phenomena such as densification and clustering in vibrated granular matter at various \(\Gamma\) values (e.g. [7, 29, 31,32,33]).

Convective flow patterns also arise with increased \(\Gamma\), the general shape of which have been known since early this century in both vertically and horizontally vibrated granular media [2, 13, 14, 34]. Convection is sensitive to grain-scale interactions and as such is affected by the interstitial medium (usually air), constitutive particle size and shape, and wall geometry and texture [17, 35,36,37]. Since the very nature of convection is net particle displacement per oscillation, it generally causes the shape and packing arrangement of a granular medium to change over many vibration cycles. This can manifest as vibration induced heaping [38, 39], diffusion [35, 39], and size segregation [40, 41], depending on vibration strength and grain-scale construction of the medium.

Motivated by the sheer variety of flow behaviour and corresponding phenomena exhibited in vibrated granular matter, in this study tests were performed on horizontally vibrated quasi-two-dimensional granular beds. The aim was to examine the influence of horizontal vibrational load on bed behaviour on both a macro-scale and at the grain-scale, and how this deviates between homogeneous and inhomogeneous material.

Tests were first performed on homogeneous, then inhomogeneous material to see how the response of the granular bed deviated with changes to material composition. The inhomogeneous bed consisted of a ternary bead mixture, the smallest two diameters of which were chosen to be ‘geometrically compatible.’ That is, the smallest beads were almost the perfect size to fit in the voids present between the second-smallest beads.

The paper is structured as follows. First, the experimental apparatus and test procedures are described. The method of framing granular matter as quasi-two dimensional is discussed by comparing results from these tests with numerous results already established in the literature. Then, phenomena witnessed in the tests are discussed individually and compared. Explanations for specific flow behaviours are presented.

2 Materials and method

2.1 Short description of apparatus

The vibrating granular media was examined as a quasi-two-dimensional problem. The experimental set up consisted of a thin transparent tank of internal dimensions 300x145x10mm affixed to an LDS V406 electrodynamic shaker. The tank was mounted on linear bearings to limit its movement to the horizontal direction only, and the shaker was connected to an oscilloscope used for determining the amplitude and frequency of vibration after appropriate calibration. A MotionBlitz Cube3 512x512 pixel high-speed camera was focused on the side wall of the tank and set to capture the motion of the granular media at 800 frames per second. These frames were used to create video footage of the tests. Figure 1 displays a schematic of the experimental set-up.

Fig. 1
figure 1

Schematic of experimental apparatus

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Clear glass beads of density 2600\(kg/m^3\) were used as the granular material in the tank. In order for a few particle paths to be tracked, ten of these beads were coloured red, a plain white background was placed behind the tank, and a 1500lm lamp was placed behind the camera and aimed at the tank.

Tests were first performed on a homogeneous system consisting of beads 2.5mm in diameter, before being performed on a ternary mixture of 1mm, 2.5mm, and 4mm diameter beads.

2.2 Tests and procedure

The clear glass beads were poured into the tank to a depth of 60mm before the red beads were placed at varying depths with tweezers. The system was vibrated at a modest \(\Gamma <0.5\) for 30s to break the initial configuration of beads and then allowed to settle. This method consistently produced a solid fraction of each bed at rest of approximately 0.613. At the start of each test, the tank was harmonically vibrated for 10s to allow it to reach a ‘steady state,’ at which time the subsequent 2s of flow behaviour was captured by the high-speed camera for analysis.

The tests were performed at 0.5mm amplitude for 4Hz, and 10Hz-80Hz in 10Hz increments. After the 0.5mm test at each frequency, the amplitude was smoothly increased to the maximum amplitude the shaker could produce without distortion to the sinusoidal waveform. At 60Hz, 70Hz, and 80Hz, the shaker was unable to vibrate at an amplitude of greater than 0.5mm. Table 1 gives a full breakdown of test conditions, where D is the constituent bead diameter of the bed.

Table 1 Test conditions
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3 Quasi-two-dimensional validity

These tests were conducted in a custom built quasi-two-dimensional tank. The tank is labelled as quasi-2D because its thickness, in the z direction, was more than an order of magnitude smaller than its width and depth, in the x, y plane, but still over twice as large as the largest particle diameter. Thus, the thickness was large enough not to impose significant constraints on the beads in the tank, too large for the granular bed to be considered fully two-dimensional, but too small for phenomena to occur in the z direction.

