Peter Adesina
email: peter.adesina@inrae.fr

Antoine Wautier
email: antoine.wautier@inrae.fr

Nadia Benahmed
email: nadia.benahmed@inrae.fr

A MICROMECHANICS-BASED CLASSIFICATION OF THE REGIMES DELINEATING THE BEHAVIOUR OF GAP-GRADED SOILS

Keywords: Gap-graded assemblies, regimes, density, triaxial compression, DEM

This study presents a micromechanical evaluation of the regimes delineating the behaviour of gap-graded granular assemblies, using discrete element simulations. Dense and loose bimodal assemblies of different fines content were prepared and subjected to drained triaxial compression until the critical state was reached. The regimes delineating the behaviour of the assemblies were evaluated, characterised and their significance discussed. While two regimes demarcated by the threshold fines content were identified based on the analysis of the macroscale characteristics of the assemblies, up to four regimes were identified based on the contributions of the particle size fractions and contact types to the total mean stress. Contrary to previous studies according to which fines control the mechanical behaviour of gap-graded assemblies from the threshold fines content, 𝑓 𝑐 𝑡ℎ , we found that the fines do not play a primary role in stress transmission until beyond a significantly larger fines content, 𝑓 𝑐 𝑒𝑞 (the equivalent fines content), which depends on density and stress state. Based on the correlation found between the critical state strength and the stress-based skeleton void ratio proposed in this study, we conclude that stress-based skeleton void ratio can be useful in understanding the mechanical response of gap-graded materials at the critical state.

Introduction

Soil mixtures containing a coarser and a finer fraction (as in silty sands, sandy gravels, etc.) are abundant in nature. It is well established that the mechanical behaviour of these mixtures depends on the proportions of the coarser and the finer fraction within the mixtures. However, there is no consensus in the literature on specific fine contents delineating different behaviour for gap-graded granular mixtures. For example, as a result of the conceptual analysis presented in experimental studies, it has been suggested that the fines play a primary role in the shear behaviour of gap-graded assemblies beyond a threshold fines content [START_REF] Thevanayagam | Undrained Fragility of Clean Sands, Silty Sands, and Sandy Silts[END_REF]; and control the behaviour beyond a limiting fines content [START_REF] Lade | Effects of nonplastic fines on static liquefaction of sands[END_REF][START_REF] Salgado | Shear Strength and Stifness of Silty Sand[END_REF]Skempton & Brogan, 1994;[START_REF] Vallejo | Interpretation of the limits in shear strength in binary granular mixtures[END_REF]. The threshold fines content is usually related to geometric properties, and in particular the measure of the minimum void ratio [START_REF] Cubrinovski | Maximum and minimum void ratio characteristics of sands[END_REF][START_REF] Yang | Determination of the Transitional Fines Content of Mixtures of Sand and Non-plastic Fines[END_REF][START_REF] Zuo | Determination of the transitional fines content of sand-non plastic fines mixtures[END_REF]. From the threshold fines content, the fines start to disperse the coarse particles. The limiting fines content corresponds to the situation where the fines completely disperse the coarse particles (floating grains in a fine matrix) (Skempton & Brogan, 1994;[START_REF] Thevanayagam | Undrained Fragility of Clean Sands, Silty Sands, and Sandy Silts[END_REF]. Since quantifying the stress or the dispersion of individual particles within a matrix is difficult in experiments, it is almost impossible to ascertain the delineating fines content of the conceptual models without the use of a micromechanical framework.

In this regard, the discrete element method (DEM) proposed by [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF] has been found an effective numerical tool for probing the micromechanics of granular materials.

DEM offers the opportunity to evaluate the assertions from the conceptual models in earlier experimental studies where the stress transmitted by individual particles cannot be determined.

In the earlier studies where DEM has been used to assess the behaviour of gap-graded assemblies, the approach adopted involved determining the proportion of the stress transmitted by each particle size fraction and contact types. However, no consensus has been reached on the specific fines content delineating different behaviour; in fact, the existence of some of the specific fines contents suggested by experimental studies is sometimes questioned (as in [START_REF] Sufian | Influence of Fabric on Stress Distribution in Gap-Graded Soil[END_REF]) or not identified.

In this study, we conducted four analyses organised into four sections to evaluate the regimes 4 delineating the behaviour of gap-graded sand-silt mixtures. In Section 3, we determined the macroscale and micromechanical behaviour of the assemblies of sand-silt mixtures with fines content, 𝑓 𝑐 , within 10%≤ 𝑓 𝑐 ≤70%; and assessed the existence of regimes delineating the characteristics of the assemblies. The macroscale characteristics considered are the void ratio, strength and dilatancy, while the micromechanical characteristic considered is the coordination number. We then conducted in Section 4, a particle-scale analysis of the contribution of the finer and the coarser fractions to the total mean stress transmitted by an assembly. This enabled a particle stress-based evaluation of the regimes delineating the behaviour of the assemblies. Following this, in Section 5, we conducted a contact-scale analysis of the contribution of each contact type to the total mean stress, again to identify existing regimes. Finally, in Section 6, we present a stress-based skeleton void ratio as an alternative void ratio index to interpret the strength properties of the granular mixtures studied. We assessed the performance of alternative void ratio indexes such as the mechanical and the void ratio (global), in interpreting the strength exhibited by the granular mixtures studied, at both the peak and the critical state. An assessment of the internal instability of the gap-graded assemblies (i.e. their susceptibility to internal erosion) is beyond the scope of this study. However, the implications of the findings with respect to internal erosion are discussed in the conclusion.

Numerical Simulation approach

The DEM simulations in this study were conducted using the open-source code YADE (Šmilauer et al, 2021). All assemblies were generated to have a bimodal grading of fine particles with diameter, 𝐷 𝑓 =0.375 mm and coarse particles with diameter 𝐷 𝑐 =3.075 mm such that the size ratio, 𝜆 = 𝐷 𝑐 /𝐷 𝑓 = 8.2. The bimodal grading is considered the simplest type of gapgraded material and was employed in this study to ensure the results obtained are solely from the interaction between the coarser and the finer fractions without the interference of the grading effect from within either of the fractions. The size ratio value was selected sufficiently large to allow for fine migration in the pore space (should water flow be considered) [START_REF] Lade | Effects of Non-Plastic Fines on Minimum and Maximum Void Ratios of Sand[END_REF][START_REF] Rahman | On equivalent granular void ratio and steady state behaviour of loose sand with fines[END_REF][START_REF] Shire | The influence of fines content and sizeratio on the micro-scale properties of dense bimodal materials[END_REF][START_REF] Thevanayagam | Undrained Fragility of Clean Sands, Silty Sands, and Sandy Silts[END_REF].

After a parametric study to determine the representative element volume (REV), each assembly contains 100,000 particles (Fig. 1). Increasing the sample size to 200,000 particles for the 𝑓 𝑐 = 30% case at both the dense and loose states did not significantly influence the stress-strain responses (see Appendix A). The slight sensitivity to sample size observed in the volumetric strains was also reported in the DEM study on REV for granular materials by [START_REF] Adesina | Determining a representative element volume for DEM simulations of samples with non-circular particles[END_REF] and can be attributed to the small variation in the initial void ratio of the assemblies.

Table 2 shows the number of the fines and the coarse particles for each percentage of fines content, 𝑓 𝑐 (%).

