Column collapse

2D

Due the number of particles required and additional complexity of computation in an additional dimension, a large number of early DEM studies were completed in 2D. To further simplify the computation only circular particles were considered - non-spherical shaped particles have not been used to model the collapse to date.

The first such study for a 2D column was completed by Zenit [2005] who used a soft-particle model. Details of particle size and density are not recorded in this publication nor covergence of the time step, but a coefficient of restitution of e= 0.75 is given and two different friction coefficients µ= 0.30,0.57. The number of particles varies with 100< N <10000 to achieve a variety of aspect ratios with different initial column radii.

(a) (b)

Figure 2.38: Scalings of the collapse of a column as modelled by Zenit [2005] using soft-particle DEM. Solid circles µ = 0.30, squares µ = 0.57, solid line scaling by Lajeunesse et al. [2005], dashed line scaling by Lube et al. [2005].(a) Final height with aspect ratio. (b) Final pile radius with aspect ratio.

Time lapse of a collapse for 5000 particles is shown in Figure 2.37. The DEM model recovers the qualitative dynamics extremely well, and while the scalings display the right characteristics as observed in Figure 2.38 the runout is severely overestimated. Potential reasons can lie with the correct selection of friction or the aforementioned system size. Another difference with experimental studies by Lube et al. [2005]; Lajeunesse et al. [2005] is the particle shape - random shapes such as that found in quartz sand can lead to significantly more jamming and less rolling and hence a shorter final runout.

Another study also performed in 2D by Staron and Hinch [2005] made use of the contact dynamics algorithm [Moreau, 1994], with an example of a collapse shown in Figure 2.39 (a). The number of particles was varied 1000< N <8000 with some polydispersity such that the smallest particle was at least 2/3 of the size of the largest one. This allowed column aspect ratios 0.21< a <17 with a mean packing density φ = 0.82. The system size was investigated with s =r0/d = 10,20,30,40 but was found to have little effect. Their simulations found that:

rf −r0 r0 = ( 2.5a a_≤2 3.25a0.75 _{a >2} (2.52) hf t0 =        a a_≤1 0.65a0.35 _{1< a <10} 1.45 a_≥10 (2.53)

(a) (b)

Figure 2.39: Collapse of a column as modelled by Staron and Hinch [2005] using hard-particle DEM. (a) Example of collapse with a = 9.1, µ = 1, e = 0.9. (b) Normalised runout scaling.

Figure 2.40: DEM simulation of axisymmetric collapse as performed by Cleary and Frank [2006]. The column shown consists of 165,000 particles witha= 1.91.

for the runout are significantly higher which is likely due to the choice of friction and the circular shape particles enhancing the movement. The stagnation of the height was not found to occur in experiments, and while no reason was given by the author it is thought that this might be a combination of the circular nature of the particles and small system size. Staron and Hinch [2005] also found the correct square root proportionality of collapse duration with aspect ratio, but as shown in Figure 2.39 (b) the initial acceleration phase was not well captured.

Axisymmetric

The only axisymmetric column collapse DEM simulation recorded is in a preprint by Cleary and Frank [2006] who use a soft-particle model. Here the authors use spherical particles with 1.9 < d < 2.1 mm and a physical column of radius r0 = 57.5 mm (a system size of s = 28) to mimic experiments by Lube et al. [2004]; Lajeunesse et al. [2004]. 165,000 particles are used for a single column height of h0 = 110 mm resulting ina= 1.91 with a typical collapse shown in Figure 2.40. The investigation aimed to relate DEM parameters to that physically observed in real systems. It is noted in its generality that using spherical particles results in a large over-estimation of the final runout and under-estimation of the final height. This is

due to the spherical particles causing failure within the pile too easily generating a greater spread, although it could be argued that this might be overcome at larger system sizes. Unlike in the experiments, no centrally undisturbed cone is observed and is attributed to the great amount of failure within the pile due to sphericity.

Under the best case scenario where µ = 0.3, µs = 0.65, µr = 0.025, kn =

1000,e= 0.4 the final scalings wererf = 292 mmhf = 243 mm which is a significant

difference from that observed by Lube et al. [2004]; Lajeunesse et al. [2004]. There has been no discussion on system size here which could notably play a large role - particularly when comparing to real experiments. When considering spherical particles it would be right to assume that a larger system size will result in a more stable pile. Its influence has been shown throughout the 2D channel experiments, but its emergence is yet to be shown in the axisymmetric case although it most likely exists. Outcomes from the study show the necessity of rolling friction in soft-sphere models where possible and to not overestimate friction between particles and the cylinder wall.