There is a small presence in the literature of vibrated quasi-2D granular beds, e.g. [42, 43], but fully two-dimensional or fully three-dimensional experimental set-ups are more established. In particular, horizontally vibrating a quasi-2D tank is an exceedingly uncommon method for examining granular material; to the best of the authors’ knowledge, it is a novel technique. It was logical to ensure it is a valid experimental approach before analysing individual results. In order to do this, similarities in flow behaviour were sought between these tests and existing results in the literature from tests which were conducted either on fully 3D beds or fully 2D beds.

The dynamic solid fraction during each test on the homogeneous bed was obtained by fitting a spline curve to images of the free surface of the bed and calculating the cross-sectional area in the x, y plane. This was extrapolated through the z plane to obtain a value for volumetric space occupied, enabling determination of the overall solid fraction of the bed.

A general trend for an initially non-dense packing of spheres to increase in density with \(\Gamma\), and a trend, specifically for beds vibrated at a fixed frequency with increasing amplitude, to first increase in density to a maximum value, then decrease as \(\Gamma\) further increases, are established results in the literature - for example [6, 29, 44]. Both of these trends appear in these tests.

Fig. 2
figure 2

Solid fraction variation in the homogeneous bed during vibrations

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Variation in solid fraction with both frequency and \(\Gamma\) is displayed in Fig. 2. After an initial period of little change, there is a trend for the solid fraction to increase with both the vibration frequency and \(\Gamma\). Further, visible in Fig. 2a from the hollow circles at 20, 30, 40, and 50Hz, it is clear that large amplitudes tended to produce a lower solid fraction in the beds tested at two different amplitudes and \(\Gamma\) values. In those tests, the amplitudes greater than 0.5mm corresponded to high \(\Gamma\) values where the solid fraction of the bed would be expected to be less than that at smaller \(\Gamma\).

Numerous other phenomena in these tests also matched existing results in the literature. The phenomena themselves are simply stated in this section, and are detailed further in Sects. 4 and 5 alongside other flow behaviour from these tests.

In the homogeneous bed, three flow regions formed depending on the nature of the vibrations - jammed, glassy, and fluid. This is consistent with, for example, [2, 6, 8, 28, 29]. The top layers of the bed tended to dilate during large amplitude vibration, and triangular areas formed by sloshing in the upper corners: these results are in line with those found in [13]. In the inhomogeneous bed, key phenomena were the presence of much stronger convection rolls in comparison to the homogeneous bed and low percolation - results consistent with, for example [40] and [45].

The results of both homogeneous and inhomogeneous tests pointed to a relationship between packing structure and material strength on the continuum scale, again in agreement with the existing literature on the subject [16, 46].

Finally, throughout the tests, indications of a nuanced relationship between bed response and vibration frequency and amplitude arose. Specifically, granular-gas and convective flow behaviour was influenced by both frequency and amplitude but, at the frequencies studied, changes in amplitude had a larger affect than changes in frequency.Footnote 1 This has been commented on previously in the literature, for example [2, 33, 48].

The consistent occurrence of these known phenomena throughout the tests indicates framing the horizontally vibrated granular bed as quasi-2D is a valid experimental approach.

4 Results from the homogeneous system of 2.5mm diameter beads

In the tests at 0.5mm amplitude, almost all of the flow field remained jammed with only the very top layer of beads showing any movement relative to the vibrations of the tank. As the amplitude was raised from 0.5mm and \(\Gamma\) increased at each frequency, the flow in the bed was more energised and regions of differing flow regimes formed. This began as the layers of beads closest to the free surface exhibited more movement than beads deeper in the bed, with this upper region showing a tendency to dilate during vibration depending on the value of \(\Gamma\) at each frequency.

Vibrations with amplitude larger than 0.5mm tended to give rise to a sloshing region in the upper layers of the bed and triangular areas of beads interacting solely via instantaneous collisions - granular-gas regions - in the top corners of the media. This phenomenon is best viewed using the velocity amplitude of vibration, v, expressed in Eq. 2. In general, an increase in size of these triangular granular-gas regions occurred with an increase in v.

