# The influence of fines content and size-ratio on the micro-scale properties of dense bimodal materials

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## Abstract

This paper considers factors influencing the fabric of bimodal or gap-graded soils. Discrete element method simulations were carried out in which the volumetric fines content and the size ratio between coarse and fine particles were systematically varied. Frictionless particles were used during isotropic compression to create dense samples; the coefficient of friction was then set to match that of spherical glass beads. The particle-scale data generated in the simulations revealed key size ratios and fines contents at which transitions in soil fabric occur. These transitions are identified from changes in the contact distributions and stress-transfer characteristics of the soils and by changes in the size of the void space between the coarse particles. The results are broadly in agreement with available experimental data on minimum void ratio and contact distributions. The results have implications for engineering applications including assessment of the internal stability of gap-graded soils in embankment dams and flood embankments.

## Keywords

Gap-graded soil Internal instability Discrete element method Partial coordination numbers Void size## 1 Introduction

- (i)
S*, the critical fines content at which the fines just fill the voids between the coarse particles, and below which \(\alpha<\) 1. S* was estimated to lie between \({\hbox {F}}_\mathrm{fine}\) = 24 and 29 % for dense and loose samples respectively.

- (ii)
\({\hbox {S}}_\mathrm{max}\), the fines content at which the fines separate the coarse particles from one another, which should be no higher than \({\hbox {F}}_\mathrm{fine}\) = 35 % [4].

The concepts of \(\alpha \) and S* have implications for engineering practice and the current research was carried out to complement a broader study of the internal stability of cohesionless soils [7]. Internal stability describes the ability of the coarse fraction of a soil to prevent the erosion of the fines under seepage [8]. Kenney and Lau [9] defined three prerequisites for internal instability: (i) a primary matrix of coarse particles which transfers stresses; (ii) loose finer particles in the voids between the primary matrix, which do not carry effective stress and can be moved by seepage; (iii) the inter-void constrictions within the primary matrix must be large enough to allow the loose finer particles to be transported from void to void by seepage. Skempton and Brogan [4] found that when \({\hbox {F}}_\mathrm{fine}<\) S* internal instability can initiate at lower hydraulic gradients than would be expected to cause failure by heave (i.e. \(\alpha < 1\)). The effect of \({\hbox {F}}_\mathrm{fine}\) on internal stability has also been recently experimentally demonstrated by Sibille et al. [10]. In their DEM study Shire et al. [6] established a link between \(\alpha \) and the empirical Kézdi criterion for internal instability [11], which is based upon a size-ratio between the coarse and fine particles. Shire and O’Sullivan [12] also showed that there is a link between the Kézdi criterion and micro-scale parameters such as coordination number and To et al. [13] used DEM to study the effect of packing arrangement on the characteristics of the primary fabric. Other researchers have shown that, for a given void ratio, cohesionless fines contribute less per unit volume to shear strength, stiffness and liquefaction resistance than coarse particles [1, 3, 14, 15].

This paper considers the effect of varying \({\hbox {F}}_\mathrm{fine}\) and the size-ratio between coarse and fine particles, \(\chi = {\hbox {D}}_\mathrm{coarse}/{\hbox {D}}_\mathrm{fine}\), on the micro-scale properties of dense, homogeneous, isotropic collections of bimodal spheres using DEM. The paper examines the relationship between \({\hbox {F}}_\mathrm{fine}, \chi \) and both the void ratio and sizes of voids between the coarse particles, following which the contact and stress distributions within the samples are analysed. The results are verified using available experimental data for bimodal materials.

## 2 Modelling

### 2.1 Simulation approach and samples analysed

Simulation input parameters

Parameter | Units | Value |
---|---|---|

Poisson’s ratio, \(\nu \) | – | 0.3 |

Shear modulus, G | GPa | 27.0 |

Particle density, \(\rho \) | kg/mm\(^{-3}\) | \(2.67\times 10^{-6}\) |

Coefficient of inter-particle friction during isotropic compression, \(\mu \) | – | 0.0 |

Coefficient of inter-particle friction for results, \(\mu \) | – | 0.3 |

Following isotropic compression the coefficient of friction was set to \(\mu =0.3\), which is approximately equal to the experimental value reported for spherical glass beads [20]. To ensure a stable state was achieved the simulations were terminated once \({\hbox {p}}^{\prime }\) and the coordination number (the average number of contacts per particle in the system), Z, remained unchanged for 20,000 timesteps. All the results presented here correspond to this end state. All the simulations were carried out in a gravity-free environment to allow the use of periodic boundaries, thus removing boundary effects [21], and allowing easy identification of those particles which participate in effective stress transfer through the sample. Further details of the simulation methodology are given in Shire [7].