The assemblies studied here were prepared by subjecting them to isotropic compression at 100 kPa within six frictionless walls. The standard linear elasto-plastic model was employed to simulate the interactions between particles. Table 1 shows the simulation parameters employed in the simulations. Following [START_REF] Jiang | The influence of particle-size distribution on critical state behavior of spherical and non-spherical particle assemblies[END_REF], the material density in the simulations was scaled to 1000 times the original value in order to increase the critical time step, ∆𝑡 𝑐𝑟 ∝

𝑟√𝜌 𝐸

⁄ , where 𝑟 is the particle radius, 𝜌 is the density and 𝐸 is the Young's modulus. A fraction (0.9∆𝑡 𝑐𝑟 ) of this critical time step was then applied to ensure numerical stability. While this density scaling approach was adopted for computational efficiency, it does not compromise the quasi-static conditions of the simulations as long as the inertial number remains sufficiently small [START_REF] Thornton | Numerical simulations of deviatoric shear deformation of granular media[END_REF][START_REF] Thornton | Quasi-static deformation of particulate media[END_REF]. Based on the standard practice adopted in prior DEM studies including [START_REF] Thornton | Numerical simulations of deviatoric shear deformation of granular media[END_REF] for assembly preparation, here, assemblies of different initial densities were generated by using a friction coefficient µ=0.03 for the dense assemblies and µ=0.5 for the loose assemblies, during isotropic compression. The void ratio of the dense assemblies is denoted 𝑒 𝑚𝑖𝑛 , while the void ratio of the loose assemblies is denoted as 𝑒 𝑚𝑎𝑥 . It is important to note that these 𝑒 𝑚𝑖𝑛 and 𝑒 𝑚𝑎𝑥 values cannot be directly mapped to the values obtained using standard procedures employed in experiments.

After isotropic compression, the assemblies were subjected to triaxial compression under a constant confining pressure of 𝜎 𝑥𝑥 =𝜎 𝑦𝑦 =100 kPa5 in both the 𝑥-direction and 𝑦-direction and a strain rate of 0.01 s -1 in the 𝑧-direction. A numerical damping coefficient of 0.05 is used.

These conditions ensured that for our simulations, the inertial number for the coarse particles, 𝐼 𝑑𝑚𝑎𝑥 = 1.60 x 10 -4 , and that for the fines, 𝐼 𝑑𝑚𝑖𝑛 = 1.95 x 10 -5 , therefore the simulations were deemed quasi-static [START_REF] Da Cruz | Rheophysics of dense granular materials : Discrete simulation of plane shear flows[END_REF]. Prior to shearing, additional number of cycles at µ=0.5 was applied to equilibrate the assemblies. Also, the unbalanced force ratio6 was checked to remain below a limit of 0.01 to guarantee static equilibrium. All assemblies were sheared at µ=0.5 regardless of the friction coefficient used during the assembly preparation stage. A total of 22 shearing simulations were conducted in this study (11 fines contents are considered for 2 relative densities) (Table 2). The initial void ratios of the gap-graded assemblies with different fines content are plotted in Fig. 2a. As earlier mentioned, the void ratio of the dense and loose assemblies studied are denoted as 𝑒 𝑚𝑖𝑛 and 𝑒 𝑚𝑎𝑥 , respectively. In both the dense and the loose assemblies, two regimes of distinct trends are observed. The first regime is characterised by a reduction in the void ratio as 𝑓 𝑐 increases. This is as a result of the fines progressively filling the voids within the coarse particles. At a certain fines content referred to as the critical fines content (Skempton & Brogan, 1994), the threshold fines content, 𝑓 𝑐 𝑡ℎ [START_REF] Lade | Effects of Non-Plastic Fines on Minimum and Maximum Void Ratios of Sand[END_REF][START_REF] Thevanayagam | Undrained Fragility of Clean Sands, Silty Sands, and Sandy Silts[END_REF], or the transitional fines content [START_REF] Yang | Determination of the Transitional Fines Content of Mixtures of Sand and Non-plastic Fines[END_REF], the voids within the coarse particles are fully filled by the fines. The second regime corresponds to 𝑓 𝑐 >𝑓 𝑐 𝑡ℎ and it is characterised by an increase in the void ratio as the 𝑓 𝑐 increases. In this regime, the fines progressively disperse the coarse particles, thereby causing an increase in the void ratio. These two regimes have been reported

in previous studies involving gap graded materials [START_REF] Cubrinovski | Maximum and minimum void ratio characteristics of sands[END_REF][START_REF] Kuerbis | Effect of gradation and fines content on the undrained response of sand[END_REF][START_REF] Lade | Effects of nonplastic fines on static liquefaction of sands[END_REF][START_REF] Li | Study of shear induced stress redistribution in gap-graded soils by discrete element method[END_REF]Y. Li et al., 2022;[START_REF] Minh | Strong force networks in granular mixtures[END_REF][START_REF] Sufian | Influence of Fabric on Stress Distribution in Gap-Graded Soil[END_REF][START_REF] Vallejo | Interpretation of the limits in shear strength in binary granular mixtures[END_REF][START_REF] Zuo | Determination of the transitional fines content of sand-non plastic fines mixtures[END_REF]. The two identified regimes are referred to as the underfilled and the overfilled categories by dam engineers (ICOLD, 2013;[START_REF] Shire | Fabric and Effective Stress Distribution in Internally Unstable Soils[END_REF], and are demarcated by the threshold fines content (i.e. the filled state).

One particular feature that is less frequently highlighted in the literature is the fact that the 𝑓 𝑐 𝑡ℎ slightly depends on the relative density of the mixture. The 𝑓 𝑐 𝑡ℎ here was attained at 𝑓 𝑐 = 0.30

for the dense assemblies whereas 𝑓 𝑐 𝑡ℎ = 0.35 for the loose assemblies. Indeed, in looser assemblies, the pores are larger and more fine grains are required to fill the voids before the coarse grains are dispersed. This is in agreement with the 3D DEM study of binary mixtures by [START_REF] Minh | Strong force networks in granular mixtures[END_REF]. In the literature, different limits within which the threshold fines content is attainable have been reported. For example, Skempton & Brogan (1994) suggested that in practice, 𝑓 𝑐 𝑡ℎ is unlikely to occur beyond 24% and 29% for dense and loose packings of sandygravels, respectively. Lade & Yamamuro (1997) suggested a limit of 20%-30% fines content for sand-silt mixtures of different gradations. Other studies show that the 𝑓 𝑐 𝑡ℎ can occur outside Skempton and Brogan's and Lade & Yamamuro's limits, depending on the grain size distribution and the particle shape considered [START_REF] Evans | Liquefaction Behaviour of Sand-Gravel Composites[END_REF][START_REF] Sarkar | Influence of particle shape and size on the threshold fines content and the limit index void ratios of sands containing non-plastic fines[END_REF][START_REF] Shire | Fabric and Effective Stress Distribution in Internally Unstable Soils[END_REF][START_REF] Sufian | Influence of Fabric on Stress Distribution in Gap-Graded Soil[END_REF][START_REF] Wang | Updated skeleton void ratio for gravelly sand mixtures considering the effect of grain size distribution[END_REF][START_REF] Zuo | Determination of the transitional fines content of sand-non plastic fines mixtures[END_REF].

The conceptual distinction between underfilled and overfilled assemblies led [START_REF] Yin | A micromechanics-based model for sand-silt mixtures[END_REF] to propose the following equation for determining the void ratio, 𝑒, for sand-silt mixtures at different fines content 𝑓 𝑐 :

𝑒 = [𝑒 ℎ𝑐 (1 -𝑓 𝑐 ) + 𝑎𝑓 𝑐 ] 1-𝑡𝑎𝑛ℎ[𝜉(𝑓 𝑐 -𝑓 𝑡ℎ )] 2 + 𝑒 ℎ𝑓 (𝑓 𝑐 + 1-𝑓 𝑐 (𝑅 𝑑 ) 𝑚 ) 1+𝑡𝑎𝑛ℎ[𝜉(𝑓 𝑐 -𝑓 𝑡ℎ )] 2 (1)
where 𝑒 ℎ𝑐 is the void ratio of the pure sand, 𝑒 ℎ𝑓 is the void ratio of the pure silt , 𝑎 is a material constant depending on the fabric structure of the soil mixture, 𝜉 is a material constant controlling the transition from a coarse grain matrix to a fine grain matrix, 𝑓 𝑡ℎ is the threshold fines content at which the coarse and fine grains contribute equally to the global void ratio, 𝑅 𝑑 is the ratio of the mean size of the coarse grains, 𝐷 50 , to the mean size of the fine grains 𝑑 50

and 𝑚 (0 < 𝑚 < 1) is a coefficient that depends on grain characteristics and fine grain packing.