In the glassy and jammed regions, innate disorder in the bulk particle arrangement led to the medium being split into discrete areas of geometrically ordered beads - lattice arrangements often called ‘granular crystals’ or ‘crystalline lattices.’ The lines bordering the perimeter of these areas were frequently visible and are labelled in this paper as ‘shear lines.’ Shear lines present in the medium disrupted bead movement and weakened the medium on the continuum-scale, from which it is clear they play a key role in the nature of the medium’s response to vibration.

Figure 3 illustrates the sloshing region in the upper layers of the bed, and Fig. 4 depicts shear lines and the triangular granular gas region. It should be noted that not all results images were suitable for PIV analysis; hence the technique is not used as a quantitative analysis technique in this paper. The images for Fig. 3 were suitable for processing with PIV, where it is used for illustration purposes only as it highlights the variation in bead movement in different areas of the bed clearly. The software used was PIVLab [49, 50].

Fig. 3
figure 3

Sloshing upper region of the bed at large amplitude. The velocity of the beads in the upper layers is significantly greater than lower in the bed, which are almost stationary at this turning point

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Fig. 4
figure 4

Shear lines in the granular bed

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4.1 Emergence of simultaneous flow states

The differing flow regions in the medium emerged from a complex, but nonetheless visible, series of events occurring from particle-particle and particle-wall interactions; the nature of which are detailed below.

Individual bead movement depends on the resistance to motion it faces from the ‘cage’ formed around it by its neighbouring beads. For non-vibrofluidised regimes, the net weight of the beads above each individual layer increases with depth, so resistance to motion from neighbouring beads also increases with bed-depth. Hence, in these tests, the beads in layers nearer the surface faced less resistance to motion and experienced greater movement in each oscillation than beads deeper in the bed.

The limits to bead movement imposed by the rigid boundaries of the tank, combined with these variations in general resistance to bead movement, created an anisotropic effect whereby the horizontal vibrations produced a net upwards-diagonal movement of beads close to the boundaries.Footnote 2 Fig. 5 presents a graphical explanation for this anisotropy using a similar explanation to that found in [28]. This movement was cyclic - there was no convection, rather, the beads oscillated diagonally up and down.

Fig. 5
figure 5

Idealised graphical explanation for the tendency of beads to move away from the rigid boundaries of the tank

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The repetitive agitation induced in the bed by large amplitude vibrations resulted in particle layer height not being spatially or temporally uniform, rather than each layer consisting of a neat row of beads sitting side-by-side throughout the duration of the vibrations. It is difficult to characterise layer height for this reason. This non-uniformity in layer construction was key to the formation of neighbouring flow regimes, since it resulted in a ‘boundary layer’ between the two adjacent areas of flow which beads continually broke away from and rejoined.

4.2 Triangular corner shape and flow formation

Despite the chaotic nature of the bead trajectories and interactions in the triangular corners of the tank, trends were plainly visible. Figure 6 displays the change in size of the triangular corner area with velocity amplitude v. There is little change in area until v exceeds 200mm/s, whereupon large variations arise. All tests conducted at 0.5mm amplitude, regardless of frequency, produced little granular-gas behaviour in the corners of the tank. The larger triangular areas were all produced by vibration amplitudes greater than 0.5mm, pointing to the vibro-fluidised granular-gas response having a larger dependency on amplitude than frequency. It must be stressed that this does not negate the effect of frequency on flow response as a whole: the largest amplitude of 6mm coupled with a frequency of 4Hz produced no movement of particles at all - this data point can be seen as the dark circle at \(v=151mm/s\).

Fig. 6
figure 6

Corner area variation with container velocity amplitude, v. There is little change in area until v exceeds 200mm/s, whereupon large variations arise. Notably more variation in corner area arises with vibration amplitudes greater than 0.5mm than with amplitudes of 0.5mm

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The net flow pattern in these triangular areas, when they formed, was rotational. This flow pattern was intrinsically linked to the shape of the triangular area as follows.

Triangular corner shape formation: Sloshing in the fluid upper region of the tank caused the formation of triangular areas between the particles and the walls. In addition to the sloshing motion of the upper region of the medium, a general upward motion at the wall was witnessed, which contributed to the formation of the triangular areas.