In selecting the samples for analysis, consideration was given to the range of \({\hbox {F}}_\mathrm{fine}\,{\hbox {and}}\,\chi \) values that merited consideration. Following the contribution of Skempton and Brogan, discussed above, the research specifically considered \({\hbox {F}}_\mathrm{fine}\) values about the critical fines content, i.e., \({\hbox {F}}_\mathrm{fine}= 20, 25, 30\) and 35 %. The definitions of \(\alpha \) and S* are based on an assumption the fines can fit within the voids formed between the coarse particles. However, there is a limit to the size ratio, \(\chi = {\hbox {D}}_\mathrm{coarse}/{\hbox {D}}_\mathrm{fine}\), at which a fine particle (with diameter \({\hbox {D}}_\mathrm{fine})\) can fit between coarse particles (with diameter \({\hbox {D}}_\mathrm{coarse})\), and below this limiting ratio granular materials cannot be considered to be gap-graded. Based on a consideration of mutually touching uniform circles, Lade et al. [18] suggested that at \(\chi \approx \) 6.5 a single fine particle can fit within the smallest possible constriction formed between three coarse particles. Ratios of \(\chi \approx 6\) have been adopted in the definition of “gap-graded” materials in studies considering the effect of non-plastic fines on soil behaviour (e.g. [3, 22, 23]). However, the smallest circle which can be inscribed between four mutually contacting circles occurs at \(\chi \approx 2.4\), and this was taken as a lower limit \(\chi \) for gap-graded soils by Choo and Burns [24]. Both these limits are based on two-dimensional considerations of inter-void constrictions. In three dimensions the largest sphere which can fit within the void body of the densest face centred cubic packing of uniform spheres (e \(=\) 0.35) occurs at \(\chi \approx 4.45\) and for the looser orthorhombic packing (e \(=\) 0.65) this occurs at \(\chi \approx 2\). It is clear from these theoretical considerations that for \(\chi \le \) 2 the material cannot be considered gap-graded. Between \(\chi = 2\,{\hbox {and}}\,\chi = 6\) there could exist an intermediate type of gap-graded behaviour. This has been shown experimentally for the variation of void ratio by Yerazunis et al. [25]. For \(\chi > 6\), when the size ratio increases, the local increase in void ratio of the fines close to coarse particle surfaces becomes less significant [26] and the fine packings between the coarse particles will progressively densify so that with increasing \(\chi \) the critical fines content, S*, increases. Consideration of these earlier results motivated use of \(\chi \) values of 2, 4, 6, 8 and 10 in the analyses.

Summary of simulations and results

Size ratio, \({\chi } = {\hbox {D}}_\mathrm{coarse}/{\hbox {D}}_\mathrm{fine}\) | Fines content, \({\hbox {F}}_\mathrm{fine}\) (%) | Number of particles | Void ratio, e | Coordination number, Z | Stress-reduction factor, \(\alpha \) |
---|---|---|---|---|---|