The fitting parameters for the dense assemblies are 𝑒 ℎ𝑐 = 0.67;𝑎 = -0.64; 𝜉 = 16.6; 𝑓 𝑡ℎ = 0.28; 𝑒 ℎ𝑓 =0.62; m=0.73 while the parameters for the loose assemblies are 𝑒 ℎ𝑐 = 0.71;𝑎 = 0.03; 𝜉 = 12.6; 𝑓 𝑡ℎ = 0.26; 𝑒 ℎ𝑓 =0.76; m=0.99. It is clear, as shown in Fig. 2a, that this phenomenological equation fits our data. 𝑒 𝑚𝑖𝑛 for the underfilled assemblies (i.e. 𝑓 𝑐 <𝑓 𝑐 𝑡ℎ = 35%) is generally higher in comparison to the overfilled assemblies. In the underfilled assemblies, we observed an initial increase in 𝑒 𝑚𝑎𝑥 -𝑒 𝑚𝑖𝑛 with the fine content which indicates that the fines do not simply fill the voids in the loose case. Indeed, they enable the formation of larger pore structures.

Fig. 3 shows the evolution of the stress ratio, 𝑞/𝑝, and the volumetric strain (𝜖 𝑣𝑜𝑙 ), against the axial strain (𝜖 𝑎 ), for the dense assemblies of binary mixtures with different fines content. The corresponding data for the loose assemblies are presented in Fig. 4. The assemblies were sheared until 𝜖 𝑎 ≈ 0.5 in order to reach the critical state. Here, the deviatoric stress, 𝑞 = 𝜎 𝑧𝑧 -(𝜎 𝑦𝑦 + 𝜎 𝑥𝑥 ) 2 ⁄ and the mean stress, 𝑝 = (𝜎 𝑧𝑧 + 𝜎 𝑦𝑦 + 𝜎 𝑥𝑥 )/3 (axisymmetric conditions). In a typical fashion to sand behaviour, the 𝑞/𝑝 for the initially dense assemblies increased until reaching a peak and thereafter softened to the critical state where the 𝑞/𝑝 fluctuates around a mean value, with more pronounced fluctuations observed for 𝑓 𝑐 < 45% (Fig. 3a &b). The stiffness of the packings as influenced by the fines content is shown in the inset of Fig. 3a-c.

These dense assemblies exhibited a dilative volumetric response ( 𝜖 𝑣𝑜𝑙 < 0) during shearing (Fig. 3d-f). At large strains (𝜖 𝑎 > 35%), 𝜖 𝑣𝑜𝑙 tend to increase from 𝑓 𝑐 = 10% to 𝑓 𝑐 = 35%

(indicating a less dilative behaviour), and then decreased with further increase in the 𝑓 𝑐 . In contrast, at small strains (𝜖 𝑎 < 2%) (inset of Fig. 3d-f), a more dilative response was observed from 𝑓 𝑐 = 10% to 𝑓 𝑐 𝑡ℎ =30% as 𝜖 𝑣𝑜𝑙 decreases. The stress ratio 𝑞/𝑝 for the initially loose assemblies (Fig. 4a-c) increased monotonically from the beginning of shearing to the critical state without significant softening observed, as expected. These loose assemblies exhibited a contractive response (𝜖 𝑣𝑜𝑙 > 0) during shearing (Fig. 4d-f). In Fig. 5a, the stress ratio 𝑞/𝑝 at the peak, (𝑞/𝑝) 𝑝𝑒𝑎𝑘 , is plotted against 𝑓 𝑐 alongside the void ratio at the peak, 𝑒 𝑝𝑒𝑎𝑘 , for comparison. Similarly, Fig. 5b shows the plot of the critical state stress ratio, (𝑞/𝑝) 𝑐𝑟𝑖𝑡 , against 𝑓 𝑐 alongside the critical state void ratio, 𝑒 𝑐𝑟𝑖𝑡 . Here, the (𝑞/𝑝) 𝑝𝑒𝑎𝑘 is the maximum 𝑞/𝑝 attained between the start of shearing and the critical state;

(𝑞/𝑝) 𝑐𝑟𝑖𝑡 and 𝑒 𝑐𝑟𝑖𝑡 are mean values determined from the start of the critical state to the end of shearing. The critical state marks the state at which there is no noticeable change in the void ratio or the volume of a sample during shearing. This generally occurred at 𝜖 𝑎 > 0.35 for the studied assemblies. As the fines progressively fill the voids within the coarse particles in the underfilled regime where 𝑓 𝑐 < 𝑓 𝑐 𝑡ℎ , the assemblies become denser such that (𝑞/𝑝) 𝑝𝑒𝑎𝑘 increases with 𝑓 𝑐 until the threshold fines content where the highest (𝑞/𝑝) 𝑝𝑒𝑎𝑘 is attained (Fig. 5a). For 𝑓 𝑐 >𝑓 𝑐 𝑡ℎ , the fines disperse the coarse particles leaving more voids within the assemblies thereby causing a monotonic decrease in (𝑞/𝑝) 𝑝𝑒𝑎𝑘 as 𝑓 𝑐 increased.

In agreement with earlier experimental studies on Ottawa sand and Carmague silty sand with 𝑓 𝑐 ≤ 20% under drained shearing [START_REF] Benahmed | An experimental investigation into the effects of low plastic fines content on the behaviour of sand/silt mixtures[END_REF][START_REF] Chang | Micromechanical modeling for behavior of silty sand with influence of fine content[END_REF][START_REF] Salgado | Shear Strength and Stifness of Silty Sand[END_REF], a marginal increase in (𝑞/𝑝) 𝑐𝑟𝑖𝑡 was observed from 𝑓 𝑐 = 10% to 𝑓 𝑐 = 20%. This is followed by a significant decrease in (𝑞/𝑝) 𝑐𝑟𝑖𝑡 until 𝑓 𝑐 = 40%, and finally a monotonic increase until 𝑓 𝑐 = 70% (Fig. 5b). The 𝑒 𝑐𝑟𝑖𝑡 follows a similar trend as the 𝑒 𝑝𝑒𝑎𝑘 , although the threshold fines content shifted from 𝑓 𝑐 𝑡ℎ =30% at the initial state (Fig. 5a) to 𝑓 𝑐 𝑡ℎ =35% at the critical state (Fig. 5b). While 𝑒 𝑝𝑒𝑎𝑘 could be used to explain the trend observed for (𝑞/𝑝) 𝑝𝑒𝑎𝑘 , (𝑞/𝑝) 𝑐𝑟𝑖𝑡 does not correlate with 𝑒 𝑐𝑟𝑖𝑡 . In Section 6, we provide a discussion on the use of an alternative void index, the stress-based skeleton void ratio, to explain the trend observed at the critical state. It is worthy of note that having a lot of fine grains filling the void space increases the peak strength but decreases the critical state strength. This might be interpreted in the capacity of fine grains to i) provide lateral support to force chains if the initial state is dense, but at the same time ii) act as ball bearings at critical state and ease shearing with limited deviatoric stress. Fig. 6a shows the maximum dilatancy rate, (-d𝜀 𝑣 d𝜀 𝑎 ) ⁄ 𝑚𝑎𝑥 against 𝑓 𝑐 . In the underfilled regime, (-d𝜀 𝑣 d𝜀 𝑎 ) ⁄ 𝑚𝑎𝑥 for the dense assemblies increased from 𝑓 𝑐 = 10% until the threshold fines content 𝑓 𝑐 𝑡ℎ =30%, as the fines progressively fill the void space (Fig. 6a). For 𝑓 𝑐 >𝑓 𝑐 𝑡ℎ , (-d𝜀 𝑣 d𝜀 𝑎 ) ⁄ 𝑚𝑎𝑥 decreased monotonically with an increase in 𝑓 𝑐 . The increase in dilatancy observed within the underfilled regime was also reported for Ottawa sand having 𝑓 𝑐 ≤ 20%, under drained triaxial shearing [START_REF] Chang | Micromechanical modeling for behavior of silty sand with influence of fine content[END_REF][START_REF] Salgado | Shear Strength and Stifness of Silty Sand[END_REF]; and was attributed to increased particle interlock as the fines occupy the voids within the coarse particles [START_REF] Kuerbis | Effect of gradation and fines content on the undrained response of sand[END_REF][START_REF] Salgado | Shear Strength and Stifness of Silty Sand[END_REF]. Indeed, for the assemblies of binary mixture studied here, there is a correlation between the maximum dilatancy angle, 𝜓 𝑚𝑎𝑥 and the initial void ratio, 𝑒 𝑚𝑖𝑛 , (Fig. 6b), where