Upward bead motion at the wall: The net oscillatory motion of the beads within the tank was always out of phase with the motion of the tank itself. This is because on the continuum-scale the granular material was not a rigid structure; on the grain-scale, the net movement of the beads was limited by the speed of particle-particle momentum transfer which ‘passes’ the wave-front through the medium. Once per cycle, this lack of synchronisation caused the tank wall to collide with the beads. This happened when the direction of motion of the tank was reversed and the wall motion was temporarily 180 degrees out of phase with the motion of the beads closest to it, with the wall now moving towards the beads. These beads then collided with their neighbouring beads and were ‘crushed’ between them and the wall. As previously described, beads faced less resistance to motion from above than from below from their neighbouring beads; an imbalance which led to upwards bead motion at the wall.

Rotating motion in the triangular area: As previously described, beads were forced upwards at the wall, and the deeper these beads were, the more resistance they faced to their motion. Over many oscillations, beads at the wall were gradually forced upwards towards the bottom corner of the triangular region, whereupon they faced very little resistance to upwards motion. At this point, beads at the wall would be ‘pinged’ upwards into the air upon the next ‘crush’ between the wall and neighbouring beads. Whilst airborne, these beads may collide with the side wall and/or other airborne beads in the triangular region. When the beads fell back to the surface of the medium, they tended to fall onto the slope of the triangular region and roll to the bottom corner. This process would be repeated throughout the duration of the oscillatory motion of the tank, giving rise to the general rotating flow pattern of beads within the triangular corner area.

4.3 Shear lines in the less agitated regions

Innate disorder in the bulk particle arrangement led to the medium being split into discrete areas of ordered crystalline lattices in regions where beads were not agitated enough for sloshing or a vibrofluidised response to occur. The lines bordering the perimeter of these areas were frequently visible and are labelled in this paper as ‘shear lines,’ highlighted in Fig. 4. These lines play an important role in the nature of the continuum-scale response to vibrations.

Shear lines consist of non-crystalline lines of bead contacts. In the glassy region, they were a barrier against uniform bead movement within the bulk structure of the medium. Although it is known that the level of disorder in granular arrangements does not have a significant affect on the propagating wave structure [51], adjacent crystalline areas bounded by shear lines oscillated with similar amplitudes to one another, but out of phase with one another. (In jammed regions, shear lines were present but this phenomenon did not occur as particles did not oscillate relative to the tank).

These lines marked clear weak points in the structure of the granular medium. After a number of oscillations, adjacent areas of beads would ‘collapse’ into one another: the shear line dividing them would cease to exist and one larger granular crystal moving uniformly would form. No link was found between the number of oscillations and life of shear lines - indeed there appeared to be no way of predicting when a shear line would fail.

However, Fig. 2 shows a clear trend for solid fraction to increase with vibration frequency in the tests conducted at an amplitude of 0.5mm, where granular-gas behaviour and sloshing were minimal. It is interesting to correlate this to the granular beds obtaining a denser configuration of beads after shear line collapse than the less ordered configuration before; indicating response to vibration is more dependent on frequency than amplitude in regions not exhibiting large relative particle movement and excitation.

Shear lines have not, to our knowledge, been specifically commented on previously in the literature but it is apparent their presence affects packing structure in such a way as to weaken the material on the continuum scale.

5 Results from the inhomogeneous system of 1mm, 2.5mm, and 4mm diameter beads

The introduction of differently sized beads - of 1mm, 2.5mm, and 4mm diameter - changed the behaviour of the system, most notably by increasing convective effects and by introducing segregation and heaping.

5.1 Test at 10Hz

When left vibrating long enough at 5mm amplitude at 10Hz, the system separated broadly into two layers of beads without the presence of convection rolls, with most of the 1mm and 2.5mm diameter beads occupying the middle and lower regions of the bed, and most of 4mm diameter beads the upper region of the bed. This is visible in Fig. 7a. No further separation took place throughout the test. The 1mm beads percolated through the mixture only until they occupied a space surrounded by 2.5mm beads; a phenomenon which reduces void ratio in the material, known as the void-filling mechanism [45]. Once there, they greatly limited possible movement of both themselves and the 2.5mm beads. The net outcome was a wedging effect in the region occupied by the 1mm and 2.5mm diameter beads, responsible for their lack of segregation.