1 | 0 | 600 | 0.573 | 6.03 | N/A |

2 | 20 | 307 | 0.507 | 5.04 | 0.94 |

25 | 367 | 0.495 | 5.22 | 0.97 | |

30 | 443 | 0.490 | 5.62 | 1.10 | |

35 | 531 | 0.480 | 5.72 | 1.06 | |

4 | 20 | 1694 | 0.332 | 1.48 | 0.41 |

25 | 2230 | 0.327 | 5.10 | 0.86 | |

30 | 2843 | 0.337 | 5.54 | 1.05 | |

35 | 3552 | 0.351 | 5.75 | 1.17 | |

50 | 5008 | 0.394 | 5.94 | 1.12 | |

6 | 20 | 5588 | 0.256 | 0.17 | 0.27 |

25 | 7288 | 0.256 | 5.68 | 0.56 | |

30 | 9358 | 0.276 | 5.80 | 1.07 | |

35 | 11750 | 0.294 | 5.87 | 1.13 | |

8 | 20 | 13338 | 0.253 | 0.05 | 0.00 |

25 | 17211 | 0.227 | 5.67 | 0.31 | |

30 | 22043 | 0.247 | 5.89 | 1.00 | |

35 | 27713 | 0.269 | 5.93 | 1.16 | |

10 | 20 | 25376 | 0.259 | 0.03 | 0.00 |

25 | 33520 | 0.211 | 5.35 | 0.09 | |

30 | 42959 | 0.231 | 5.88 | 1.02 | |

35 | 54033 | 0.255 | 5.92 | 1.16 |

### 2.2 Calculation of \(\alpha \)

*n*: sample porosity; \({\hbox {N}}_\mathrm{p,fine}\): number of fine particles and \(\alpha \) is:

## 3 Results

### 3.1 Analysis of void space

The smallest voids in the monodisperse random packing are \({\hbox {D}}_\mathrm{void}=0.2245 \,{\hbox {D}}_\mathrm{coarse}\), which is equal to the minimum void between the densest possible regular packings (close-packed cubical and hexagonal). However, the majority of the voids are larger than this, with most (\(\sim \)70 %) falling within the range of \({\hbox {D}}_\mathrm{void}=0.3{-}0.5\,{\hbox {D}}_\mathrm{coarse}\). 15 % of the voids have \({\hbox {D}}_\mathrm{void} > 0.5\,{\hbox {D}}_\mathrm{coarse}\). When \(\chi =2, {\hbox {D}}_\mathrm{fine}=0.5 {\hbox {D}}_\mathrm{coarse}\), and therefore \({\hbox {D}}_\mathrm{fine} > {\hbox {D}}_\mathrm{void}\) for the majority of the voids, meaning that the fines will not be able to sit between the coarse particles under reduced stress, confirming that materials with \(\chi = 2\) should not be considered to be gap-graded. As \(\chi \) increases, \({\hbox {D}}_\mathrm{fine} < {\hbox {D}}_\mathrm{void}\) meaning single fines and groups of fines are able to fit more efficiently within voids. When the gap-ratio is large \((\chi =10)\) and \({\hbox {F}}_\mathrm{fine} \approx S^*\,({\hbox {F}}_\mathrm{fine}= 25\,\%)\) the void size distribution is similar to the sample containing no fines indicating that the coarse particles form a dense network very similar to that if there were no fines present.

### 3.2 Contact density

The extent to which the finer particles carry a reduced stress, i.e. the \(\alpha \) value, is influenced by the contact network, and the connectivity (i.e. number of contacts per particle) of the finer particles. Pinson et al. [19] identified contacts between coarse and fine particles in bimodal packings of spheres with \(\chi = 2\) and 4 using a liquid bridge technique. The resultant connectivity data can be compared with the DEM data generated in this study. The DEM samples are somewhat denser than the experimental samples. However, while the experimental void ratio was measured for the whole sample, connectivity was measured away from the sides of the container in order to avoid wall effects and therefore void ratio is probably overestimated in the experiments.