𝜓 𝑚𝑎𝑥 = sin -1 [((-d𝜀 𝑣 d𝜀 𝑎 ) ⁄ 𝑚𝑎𝑥 ) (2 + (-d𝜀 𝑣 d𝜀 𝑎 ) ⁄ 𝑚𝑎𝑥 ) ⁄
] following [START_REF] Xiao | Strength-dilatancy relation of sand containing non-plastic fines[END_REF]. A linear relationship was also established between the peak friction angle, 𝜙 𝑝 and the 𝜓 𝑚𝑎𝑥 (Fig. 6c), where the friction angle, 𝜙 = sin -1 [(𝜎 𝑧𝑧 -𝜎 𝑦𝑦 ) (𝜎 𝑧𝑧 + 𝜎 𝑦𝑦 ) ⁄ ]. This relationship has been found in earlier studies for clean sands [START_REF] Adesina | Understanding the interplay between particle shape, grading and sample density on the behaviour of granular assemblies: A DEM approach[END_REF][START_REF] Bolton | The strength and dilatancy of sands[END_REF][START_REF] Vaid | The strength and dilatancy of sand[END_REF], and was also reported for sand-silt mixtures having 𝑓 𝑐 ≤ 20% in the experimental study on sand with non-plastic fines [START_REF] Xiao | Strength-dilatancy relation of sand containing non-plastic fines[END_REF].

Fig. 6 (a) Effect of fines content, 𝑓 𝑐 , on the maximum dilatancy rate, (-d𝜀 𝑣 d𝜀 𝑎 ⁄ ) 𝑚𝑎𝑥 .

Relationship between (b) maximum dilatancy angle, 𝜓 𝑚𝑎𝑥 and initial void ratio, 𝑒 𝑚𝑖𝑛 (c) peak friction angle, 𝜙 𝑝 , and maximum dilatancy angle, 𝜓 𝑚𝑎𝑥 for dense assemblies of binary mixture.

The coordination number, i.e. the average number of contacts per particle in a granular system is the simplest measure of the connectivity of the contact network which is related to the structural stability of the system [START_REF] Thornton | Granular Dynamics, Contact Mechanics and Particle System Simulations: A DEM study[END_REF]. Here, we employ the mechanical coordination number as a micromechanical parameter to identify and characterise the regimes delineating the behaviour of gap-graded materials. The mechanical coordination number, 𝑍 𝑚 , is defined as the average number of contacts per particle when the particles which do not contribute to force transmission (i.e., particles with less than two contacts which are referred to as rattlers) are excluded7 . [START_REF] Thornton | Numerical simulations of deviatoric shear deformation of granular media[END_REF] defined the 𝑍 𝑚 as:

𝑍 𝑚 = (2𝑁 𝑐 -𝑁 1 ) 𝑁 𝑝 -𝑁 1 -𝑁 0 (2)
where 𝑁 𝑐 is the total number of contacts, 𝑁 𝑝 is total number of particles and 𝑁 0 and 𝑁 1 are the number of particles with zero and one contact in the granular system, respectively. Figs. 7a-c show 𝑍 𝑚 for all particles, 𝑍 𝑚𝑐 for the coarse particles only, and 𝑍 𝑚𝑓 for the fine particles only, respectively, alongside the void ratio, 𝑒, at the initial, the peak and the critical states. While 𝑍 𝑚 considers all particles, 𝑍 𝑚𝑐 and 𝑍 𝑚𝑓 considers only the contacts made by the coarse particles and the fine particles, respectively 8 . Figs. 7d-e show the number of fines and coarse particles that do not contribute to force transmission (i.e. the fines-rattlers and coarse-rattlers) at the initial, the peak and the critical states. Generally, two trends demarcated by the 𝑓 𝑐 𝑡ℎ can be observed in Fig. 7a; a decrease in the 𝑍 𝑚 for 𝑓 𝑐 <𝑓 𝑐 𝑡ℎ (although the decrease is less significant for the dense assemblies at the initial state); and a monotonic increase for 𝑓 𝑐 >𝑓 𝑐 𝑡ℎ . The trends observed here are similar to the data shown in the DEM study by [START_REF] Liu | A semi-empirical reevaluation of the influence of state on elastic stiffness in granular materials[END_REF] for the 𝑍 𝑚 of sand-silt mixtures having particle size ratio, 𝜆 = 8.4 and 18.1, and at the initial states.

Generally, an initial increase in the 𝑓 𝑐 resulted in no change in the 𝑍 𝑚𝑐 and the 𝑍 𝑚𝑓 until 𝑓 𝑐 𝑡ℎ , after which a monotonic increase followed (Figs. 7b &c). The monotonic decrease in the number of fines-rattlers from 𝑓 𝑐 𝑡ℎ (Fig. 7d) indicates an increase in the number of fines mobilised in force transmission as 𝑓 𝑐 increases. As expected, the number of coarse-rattlers decreased with an increase in 𝑓 𝑐 until 𝑓 𝑐 𝑡ℎ , beyond which no rattler exists among the coarse particles (Fig. 7e). where 𝑍 𝑖𝑛𝑖 and 𝑍 𝑐𝑟𝑖𝑡 are the coordination number at the initial states and at the critical state, respectively. Fig. 8b shows the excess friction angle, which indicates the strain softening exhibited by the assemblies. From these figures, we observe the emergence of three groups of 𝑓 𝑐 values with similar ∆𝑍 and 𝜙 𝑝 -𝜙 𝑐 values. This suggests the existence of three regimes, the first being 𝑓 𝑐 < 𝑓 𝑐 𝑡ℎ where ∆𝑍 ≈ 0 and strain softening is relatively minimal. In the second regime delineated by 𝑓 𝑐 𝑡ℎ ≤ 𝑓 𝑐 < 45%, ∆𝑍 is highest, suggesting that the highest amount of contacts were lost within this regime during shearing. This is accompanied by significant softening occurring within the regime, in agreement with the DEM study by [START_REF] Adesina | Understanding the interplay between particle shape, grading and sample density on the behaviour of granular assemblies: A DEM approach[END_REF], where a linear relationship is established between the degree of softening and the contact loss during shearing by linearly-graded granular assemblies. The significant loss of contacts exhibited by the assemblies in this regime can be linked to the large fluctuations in their shearing responses at the critical state (see Figs. 3 & 4). It has been shown in the DEM study by [START_REF] Adesina | Determining a representative element volume for DEM simulations of samples with non-circular particles[END_REF] that the magnitude of fluctuations in the shearing responses of granular assemblies at the critical state increases with a decrease in the total number of contacts in the assemblies. This observation can also be related to the size of the REV, which depends on the fine contents. While the condition is largely met for small and large fine contents with 100,000 grains, this is less the case for intermediate fine contents in which local microstructure rearrangements reflect more in the macroscopic response. This is observed in Fig. 16 through the evolution of the boundary term in the stress decomposition. In Appendix A, we showed that our sample size of 100,000 particles can be considered as a REV.