The top layers oscillated with larger amplitude than the rest of the bed and sloshed from side to side, which, as discussed in section 4.2, is a result of the reduced resistance to motion at the free surface. The sloshing region in the mixed bed had both depth and amplitude slightly greater than the equivalent tests on the homogeneous bed - this is clear when comparing Fig. 7a and b.

Fig. 7
figure 7

Comparison between homogeneous and inhomogeneous bed behaviour at amplitude 5mm, frequency 10Hz. Triangular corner area is larger in the inhomogeneous bed. The segregation is clear to see in the inhomogeneous bed, with the mid and lower layers consisting mostly of 1mm and 2.5mm diameter beads; many of the 4mm beads having risen to the upper layers

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5.2 Tests at 40Hz and 70Hz

When vibrated at 40Hz and 70Hz, the inhomogeneous bed exhibited considerably stronger convective forces than those in the homogeneous system, which resulted in convection rolls in the plane of the vibrations occupying the full depth of the bed, as illustrated in Fig. 8. Figure 9 shows images from the 70Hz, 0.5mm test, where the movements of the red coloured beads were traced. These tests at 70Hz were carried out for 7s with frames captured at 1s intervals. The large, slow, anticlockwise convection roll can be seen from the movement of these beads. There were two characteristics of note regarding these convection rolls. Firstly, their depth was the same at both 40Hz and 70Hz frequency 0.5mm amplitude tests, but the rotation was fastest in the 1mm amplitude tests at 40Hz. Secondly, they were absent in the 10Hz test.

Fig. 8
figure 8

Illustration of convection rolls seen in the inhomogeneous beds at 40Hz and 70Hz frequencies. Note that the triangular area in the top-left corner is still present, but small. Also note the distinct slope of the free surface

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Fig. 9
figure 9

Particle movement in the 70Hz, 0.5mm test. The large, slow convection roll can be seen through the movement of the tracer beads, particularly the three beads furthest from the centre of the roll

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Table 2 displays the tests run on the inhomogeneous beds, listed in order of the visible speed of convection rolls present in each. Convection rolls were present and were of the same depth in the three test conditions with the highest \(\Gamma\) values, and of those, the fastest convection rolls were present in the test with the largest amplitude. This points to convective flow response being dependent on both amplitude and frequency of vibration, but having a greater dependency on amplitude - although it is acknowledged that more data points are required for this relationship to be explored fully.

Table 2 Vibration details for the inhomogeneous beds, listed in order of visible speed of convection rolls
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A heaping effect, albeit small, occurred in the system due to these convection rolls, which in turn had a small contribution to the convective process itself. Towards the middle of the tank, the beads rose from the base of the tank before moving outwards towards the side walls. The slope of this heap had a low gradient, but was enough for a few 4mm beads at the free surface to roll part of the way towards the edges of the tank, before rejoining the rest of the beads in the main roll and being transported back towards the centre, for the process to be repeated.

6 Discussions

Given both the homogeneous and inhomogeneous beds were subjected to comparable vibration conditions, some discussion is warranted regarding plausible reasons for their markedly different behavioural responses.

The homogeneous bed did not exhibit convective flow for any of the imposed test conditions; its dominant responses being sloshing and a triangular granular-gas area in the upper region of the bed, and, in regions or test conditions not excited enough for these to occur, the merging of discrete neighbouring crystalline areas upon failure of shear lines. Conversely, the ternary mixture was unable to form an ordered crystalline arrangement and exhibited greater movement in the sloshing region at 10Hz, and strongly convective flow at 40Hz and 70Hz.

The fundamental differences in internal material structure between the homogeneous and inhomogeneous beds inevitably led to variations in micro-scale particle-particle interactions. It is logical, therefore, that these variations in particle interactions combined to produce the differing macro-scale responses of each granular bed as a whole.

6.1 Grain-scale interactions of homogeneous material

A useful insight into the grain-scale interactions taking place within a homogeneous regime of round particles can be found by examining the nature of the shear lines exhibited by the homogeneous bed. Areas of crystalline structure retained their ordered geometric arrangement and merged with neighbouring areas upon the collapse of shear lines bordering them.