The distributions of connectivity for \(\chi = 2\) and 4 are given in Fig. 6. Figure 6a, b give the connectivity for fine to fine \(({\hbox {C}}^\mathrm{fine-fine})\) and fine to coarse \(({\hbox {C}}^\mathrm{fine-coarse})\) contacts respectively for \(\chi = 2\) and \({\hbox {F}}_\mathrm{fine} = 25\), 30 and 35 % for the DEM simulations. Equivalent data for \(\chi = 4\) are presented in Fig. 6c, d. For both \(\chi \) values experimental data are included in the figures; for \(\chi = 2\) the experimental data considers \({\hbox {F}}_\mathrm{fine}=28\,\%\), while experimental data for \({\hbox {F}}_\mathrm{fine}=28\) and 50 % were available for \(\chi = 4\). In all cases the experimental and DEM data show the same upper limits to the distribution of connectivities and the proportions of particles with 0 contacts are broadly similar for equivalent \({\hbox {F}}_\mathrm{fine}\) values. For \(\chi =2\) the experimental \({\hbox {C}}^\mathrm{fine-fine }\) distribution with \({\hbox {F}}_\mathrm{fine}=28\,\%\) is similar to the DEM distributions for \({\hbox {F}}_\mathrm{fine} = 30\) and 35 % (Fig. 6a), although the experimental data has a greater proportion of particles with C\(^\mathrm{fine-fine} > 4\) and fewer with \({\hbox {C}}^\mathrm{fine-fine} = 0\). The DEM sample with \({\hbox {F}}_\mathrm{fine}=25\,\%\) shows fewer fine to fine contacts per particle, specifically there are many more particles with \({\hbox {C}}^\mathrm{fine-fine}=0\) in this sample. The particles with \({\hbox {C}}^\mathrm{fine-fine}=0\) are likely to be either trapped between two coarse particles or isolated within the voids between the coarse particles. As shown in Fig. 6b, the experimental and DEM distributions of \({\hbox {C}}^\mathrm{fine-coarse}\) show good agreement for all three DEM samples despite the difference in void ratio.

As shown in Fig. 6c, for \(\chi =4,\,{\hbox {C}}^\mathrm{fine-fine}\) increases as \({\hbox {F}}_\mathrm{fine}\) attains and then exceeds the critical fines content at which the fines fill the voids. For \({\hbox {F}}_\mathrm{fine} = 50\,\%\), the \({\hbox {C}}^\mathrm{fine-fine}\) distributions are very similar for both experimental and DEM data. While the DEM data for \({\hbox {F}}_\mathrm{fine}=25\,\%\) and \({\hbox {F}}_\mathrm{fine} = 30\,\%\) show far fewer fine to fine contacts per particle than the experimental data for \({\hbox {F}}_\mathrm{fine}= 28\,\%\), there is a close agreement between the experimental data for \({\hbox {F}}_\mathrm{fine}=28\,\%\) and the DEM data for \({\hbox {F}}_\mathrm{fine} =35\,\%\).

For \({\hbox {F}}_\mathrm{fine} \ge 25\,\%\) there is little variation of Z with \(\chi \). This is because the fines completely fill the voids between the coarse particles and so have many interparticle contacts, regardless of \({\hbox {F}}_\mathrm{fine}\) or \(\chi \). With reference to Skempton and Brogan [4], for materials at their highest relative density, the critical fines content S* at which fines just fill the voids between coarse particles occurs at \({\hbox {F}}_\mathrm{fine} \approx 24\,\%\) and can be identified by an increase in Z with \({\hbox {F}}_\mathrm{fine}\).

### 3.3 Stress reduction in finer particles, \(\alpha \).

Figure 7b and Table 2 show the variation of the stress-reduction factor, \(\alpha \), with \(\chi \) and \({\hbox {F}}_\mathrm{fine}\). In samples with \({\hbox {F}}_\mathrm{fine} \ge 30\,\%, \alpha \approx 1\), indicating that the coarse and fine particles contribute approximately equally to stress transfer. The same is true of samples with \(\chi = 2\) regardless of \({\hbox {F}}_\mathrm{fine}\), as the fines are unable to completely fit within the voids. For samples with \({\hbox {F}}_\mathrm{fine} = 20\,\%\) and \(\chi \ge 6, \alpha \approx 0\) and the fines are completely loose within the voids and play almost no role in stress transfer, hence \({\hbox {F}}_\mathrm{fine}< S^*\).

The value \({\hbox {Z}}^\mathrm{coarse-coarse}< 4\) has been highlighted on Figure 8a, as when \(Z^\mathrm{coarse-coarse }< 4\) coarse particles cannot be considered to be forming a mechanically stable matrix on their own and must therefore be separated from one another by fines (i.e. be overfilled). Figure 8a shows that the fine and coarse particles carry approximately equal stress (i.e. \(\alpha \approx 1\)) in every sample with \({\hbox {Z}}^\mathrm{coarse-coarse}< 4\), confirming the hypothesis that overfilled samples must be internally stable. The stress-reduction \(\alpha \) values drop rapidly as \({\hbox {Z}}^\mathrm{coarse-coarse}\) increases beyond 4 and the coarse particles are able to form a stress-transmitting primary matrix, leaving fines transmitting little stress.