In the third regime (i.e. 𝑓 𝑐 ≥ 45%), relatively minimal ∆𝑍 is observed; the strain softening within this regime transitions from moderate to minimal. ∆𝑍 values in the last two regimes are higher for dense assemblies in comparison to the loose assemblies, for the same 𝑓 𝑐 value; ∆𝑍<0

for the loose assemblies in regime 3 indicating an ultimate contact gain. While Skempton & Brogan (1994) and [START_REF] Thevanayagam | Undrained Fragility of Clean Sands, Silty Sands, and Sandy Silts[END_REF] provided a conceptual delineation of the regimes based on void ratio, here, we provide a mechanistic delineation of the regimes with our analysis of contact evolution and strain softening. In Table 3, we present a summary of the unique trends observed for each of the three regimes identified based on our analysis of the macromechanical and the micromechanical characteristics of the studied gap-graded assemblies. Note: When the data in Fig. 8 are considered, the boundary between Regime 2 and Regime 3 range between 𝑓 𝑐 ∈ [35% ; 45%] depending on density. This range will also depend on the particle size distribution considered.

Particle based stress-transmission

Using the Love-Weber particle stress tensor as described in [START_REF] Nicot | On the definition of the stress tensor in granular media[END_REF], the mean stress per particle, 𝜎 𝑖𝑗 𝑝 , is calculated as:

𝜎 𝑖𝑗 𝑝 = 1 𝑉 𝑝 𝑅 𝑝 ∑ 𝑓 𝑗 𝑐 𝑁 𝑐 𝑝 𝑐=1 𝒏 𝑖 (3)
where the summation runs over all contacts 𝑐 made by the particle 𝑝. 𝑉 𝑝 is the particle volume, 𝑅 𝑝 is the radius of the particle, 𝑁 𝑐 𝑝 is the number of contacts involving the particle, 𝑓 𝑗 𝑐 is the contact force vector and 𝒏 𝑖 is the unit branch vector between two particles in contact. The stress tensor at the material point scale (i.e. for a large number of particles), 𝜎 𝑖𝑗 , is given as a weighted average of the particle stresses (by particle volume):

𝜎 𝑖𝑗 = 1 𝑉 ∑ 𝑉 𝑝 𝑁 𝑝 𝑝=1 𝜎 𝑖𝑗 𝑝 ( 4
)
where 𝑉 is the volume of the entire system and 𝑁 𝑝 is the number of particles in the system.

𝜎 𝑖𝑗 can be decomposed to the contribution of each particle category, i.e. fine and coarse, as:

𝜎 𝑖𝑗 = 1 𝑉 ∑ 𝑉 𝑝 𝜎 𝑖𝑗 𝑝 𝑁 𝑝 𝑐 𝑝=1 ⏟ 𝜎 𝑖𝑗 𝑐 + 1 𝑉 ∑ 𝑉 𝑝 𝜎 𝑖𝑗 𝑝 𝑁 𝑝 𝑓 𝑝=1 ⏟ 𝜎 𝑖𝑗 𝑓 ( 5
)
where 𝑁 𝑝 𝑐 and 𝑁 𝑝 𝑓 are the number of coarse and fine particles in the system, respectively9 . The mean effective stress, 𝑝, for the entire system is calculated as:

𝑝 = 𝜎 𝑥𝑥 + 𝜎 𝑦𝑦 + 𝜎 𝑧𝑧 3 (6)
The proportion of the total mean stress transmitted by the finer fraction, 𝛼 𝑓 , is calculated as 𝑝 𝑓 𝑝 ⁄ , and similarly, the proportion of the total mean stress transmitted by the coarser fraction, 𝛼 𝑐 = 1 -𝛼 𝑓 , is calculated as 𝑝 𝑐 𝑝 ⁄ , where 𝑝 𝑓 and 𝑝 𝑐 are the mean stress transmitted by the finer and the coarser populations, respectively.

Fig. 9 shows the evolution of the 𝛼 𝑓 during shearing for the dense and loose assemblies of binary mixtures10 . The 𝛼 𝑓 for the dense assemblies having 𝑓 𝑐 ≥ 30% decreased significantly (with more pronounced decrease observed for 35%≤ 𝑓 𝑐 ≤50% which corresponds with regime 2 in Table 3) from the start of shearing until 𝜖 𝑎 ≈ 0.08 beyond which the change in the 𝛼 𝑓 became minimal (Fig. 9a). 𝛼 𝑓 for the loose assemblies generally exhibited minimal changes from the start to the end of shearing (Fig. 9b). In agreement with the data presented by [START_REF] Sufian | Influence of Fabric on Stress Distribution in Gap-Graded Soil[END_REF] for sand-silt mixtures subjected to constant mean stress triaxial compression, the observation made here indicates the transmission of stresses from the fine particles to the coarse particles or a concentration of stress on the coarse particles, in the dense assemblies, at the early stage of shearing. This phenomenon occurred such that for each 𝑓 𝑐 , a unique 𝛼 𝑓 exists for both the initially dense and loose assemblies at large strains (i.e. at the critical state). Fig. 10 shows the contribution of the finer fraction, 𝛼 𝑓 , and the coarser fraction, 𝛼 𝑐 , to the total mean stress for the studied assemblies. At all stages of shearing considered (Fig. 10a-d), the fine particles did not contribute to stress transmission (i.e. 𝛼 𝑓 ≈ 0) until the 𝑓 𝑐 𝑡ℎ beyond which there was a progressive increase in the 𝛼 𝑓 and a consequential reduction in 𝛼 𝑐 . The primary stress-transmitting skeleton at 𝑓 𝑐 < 𝑓 𝑐 𝑡ℎ is therefore the coarser fraction. Based on a conceptual analysis proposed in the experimental studies on granular mixtures [START_REF] Thevanayagam | Undrained Fragility of Clean Sands, Silty Sands, and Sandy Silts[END_REF][START_REF] Thevanayagam | Intergranular state variables and stress-strain behaviour of silty sands[END_REF], the mechanical behaviour of the mixtures should be primarily controlled by the fines at 𝑓 𝑐 ≥ 𝑓 𝑐 𝑡ℎ . [START_REF] Vallejo | Interpretation of the limits in shear strength in binary granular mixtures[END_REF] in his experimental study on binary granular mixtures suggests that at 𝑓 𝑐 >60%, the fines generally become the primary stresstransmitting matrix while the coarser fraction plays basically no role in stress transmission. The evidence from our micromechanical analysis shows that while the fines began to transmit stress at the 𝑓 𝑐 𝑡ℎ , they did not become the primary stress-transmitting fabric until 𝑓 𝑐 becomes higher than an equivalent fines content 𝑓 𝑐 𝑒𝑞 . Here, 𝑓 𝑐 𝑒𝑞 is the estimated 𝑓 𝑐 at which the total mean pressure is equally shared between the finer and the coarser fraction. For the studied binary mixtures, 𝑓 𝑐 𝑒𝑞 is estimated to range from 48% to 53% for the dense assemblies at the initial state and the peak; and reaches 55% for the loose assemblies at the initial state and for all assemblies at the critical state. Contrary to [START_REF] Vallejo | Interpretation of the limits in shear strength in binary granular mixtures[END_REF], while the fines transmit the largest fraction of the total mean stress for 𝑓 𝑐 > 𝑓 𝑐 𝑒𝑞 , the coarse particles continue to make a secondary and yet significant contribution to the total mean stress for much larger fine contents (at least up to 𝑓 𝑐 =70% studied here).