Figure 10 depicts a virtual experiment where a square arrangement of spheres has a horizontal force applied to the left-hand particle of the middle row. One force chain emerges, linking the particles in the middle row only. As these particles move to the right, frictional forces at the contacts between the middle row and its neighbouring rows oppose the movement: these are the only interactions between rows which oppose the initial force applied to the middle row. After some change in time, \(\Delta t\), and after relative sliding and rolling of particles on differing layers, the rows rearrange to a hexagonal crystalline structure and particles interact with neighbouring rows via a combination of normal and frictional forces. This is, in principle, a simplified force-diagram representation of shear line failure and granular crystal formation in a homogeneous bed.

Fig. 10
figure 10

Simplified force diagram illustrating grain-scale interactions in the homogeneous bed. This figure is simply intended to illustrate force network changes from packing rearrangement of layers. In the interest of clarity, only forces opposing the motion induced by the original normal force are shown (i.e. bead weight, vertical normal forces at contacts, and other forces inherent in the system like the cumulative weight from layers stacked above are not labelled in this image. However, frictional forces caused by vertical normal forces are shown, because they directly oppose particle motion)

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Normal reaction forces are inherently stronger than the frictional forces they create through the Coulomb model of friction. Hence, it is clear that the crystalline structure is the strongest arrangement of homogeneous round particles, the square arrangement the weakest, and any arrangement between these two extremes will have a corresponding strength also between these two extremes. Shear lines, being non-crystalline lines of bead contacts, are weaker than the crystalline arrangements they border; thus, there is less physical resistance to bead movement along these lines than within the areas they bound.

In the tests presented here, the initial random internal arrangement of particles in the homogeneous granular bed lay somewhere between perfectly square and perfectly crystalline. Vibrational loads imposed on the bed unsettled the disordered configuration of beads, but only enough for internal failure along shear lines with relative sliding of neighbouring crystalline regions. This internal rearrangement within the granular medium led to a more ordered particle structure; a structure stronger and more stable in the absence of shear lines than a less ordered arrangement with shear lines present.

It follows that, in the homogeneous bed, the initial grain-scale response to an external load consisted more of frictional forces opposing particle movement and local geometric rearrangement of constituent grains than normal reaction forces. The low rolling resistance exhibited by round particles is known to promote the prevalence of this grain-scale response [52]. Once a crystalline arrangement formed, many normal forces would be introduced in the bed which were previously not present or smaller in magnitude, and the arrangement would not break unless amplitude was increased. Under these larger amplitude conditions, convection would not be induced; rather, the bed would exhibit granular-gas regions and sloshing.

6.2 Grain-scale interactions of mixed material

The phenomena of note in the mixed bed vibrated at 10Hz were low percolation and large amplitude sloshing in the upper region, and at 40Hz and 70Hz was strong convection rolls. The explanation proposed in this paper is that these phenomena can all ultimately be linked to the ‘geometric compatibility’ of the constituent beads of the inhomogeneous bed.

Geometrically, the voids which exist in rectangular packings of 2.5mm beads are very nearly the perfect size for 1mm diameter beads to fit into. As stated in Sect. 5.1, this limited the separation exhibited at 10Hz as the 1mm beads percolated through the bed until they occupied a space surrounded by 2.5mm beads and a jamming effect arose. The knock-on effect of this was that the upper sloshing region, which was approximately the same depth as that of the equivalent homogeneous bed, consisted primarily of 4mm diameter beads.

Since they physically occupy greater volumetric space than 2.5mm diameter beads, there were fewer beads in the sloshing region in the mixed bed when compared to the homogeneous equivalent. The particle contact network in the sloshing region of the mixed bed was therefore also comprised of fewer beads than the respective homogeneous bed. It follows that each particle-particle contact in the sloshing region of the mixed bed must transmit a larger proportion of the vibrational load than the equivalent homogeneous bed. This, in turn, lead to a greater likelihood that the maximum normal force was reached for relative sliding and rolling between the 4mm beads to occur during each vibration cycle - resulting in the larger amplitude sloshing motion present in the mixed bed.

Regarding the large convection rolls in the tests at 40Hz and 70Hz: it is not unusual for mixed granular systems subjected to vibrations to exhibit low percolation and large convection rolls. In tests consisting of vertically vibrated granular beds constituting up to eleven different species of particles, it has been found that as the number of particle sizes in the material increases, percolation decreases and convective effects increase [40]. What is interesting in the tests presented here is that there was only a ternary mixture, two species of which were ‘geometrically compatible’ as discussed above. This indicates not only number of species, but specific constituent particle size playing a large role in flow behaviour.