Figure 8b, considers the relationship between \(\alpha \) and the skeleton void ratio, e\(_\mathrm{sk}\) (Eq. 1). When \({\hbox {e}}_\mathrm{sk} \ge {\hbox {e}}_\mathrm{coarse,max}\), the experimental maximum void ratio for monodispersed spheres alone [34], the coarse particles must be separated from one another by fine particles and \(\alpha \approx 1\). The \({\hbox {e}}_\mathrm{sk} = {\hbox {e}}_\mathrm{coarse,max}\) condition was termed the limit void ratio by Salgado et al. [1]. who found a distinct change in the stress-strain response of silty sands when this was exceeded. Considering effective stress transfer, when \({\hbox {e}}_\mathrm{sk} < {\hbox {e}}_\mathrm{coarse,max}\) the coarse particles must be in contact with one another and therefore dominate stress transfer, as shown by \(\alpha < 1\). This effect becomes more prominent as \({\hbox {e}}_\mathrm{sk}\) reduces further below \({\hbox {e}}_\mathrm{coarse,max}\). Considering both Fig. 8a and b, it is clear that even at low fines contents (20 %) samples with \(\chi = 2\) cannot form a stable fabric comprised of coarse particles alone, whereas samples with \(\chi = 4\) form an intermediate fabric in which the coarse particles transfer more stress than fine.

As noted above the void size distributions (Fig. 5) suggest that for large \(\chi \) values and when \({\hbox {F}}_\mathrm{fine} \approx {\hbox {S}}^*\) (i.e. \(\chi = 10\) and \({\hbox {F}}_\mathrm{fine}=25\,\%)\), the coarse matrix is similar to a monodisperse material. This is confirmed by the very low stress in the fines (\(\alpha = 0.09)\). The high coordination number (Z \(=\) 5.35), suggests that this is just at the point where the fines fill the voids. As \({\hbox {F}}_\mathrm{fine}\) increases to 30 % the void diameters between the coarse particles increase noticeably as fines begin to separate them and the stress in the fines also increases to \(\alpha = 1.02\). For the samples with \(\chi = 2\) the coarse voids are significantly larger because, as discussed in Sect. 3.2, the fines are larger than the majority of the voids in the monodisperse sample. For these samples \(\alpha \approx 1\).

The variation of \({\hbox {P}}{\hbox {(strong)}}_\mathrm{fine}\) with \(\chi \) and \({\hbox {F}}_\mathrm{fine}\) is shown in Fig. 9. The pattern is similar to the relationship between \(\alpha , \chi \) and \({\hbox {F}}_\mathrm{fine}\) presented in Fig. 7b, where for \({\hbox {F}}_\mathrm{fine} \ge 30\,\%, {\hbox {P}}{\hbox {(strong)}}_\mathrm{fine} > 0.5\) and therefore fines play a significant role in supporting the fabric of the samples. For samples with \({\hbox {F}}_\mathrm{fine} \le 25\,\%, {\hbox {P}}{\hbox {(strong)}}_\mathrm{fine}\) reduces with increasing \(\chi \) as the fines are able to fit more efficiently in the voids and so are less likely to interact with the coarse particles, which dominate the strong force chains (for each sample the probability of a coarse particle forming part of a strong force chain is greater than 80 %).

This supports the hypothesis of Rahman et al. [22] that the role which cohesionless fines play in stress transfer diminishes with both \(\chi \) and \({\hbox {F}}_\mathrm{fine}\) when \({\hbox {F}}_\mathrm{fine} < {\hbox {S}}^*\), (they refer to a threshold fines content equivalent to S*). For soils with \(\chi = 2\) the concept of a threshold fines content has little meaning. For soils with \(\chi \ge 4\) care must be taken in defining this threshold content—when \({\hbox {F}}_\mathrm{fine} < {\hbox {S}}^*\) the role of the fines is primarily dependent on \(\chi \), in particular for the range \(4< \chi < 6\). However, for \({\hbox {S}}^*< {\hbox {F}}_\mathrm{fine} < {\hbox {S}}_\mathrm{max}\) the fines play a lesser role in stress transfer and this is dependent on both \(\chi \) and \({\hbox {F}}_\mathrm{fine}\) as shown in Fig. 9 for \({\hbox {F}}_\mathrm{fine} = 25\,\%\). An added complication is also that the values of S* and \({\hbox {S}}_\mathrm{max}\) are density-dependent [6].