Since particles stresses result from contact forces, we estimate in Fig. 11, the proportion of the total number contacts shared by the fines12 , 𝐶 𝑛 𝑓 , and the coarse particles, 𝐶 𝑛 𝑐 . Comparing these geometric statistics to the stress contributions enables a better understanding of the trends observed in Fig. 10. Again, the trends in Fig. 11 can be classified into two regimes delineated by the threshold fines content, 𝑓 𝑐 𝑡ℎ . For 𝑓 𝑐 < 𝑓 𝑐 𝑡ℎ , 𝐶 𝑛 𝑓 generally increased with 𝑓 𝑐 but remained lower than 𝐶 𝑛 𝑐 , indicating that in this regime, the coarse particles are involved in a larger proportion of the total number of contacts than the fines. For 𝑓 𝑐 ≥ 𝑓 𝑐 𝑡ℎ , 𝐶 𝑛 𝑓 increased monotonically with 𝑓 𝑐 and was higher than 𝐶 𝑛 𝑐 ; in this case, the fines became dominant in terms of their proportion of the total number of contacts. By comparing Fig. 10 and 11, we observe that although the fines became dominant in terms of their proportion of the total number of contacts from 𝑓 𝑐 = 𝑓 𝑐 𝑡ℎ , they did not primarily contribute to the total mean stress until 𝑓 𝑐 > 𝑓 𝑐 𝑒𝑞 .

This indicates that most of the contacts involving fine grains do not carry large forces for 𝑓 𝑐 𝑡ℎ < 𝑓 𝑐 < 𝑓 𝑐 𝑒𝑞 . Figs. 12 and 13 show the cumulative distributions of the individual stresses sustained by the particles normalised by the average particle stresses in the whole assembly (𝑃 𝑝 𝑃 𝑝 ̅̅̅̅ ⁄ ), for the coarser and finer fraction respectively, at the initial states, the peak and the critical state. This is aimed at understanding the range and state of particle stresses existing within the particle size fractions in each assembly at different shearing stages. A wider range of stresses exist within the finer fraction (Fig. 13) than the coarser fraction (Fig. 12). For the coarser fraction, the proportion of the particles with stress values below the average stress (i.e. 𝑃 𝑝 𝑃 𝑝 ̅̅̅̅ ⁄ ≤ 1) was similar for all 𝑓 𝑐 values considered. Below the average stress, the cumulative distribution shifts to the right as 𝑓 𝑐 increased; the reverse was observed above the average stress. The width of the distributions for the coarser fraction tend to become smaller as 𝑓 𝑐 increased (Fig. 12a-d).

This indicates a more even stress distribution within the coarser particles as 𝑓 𝑐 increased. The proportion of the fines with stress values below the average (i.e. 𝑃 𝑝 𝑃 𝑝 ̅̅̅̅ ⁄ ≤ 1) decreased as 𝑓 𝑐 increased (Fig. 13).

The proportion of the fines bearing no stress (i.e. 𝑃 𝑝 𝑃 𝑝 ̅̅̅̅ ⁄ = 0), referred to as the non-active fines, 𝑅 𝑓 , are illustrated by the vertical line at 𝑃 𝑝 𝑃 𝑝 ̅̅̅̅ ⁄ = 0 in Fig. 13 and are plotted in Fig. 14a. Fig. 14b shows the change in the percentage of the non-active fines present within the assemblies during shearing, ∆𝑅 𝑓 (i.e. the difference in the 𝑅 𝑓 at the initial states and the critical state). Within the assemblies having 𝑓 𝑐 <𝑓 𝑐 𝑡ℎ , virtually all the fines (98.3% to 99.9%) are nonactive. This confirms that at 𝑓 𝑐 <𝑓 𝑐 𝑡ℎ , the finer fraction does not contribute to stress transfer. For 𝑓 𝑐 𝑡ℎ < 𝑓 𝑐 < 45%, 𝑅 𝑓 decreases more significantly than for 𝑓 𝑐 > 45% (Fig. 14a). The drop in the proportion of non-active fine (within 𝑓 𝑐 𝑡ℎ < 𝑓 𝑐 < 45%) depends significantly on the relative density of the assemblies (dense or loose) and the stage of shearing (initial, peak or critical state). Three regimes are identified following the analysis here. For 𝑓 𝑐 <𝑓 𝑐 𝑡ℎ , the fines are largely dormant whatever the sample state13 . Also, ∆𝑅 𝑓 ≈ 0 (Fig. 14b), indicating shearing had no effect on the activity of the fines. For 𝑓 𝑐 𝑡ℎ ≤ 𝑓 𝑐 < 45%, the proportion of active fines is very sensitive to the packing density and the stage of shearing (Fig. 14a). Also, ∆𝑅 𝑓 is highest (Fig. 14b),

suggesting that more of the active fines initially present within the assemblies became nonactive by the end of shearing (38%-85% and 11-12% for dense and loose assemblies, respectively). For 𝑓 𝑐 ≥ 45%, the proportion of non-active fines is rather low and exhibit a limited sensitivity to the packing density and the stage of shearing.

Contact based stress-transmission

Using the Love-Weber contact based stress tensor including Bagi boundary term [START_REF] Bagi | Microstructural stress tensor of granular assemblies with volume forces[END_REF], the contributions of all contact types to the total mean stress transmitted by an entire granular system is given as:

𝜎 𝑖𝑗 𝑐 = 1 𝑉 ∑ 𝑓 𝑗 𝑐 𝑁 𝑐 𝑐=1 𝑙 𝑖 (11)
where 𝑁 𝑐 is the total number of contacts in the granular system, 𝑉 is the volume of the system, 𝒍 is the branch vector (‖𝒍‖ = 𝑅 1 + 𝑅 2 for a sphere 1 to sphere 2 contact; ‖𝑙‖ = 𝑅 𝑝 for a sphere 𝑝 to wall contact), and 𝒇 𝒄 is the contact force vector. This stress tensor can be decomposed based on the contact type as:

𝜎 𝑖𝑗 𝑘 = 1 𝑉 ∑ 𝑓 𝑗 𝑐 𝑁 𝑐 𝑘 𝑐=1 𝑙 𝑖 ( 12
)
where k ∈ {c-c, c-f, f-f, s-wl}. c-c, c-f, f-f, and s-wl denote coarse to coarse, coarse to fine, fine to fine and sphere to wall contacts, respectively. The proportion of the total mean stress from each contact type is given by 𝛼 𝑘 = 𝑝 𝑘 𝑝 ⁄ . Note that the boundary term is expected to vanish as soon as the sample domain is sufficiently large (which should be the case when the REV condition is met).

Fig. 15 shows the evolution of the proportion of the stress transmitted by each contact type during shearing, for the dense and loose assemblies. At large strains (i.e., 𝜖 𝑎 >0.3), the contribution of the coarse to coarse contacts to the total mean stress, 𝛼 𝑐-𝑐 , decreases with 𝑓 𝑐 such that 𝛼 𝑐-𝑐 ≈0 for 𝑓 𝑐 > 40% (Fig. 15a&d). The contribution of the fine-fine contacts to the total mean stress, 𝛼 𝑓-𝑓 , increases with 𝑓 𝑐 for 𝑓 𝑐 >𝑓 𝑐 𝑡ℎ while 𝛼 𝑓-𝑓 ≈0 for 𝑓 𝑐 <𝑓 𝑐 𝑡ℎ (Fig. 15c&f).

