Discussion: 3D assessment of fracture of sand particles usingdiscrete element method

M. B. CIL*, K. A. ALSHIBLI*, G. R. MCDOWELL{ and H. LI{

ICE Publishing: all rights reserved

CONTRIBUTION BY G. R. MCDOWELL AND H. LIThe article by Cil & Alshibli gives each agglomerate a singlebond strength, which is taken from a normal distribution ofmean 475 MPa and standard deviation 150 MPa (Cil &Alshibli, 2012). This gives a coefficient of variation of 0?32.

The authors stated that ‘particle size and packing densityof spherical sub-particles have minor effects on thebehaviour of the agglomerate’ but have not quantified this,nor have they quantified the effect of the orientation of theagglomerate on the strength of the agglomerate for a singlebond strength. If there is no effect, then one would expect thestrength of the agglomerate to be proportional to the (single)bond strength. The coefficient of variation for the agglom-erate strengths should therefore be the same as that for thebond strengths and equal to 0?32 (this assumes the geometryat failure is the same for each single bond strength, whichwill certainly be the case if failure occurs at the same strain,but this additional complication is ignored here and it isassumed that the geometry at failure is approximatelyunchanged). The average agglomerate strength is118?43 MPa. This would give a standard deviation inagglomerate strength of 0?32 6 118?43 5 37?89 MPa.

Figure 3 of the original paper is reproduced here asFig. 6, with a normal distribution with a mean of118?43 MPa and standard deviation of 37?89 MPa plottedin the figure along the authors’ Weibull ‘DEM fit line’. Thenormal distribution fit is a better fit to the authors’ DEMdata than their proposed Weibull distribution. Thisdemonstrates that, in this case, it is not necessary orrelevant to try and fit the data with a Weibull distributionand that the distribution of strengths is a result of theallocated normal distribution of single bond strengths.

AUTHORS’ REPLYWe thank Professor McDowell and Mr Li for theirvaluable comments and criticism. It was mentioned in theoriginal paper (Cil & Alshibli, 2012) that the minimumdiameter of the spherical sub-particles, the bondingstrength between spherical sub-particles, the fabric andshape of the agglomerate were altered to capture thevariation in tensile strength of single-particle crushingobserved in laboratory experiments. The tested silica sandparticles were fairly uniform (0?6–0?85 mm diameter) andsub-spherical to spherical in shape. Yet laboratory single-particle crushing experiments exhibited a significant varia-tion in the tensile strength. Such variation might be

attributed to structural defects, crystal structure orienta-tion, particle shape or particle–platen interaction.

To reproduce such a complex system using an agglom-erate of bonded spheres within the framework of thediscrete element method (DEM), the authors incorporatedthe influence of the parameters mentioned above. Theeffects of the parameters mentioned by McDowell and Liwere quantified by conducting a series of simulations byvarying a specific model parameter. The influence ofparticle size was examined by changing the minimumradius (Rmin) while maintaining a constant radius ratio ofRmax/Rmin to preserve the same agglomerate structure.Fabric is defined by the size (radius) and location of thesub-spheres that were used to generate the agglomerate.Two agglomerates with the same number and particle sizedistribution of sub-spheres (i.e. same packing density orvoid ratio) will have different fabrics if the coordinates ofthe sub-spheres change within the agglomerate. Theinfluence of particle size is given in Table 2, which showsthat it has a relatively small standard deviation andcoefficient of variation (cv).