As was the case in the jammed region for low \(\Gamma\) vibrations, the small beads in the system occupied the available space between larger beads. Similar to Figs. 10 and 11 is a virtual experiment illustrating a simplified force-diagram of initial force chain formation in a jammed region of the inhomogeneous bed.

Fig. 11
figure 11

Simplified force diagram illustrating grain-scale interactions between two geometrically compatible species in a square arrangement. This figure is simply intended to illustrate initial force networks from a load applied to a single particle. In the interest of clarity, only forces opposing the motion induced by the original normal force are shown (i.e. bead weight, vertical normal forces at contacts, and other forces inherent in the system like the cumulative weight from layers stacked above are not labelled in this image. However, frictional forces caused by vertical normal forces are shown, because they directly oppose particle motion). Contact bridges form across the smaller particles, increasing the complexity of the initial force network formed in the bed in comparison to a homogeneous system

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The geometric compatibility between the 2.5mm beads and the 1mm beads leads to formation of ‘contact bridges’ transmitting normal forces between larger particles in different layers. Viewed across the bed as a whole, these force chains would increase the radius of influence of each individual bead, since they form complex pathways of normal forces through the bed.

It follows that, in the inhomogeneous bed, the initial grain-scale response to an external load featured a high prevalence of normal forces between neighbouring beads in all layers, resulting in stronger grain scale interactions throughout when compared to a homogeneous bed. Put another way, variation in bead diameter enables any bead to directly push a greater number of neighbouring beads (which, in turn, push their neighbouring beads, and so on) than in the equivalent homogeneous system.

Finally, the packing structure in a mixed bed is inherently more complex than that of a homogeneous bed: in one moment, smaller particles may be occupying spaces where the larger particles would otherwise have been pushed into and in another moment, the more rigid granular structure allows for the presence of voids which would not usually be available for larger particles to fall into [40]. These factors combine to produce a convective response to vibrations strong enough to overcome the jamming effect, enabling a deep section of the bed to exhibit fluid-like convective flow.

7 Summary and concluding remarks

In this research, granular bed response to various vibration conditions was examined with high-speed imaging. Beds were horizontally vibrated in a quasi-two-dimensional arrangement, firstly with homogeneous granular media and then with a ternary mixture to see how bed response deviated with changes to material composition. Large differences in flow response between the homogeneous and inhomogeneous materials were clear; the primary differences summarised as follows.

Allowing for minor manufacturing variations, each bead being identical gives homogeneous material the ability to geometrically rearrange from initial disorder into an ordered crystalline lattice structure. This process arises with shear line failure - when neighbouring areas of crystalline structure slide relative to one another along non-crystalline lines which form their joint boundary. Analysis of shear line failure and subsequent bed behaviour indicates the grain-scale interactions predominantly responsible for initial continuum-scale bed response is relative sliding of beads and, by extension, frictional forces between beads along these boundaries.

The presence of differently sized particles in the ternary mixture increases packing complexity and prevents the shear line presence and geometric rearrangement seen in the homogeneous material. This inherent difference in packing structure, coupled with the ‘geometric compatibility’ between the 1mm and 2.5mm beads, produced low percolation and large amplitude sloshing in the upper region at 10Hz, and a large convective response in the full depth of the bed at 40Hz and 70Hz. These macro-scale bed responses are attributed to the differing bead diameters in the ternary mixture enabling a high prevalence of normal forces between neighbouring beads. This, in turn, leads to stronger interparticle interactions and a wider radius of influence of each individual bead in the bed.

This raises interesting questions about what specific micro-scale characteristics are required of a granular material to exhibit certain macro-scale behaviour. The idea of geometric compatibility proposed here is rooted within the idea that packing structure complexity directly influences the nature of particle-particle interactions, and hence, the macro-scale properties of the granular material. Previous findings show that increasing the number of species in the system increases the convective response of the material: results from these tests indicate this is because having a good mix of particle sizes in the system inherently enhances packing complexity, raising the probability that there will be sets of species which predominantly interact with their neighbours via normal forces.