## 4 Conclusions

- (a)
The DEM data are physically reasonable. It was found that the minimum void ratio which can be obtained by bimodal samples falls as the size-ratio, \(\chi = {\hbox {D}}_\mathrm{coarse}/{\hbox {D}}_\mathrm{fine}\) increases, in agreement with experimental results presented by Lade et al. [18]. The connectivity distributions for fine-fine and fine-coarse contacts obtained from DEM simulations are broadly similar to the experimental results of Pinson et al. [19].

- (b)
The size ratios \((\chi )\) for which a soil can be considered gap-graded was considered. Bimodal soils with a size ratio of \(\chi \le 2\) have stress transfer and coordination number characteristics that are similar to those for uniform soils, as the fines cannot completely fit between the coarse particles. Fine and coarse particles play an approximately equal role in stress transfer \((\alpha \approx 1)\) regardless of \({\hbox {F}}_\mathrm{fine}\). Such soils should not be considered gap-graded in terms of stress-transmission.

- (c)
For \(\chi \ge 6\) particle-scale evidence is shown for two critical fines contents at which transitions in soil fabric occur:

*(i)*S*, at which the fines just fill the voids between coarse particles can be seen by an increase in the coordination number, Z;*(ii)*\({\hbox {S}}_\mathrm{max}\), where the fines begin to separate the coarse particles from one another is distinguished by a reduction in the coarse to coarse coordination number to \({\hbox {Z}}^\mathrm{coarse-coarse}< 4\). At the macro-scale this transition can be demonstrated by the skeleton void ratio, \({\hbox {e}}_\mathrm{sk}\), increasing above \({\hbox {e}}_\mathrm{coarse,max}\). For \(\chi \ge 6\) stress transfer is dependent on both \({\hbox {F}}_\mathrm{fine}\) and \(\chi \). In particular: (i) when \({\hbox {F}}_\mathrm{fine}<\) S* the fines play only a minor role in stress transfer and \(\alpha \approx 0\); (ii) when \({\hbox {F}}_\mathrm{fine} > {\hbox {S}}_\mathrm{max}\) the coarse and fine play approximately equal roles and \(\alpha \approx 1\); (iii) when \({\hbox {S}}^*<{\hbox {F}}_\mathrm{fine} < {\hbox {S}}_\mathrm{max}\) an increase in \({\hbox {F}}_\mathrm{fine}\) leads to an increase in \(\alpha \), and an increase in \(\chi \) leads to a reduction in \(\alpha \). - (d)
For \(\chi = 4\) at low fines contents behaviour which is intermediate between \(\chi = 2\) and \(\chi = 6\) is observed, with fines playing a reduced but significant role in stress transfer.

- (e)
When \({\hbox {F}}_\mathrm{fine} = 25\,\%\) the fines have a similar coordination number to the samples with \({\hbox {F}}_\mathrm{fine} \ge 30\,\%\) but \(\alpha \) is lower. This shows that the fines play a supporting role to the coarse primary, stress-transmitting, matrix.

- (f)
The influence of \({\hbox {F}}_\mathrm{fine}\) and \(\chi \) on \(\alpha \) and on \({\hbox {P}}{\hbox {(strong)}}_\mathrm{fine}\), the probability of a fine particle forming part of a strong force chain, are similar.

- (g)
Measurement of the void size distribution between the coarse particles shows that as more fine particles form part of the stress-transmitting matrix the size of voids between coarse particles increases. When \(\alpha \approx 0\) the void size distribution reduces to that for a monodisperse sample. This occurs when \(\chi \ge 6\) and \({\hbox {F}}_\mathrm{fine} \le 25\,\%\).

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