Changing the fabric of the agglomerate will influence thedistribution of force chains within the agglomerate.Therefore, we believe it unnecessary to change the

Manuscript received 9 January 2013; accepted 9 January 2013.Published online at www.geotechniqueletters.com on 1February 2013.* Department of Civil & Environmental Engineering, University ofTennessee, Knoxville, TN, USA{University of Nottingham, Nottingham, UK

Table 2. Effect of particle size, fabric and void ratio (or packingdensity) on the tensile strength of agglomerates

Tensile strength: MPa

Minimum radius, Rmin: mm0?02 124?490?03 116?310?04 134?300?05 133?120?06 126?54Mean, m 126?95Standard deviation, sd 7?27Coefficient of variation, cv 0?06

Agglomerate fabricFabric 1 101?15Fabric 2 126?54Fabric 3 114?69Fabric 4 121?24Fabric 5 95?55m 111?83sd 13?16cv 0?12

Void ratio, e0?47 106?590?54 117?570?61 111?930?72 101?90m 109?50sd 6?75cv 0?06

Cil, M. B. et al. (2013) Geotechnique Letters 3, 13–15, http://dx.doi.org/10.1680/geolett.13.00004

13

orientation of the agglomerate in this case since the fabricchanges from one agglomerate to another. This has aneffect equivalent to changing the orientation of anagglomerate without changing its fabric. The influence offabric on the tensile strength of agglomerate was examinedby changing the seed of the random number generator inPFC3D. An agglomerate with a porosity of 0?35 (void ratioof 0?54) was generated and used in the simulations. Theeffect of packing density was also investigated by varyingporosity values between 0?32 and 0?42, which correspondto void ratios of 0?47 to 0?72, respectively. The results(listed in Table 2) clearly demonstrate that packing density

has a minor effect on the tensile strength of the agglomerateand that the fabric does influence its tensile strength.

The contributors state that ‘the distribution of strengthsis a result of the allocated normal distribution of singlebond strengths’. We disagree with this comment since itignored the influence of other mentioned factors andsimplified a highly complex and non-linear model based onsingle bond strength. To further investigate the effect ofonly changing the single bond strength, we conducted aseries of DEM simulations on agglomerates with identicalparticle size distribution of sub-spheres, fabric, shape andpacking density while changing the single bond strength

Digitised DEM data

2

1

03.0 3.5 4.0 4.5 5.0 5.5 6.0

ln [l

n(1/P

s)]

–1

–2

–3

3

–4

Weibull distribution fit line

Weibull distribution fitR2 = 0.918

Normal distribution fitR2 = 0.921

Normal distribution fit line

lns

Fig. 6. The comparison of Weibull distribution fit and normal distribution fit of digitised DEM simulations

ln [l

n(1/P

s)]

2

Lab testsDEM published in original paperDEM using bond strength = 475 ± 150 MPaLab tests fit lineWeibull distribution fit for DEM resultsFit line for DEM using bond strength = 475 ± 150 MPaNormal distribution fit

DEM using bond strength = 475 ± 150 MPay = 3.93× –17.94R2 = 0.970

Normal distribution fit suggested by contributorsy = 4.02× –18.16R2 = 0.968

Weibull distribution fit for DEMy = 3.30× –16.10R2 = 0.921

Lab tests fity = 3.26× –16.06R2 = 0.938

0

–2

–4

4

lns

53 6

4

–6

Fig. 7. Comparison of Weibull and normal distribution with laboratory and previous DEM results

14 Cil, Alshibli, McDowell and Li

according to a normal distribution with a mean of475 MPa and standard deviation of 150 MPa. The resultsof the analysis are depicted in Fig. 7, which demonstratesthat changing only the single bond strength is insufficientto reproduce the Weibull modulus and match strengthvariation in experimental measurements, and that it isnecessary to incorporate other parameter effects in theanalysis. We agree with the contributors’ comment thatthe normal distribution fit line yields a fit as good as the

Weibull distribution as demonstrated by the fit lines shownin Fig. 7. However, it is not enough to change only thesingle bond strength to match experimental measurements.

REFERENCE

Cil, M. B. & Alshibli, K. A. (2012). 3D assessment of fracture of sandparticles using discrete element method. Geotechnique Lett. 2, No.3, 161–166, http://dx.doi.org/10.1680/geolett.12.00024.

Discussion: 3D assessment of fracture of sand particles using discrete element method 15