The contribution of the coarse to fine contacts, 𝛼 𝑐-𝑓 , initially increases with 𝑓 𝑐 until 𝑓 𝑐 =40% and thereafter decreases with further increase in 𝑓 𝑐 (Fig. 15b&e). During the early stage of shearing (i.e., 𝜖 𝑎 < 0.1), we observed a decrease in 𝛼 𝑐-𝑐 and 𝛼 𝑓-𝑓 which is accompanied by a commensurate increase in the 𝛼 𝑐-𝑓 . This indicates a redistribution of stress from the c-c and f-f contacts to the c-f contacts. Although the coarse particles lost stress as indicated in the decrease in 𝛼 𝑐-𝑐 , they are also gained stress from the fines as indicated in both the increase in 𝛼 𝑐-𝑓 and the decrease in 𝛼 𝑓-𝑓 . Ultimately, there was a redistribution of stress from the fines to the coarse particles as indicated in Fig. 9a. 16 shows the proportion of the mean stress transmitted by each contact types at the initial states, the peak and the critical state. The trends observed between the contact types 14 can be classified into four regimes based on the 𝑓 𝑐 . In the first regime (𝑓 𝑐 <𝑓 𝑐 𝑡ℎ ), the contribution of the fine to fine contacts to the total mean stress, 𝛼 𝑓-𝑓 ≈0, indicating that the f-f contacts do not contribute to mean stress below the 𝑓 𝑐 𝑡ℎ . In this regime, the condition 𝛼 𝑐-𝑐 > 𝛼 𝑐-𝑓 > 𝛼 𝑓-𝑓 generally holds, hence, the regime is referred to as coarse-dominated, following [START_REF] Vallejo | Interpretation of the limits in shear strength in binary granular mixtures[END_REF] and [START_REF] Sufian | Influence of Fabric on Stress Distribution in Gap-Graded Soil[END_REF]. Within the second regime referred to as transitional coarse-dominated was defined in Section 4 as the 𝑓 𝑐 at which the total mean stress is equally shared between the coarser and the finer fractions) and referred to as transitional fines-dominated, the condition 𝛼 𝑐-𝑓 > 𝛼 𝑓-𝑓 > 𝛼 𝑐-𝑐 holds. The transitional fines content, 𝑓 𝑐 𝑡𝑟 , delineating the transitional regimes (i.e. the transitional-coarse dominated and the transitional-fines dominated regimes)

(
was found around 𝑓 𝑐 = 40% for the assemblies and the shearing stages considered here (Fig. 16a-d). The transitional regime based on the micromechanical analysis ranged from 𝑓 𝑐 𝑡ℎ to 𝑓 𝑐 𝑒𝑞 and is therefore within 𝑓 𝑐 =30%-48% and 𝑓 𝑐 =35%-55% for the dense and loose assemblies, respectively. For 𝑓 𝑐 >𝑓 𝑐 𝑒𝑞 (i.e., in the fourth regime), the condition 𝛼 𝑓-𝑓 > 𝛼 𝑐-𝑓 > 𝛼 𝑐-𝑐 holds, hence, the regime is fines-dominated. While the classification in the experimental study by [START_REF] Vallejo | Interpretation of the limits in shear strength in binary granular mixtures[END_REF] was based on void ratio, we present here a similar classification based on the micromechanical analysis conducted, with distinct delineations in agreement with the 3-D DEM study on binary mixtures having size ratio, λ=4, reported by [START_REF] Minh | Strong force networks in granular mixtures[END_REF].

14 Excluding the contributions to the mean stress from the contacts between the spherical particles and the walls, 𝛼 𝑠-𝑤𝑙 , which were generally less than 0.15. We observed that 𝛼 𝑠-𝑤𝑙 values for the underfilled assemblies (𝑓 𝑐 <𝑓 𝑐 𝑡ℎ ) were generally higher than the overfilled assemblies. This suggests that a greater number of particles is required to ensure a representative element volume (REV) in underfilled assemblies in comparison to the overfilled assemblies. 15 The proportions of all contact types belonging to the wall, 𝐶 𝑤𝑙 𝑡 were less than 0.17 and 0.08 for 𝑓 𝑐 <𝑓 𝑐 𝑡ℎ and for 𝑓 𝑐 >𝑓 𝑐 𝑡ℎ , respectively.

From the threshold fines content, the 𝐶 𝑐-𝑐 𝑡 values were significantly low (ranging from 2.1e-2 -9e-5) at all states, and was zero for 𝑓 𝑐 =70% at the critical state (Fig. 17d). This show that the fines disperse the coarse particles (i.e. 𝐶 𝑐-𝑐 𝑡 becomes minimal) from the threshold fines content, 𝑓 𝑐 𝑡ℎ , which is in agreement with the 3D DEM study on binary mixtures by [START_REF] Minh | Strong force networks in granular mixtures[END_REF],

and not beyond 𝑓 𝑐 𝑡ℎ as suggested in the experimental study on sandy-gravels by Skempton & Brogan (1994). The suggestion by [START_REF] Salgado | Shear Strength and Stifness of Silty Sand[END_REF] in their experimental study, that the fines controls the mechanical behaviour of sand-silt mixtures when the sand particles are completely floating (𝐶 𝑐-𝑐 𝑡 ≈ 0) in the silts is not supported by the data shown here. While the fines disperse the coarse particles from 𝑓 𝑐 𝑡ℎ , they did not become the primary stress-transmitting matrix until 𝑓 𝑐 𝑒𝑞 is reached as explained in Section 4. In the transitional regime, there is the possibility that some fines get trapped between the coarse particles while being loosely connected to the rest of the fines in the assemblies.

Also, while for 𝑓 𝑐 <𝑓 𝑐 𝑡ℎ , the 𝐶 𝑐-𝑓 𝑡 ≈0 in the dense assemblies (Fig. 17a), in the loose assemblies (Fig. 17b), 𝐶 𝑐-𝑓 𝑡 >0. This suggests that, in agreement with [START_REF] Thevanayagam | Undrained Fragility of Clean Sands, Silty Sands, and Sandy Silts[END_REF] and [START_REF] Shire | Micro-scale Modelling of Granular Filters[END_REF], the fines in the dense assemblies are confined within the voids between the coarse particles with little interaction with the coarse matrix; in the loose assemblies, the fines interact more with the coarse particles. The fact that the fines are interacting more with the coarse particles in the loose case may indicate the fines are trapped within the coarse particles thereby creating larger voids which may require more fines to fill in comparison to the dense assemblies. This observation is consistent with [START_REF] Salgado | Shear Strength and Stifness of Silty Sand[END_REF] and [START_REF] Lade | Effects of nonplastic fines on static liquefaction of sands[END_REF] who found more fines on the surfaces of the coarse particles in their loose assemblies, in comparison to the dense assemblies, and suggested that this is responsible for the larger drop in the 𝑒 𝑚𝑖𝑛 than in the 𝑒 𝑚𝑎𝑥 for a given increase in the fines content. The drop in the 𝑒 𝑚𝑎𝑥 is mitigated than in the 𝑒 𝑚𝑖𝑛 because the fines interacting with the coarse particles helps to create larger voids hence the lower drop in the void ratio in the loose case.

Stress-based skeleton void ratio

The idea of sand skeleton void ratio was introduced in the experimental study by [START_REF] Kuerbis | Effect of gradation and fines content on the undrained response of sand[END_REF] in order to understand the behaviour of silty sands with various fines content under undrained triaxial shearing. This proposition, which was based on a conceptual analysis of the fabric of soil mixtures, have been applied in other experimental and numerical studies [START_REF] Benahmed | An experimental investigation into the effects of low plastic fines content on the behaviour of sand/silt mixtures[END_REF][START_REF] Chang | Micromechanical modeling for behavior of silty sand with influence of fine content[END_REF][START_REF] Lade | Effects of nonplastic fines on static liquefaction of sands[END_REF][START_REF] Ni | Contribution of fines to the compressive strength of mixed soils[END_REF][START_REF] Pitman | Influence of fines on the collapse of loose sands[END_REF][START_REF] Rahman | On equivalent granular void ratio and steady state behaviour of loose sand with fines[END_REF][START_REF] Thevanayagam | Effect of Fines and Confining Stress on Undrained Shear Strength of Silty Sands[END_REF][START_REF] Thevanayagam | Undrained Fragility of Clean Sands, Silty Sands, and Sandy Silts[END_REF][START_REF] Vaid | Liquefaction of silty soils[END_REF]. In order to understand the critical state behaviour of the gap-graded assemblies subjected to drained shearing in this study, we propose a skeleton void ratio based on the stress sustained by individual particles. The skeleton void ratio, 𝑒 𝑠𝑘𝑒𝑙 , is here defined as the void ratio of a granular system when the fines with particle stress, 𝑃 𝑝 , lower than a threshold stress value are regarded as part of the void space and are therefore excluded. Note that we propose to exclude only the fines (and not the coarse particles) for our skeleton void ratio calculation. This is based on the fact that loosely stressed fine particles may be eroded through the constrictions of the soil matrix whereas coarse particles cannot as they are usually larger than the constriction size [START_REF] Garner | Understanding internal erosion: a decade of research following a sinkhole event[END_REF][START_REF] Icold | ICOLD Bulletin 164 on Internal erosion of existing dams, leeves and dikes, and their foundations[END_REF][START_REF] Skempton | Experiments on piping in sandy gravels[END_REF][START_REF] Thevanayagam | Intergranular state variables and stress-strain behaviour of silty sands[END_REF]. In Figure 17 we show the variation of the proportion of the total mean stress transmitted by the "skeleton" finer fraction, 𝛼 𝑓 𝑠𝑘𝑒𝑙 , for different stress thresholds, where the threshold stress is defined as:

Threshold stress = 𝑥 ⏟ threshold coefficient × 𝑃 𝑝 ̅̅̅̅ ⏟ Average particle stress (10) In Fig. 18, the threshold coefficient, 𝑥, ranges from 0 to 2.

-Unsurprisingly, the contribution of the fine remains negligible whatever the threshold value for 𝑓 𝑐 <𝑓 𝑐 𝑡ℎ (the contribution for all the fines, 𝛼 𝑓 , is already negligible).

-For 𝑓 𝑐 𝑡ℎ ≤ 𝑓 𝑐 ≤ 45%, generally no change is observed in 𝛼 𝑓 𝑠𝑘𝑒𝑙 until 𝑥 = 0.1, which means that the fine grains transmitting a pressure 𝑃 𝑝 lower than 10% of the mean pressure 𝑃 𝑝 ̅̅̅̅ , contribute only marginally to the stress transmitted by the finer fraction.

-For 𝑓 𝑐 >45%, no change was observed in 𝛼 𝑓 𝑠𝑘𝑒𝑙 until 𝑥 > 0.25.

From the above analysis, we can retain x=0.1 as a reasonable threshold to define the skeleton fines for all fines content. Fig. 19 shows the proportion of the fines left in an assembly after each exclusion. These "skeleton" fines are included in the solid volume while computing the stress-based skeleton void ratio, 𝑒 𝑠𝑘𝑒𝑙 plotted in Fig. 20. For 𝑓 𝑐 ≥𝑓 𝑐 𝑡ℎ , the proportion of the fines included as part of the assembly skeleton decreases monotonically as the threshold stress increases (Fig. 19a-d).

The mechanical void ratio, 𝑒 𝑚𝑒𝑐ℎ has been computed in the literature as an alternative void ratio index to the void ratio, 𝑒, of granular materials [START_REF] Liu | A semi-empirical reevaluation of the influence of state on elastic stiffness in granular materials[END_REF][START_REF] Otsubo | Particle Scale Analysis of Soil Stiffness and Elatic Wave Propagation[END_REF].

𝑒 𝑚𝑒𝑐ℎ excludes the particles with 𝑐 ≤ 1. In Fig. 20, we compare the evolutions of 𝑒, 𝑒 𝑚𝑒𝑐ℎ and 𝑒 𝑠𝑘𝑒𝑙 (for 𝑥 = 0.1), 𝑒 𝑠𝑘𝑒𝑙 ; 𝑥 = 0.1, for the different fines contents. These evolutions are put in parallel with the peak and the critical state stress ratios.

As shown in Figure 19, 𝑒 𝑠𝑘𝑒𝑙 ; 𝑥 = 0.1, shows a negative correlation with the critical state strength, (𝑞/𝑝) 𝑐𝑟𝑖𝑡 (Fig. 20b &d), and no correlation with the peak strength, (𝑞/𝑝) 𝑚𝑎𝑥 (Fig. 20a). Instead, at the peak, (𝑞/𝑝) 𝑚𝑎𝑥 correlates negatively with 𝑒, (Fig. 20a &c), in agreement with established relationship between the initial void ratio and the peak strength in the literature [START_REF] Adesina | DEM study on the effect of particle shape on the shear behaviour of granular materials[END_REF][START_REF] Holubec | Effect of particle shape on the Engineering Properties of Granular Soils. Evaluation of Relative Density and Its Role in Geotechnical Projects Involving Cohesionless Soils[END_REF][START_REF] Ng | Triaxial Test Simulations with Discrete Element Method and Hydrostatic Boundaries[END_REF]. In order to determine the best performing void ratio index among the indexes considered here, we estimated the goodness of the fit, 𝑅 2 , for the relationship between the void ratio values determined using the indexes and the strength exhibited by the assemblies (Table A.1 in the Appendix C). It is obvious that while the best performing void ratio index for (𝑞/𝑝) 𝑚𝑎𝑥 is 𝑒 (𝑅 2 =0.90 for underfilled and 𝑅 2 =0.97 for the overfilled assemblies). This finding is consistent with the fact that the microstructure of granular assemblies (including the force chain network) evolve rapidly under shearing at the critical state [START_REF] Deng | Lifespan dynamics of cluster conformations in stationary regimes in granular materials[END_REF][START_REF] Wautier | Micro-inertia origin of instabilities in granular materials[END_REF]. Consequently, the nonactive and the marginally active fines do not have the chance to interact with the force chains and should therefore not be considered as part of the void when assessing the mechanical behaviour. On the contrary, at the peak, although the non-active and the marginally active fines do not contribute significantly to the total mean stress, they serve as a support for the force chain network and as a result constitute an important part of the microstructure of the assemblies [START_REF] Tordesillas | Force cycles and force chains[END_REF][START_REF] Wautier | Micro-inertia origin of instabilities in granular materials[END_REF]Zhu et al., 2016a;Zhu et al., 2016b).

Therefore, they have to be taken into account in the void ratio calculation.

Summary and Conclusions

This study presents a micromechanical evaluation of the regimes delineating the behaviour of gap-graded granular assemblies of different fines content, using discrete element simulations.

The existence of the regimes delineated by milestone fines content was assessed using the macromechanical and micromechanical characteristics of the assemblies, and the contributions of the particle size fractions and contact types to the total mean stress. The key findings and conclusions are presented below.

(i) Two regimes were identified based on the macroscopic characteristics of the assemblies, three regimes based on particle scale analysis and four regimes based on contact scale analysis. We provide distinct delineations of the regimes by fines content, in an original manner. Fig. 21 presents a summary of all the regimes identified in this study. Table 3 highlights the unique trends characterising the three regimes identified based on the macromechanical and the micromechanical analyses conducted. We showed that the boundaries delineating the identified regimes depend on density and stress state. We acknowledge that the boundaries can also vary with a change in particle size ratio (and more generally a change in coarse and fine particle size distributions) which is beyond the scope of our study.

(ii)

The micromechanical analyses adopted overcome the limitation of experimental studies where it is difficult to ascertain the claims emanating from the conceptual analysis presented in the studies, since individual particles stresses or particle dispersion cannot be easily determined in experiments. The claims that the fines become the primary stress-transmitting matrix at the threshold fines content or at a limiting fines content; or that stress is equally sheared by the coarse and the fine particles in the transitional zone [START_REF] Vallejo | Interpretation of the limits in shear strength in binary granular mixtures[END_REF] or at 𝑓 𝑐 >35% [START_REF] Shire | Fabric and Effective Stress Distribution in Internally Unstable Soils[END_REF] are not supported by the evidence provided here. Instead, we found that the coarse particles are dispersed by the fines (i.e. where coarse to coarse contacts are minimal or non-existent) from the threshold fines content. In addition, the fines do not contribute to the total mean stress below the threshold fines content, they play a secondary role in the transitional zone (𝑓 𝑐 ∈ [30% ; 55%] depending on density and stress state), and only become the primary stress-transmitting matrix beyond the transitional zone (i.e. when 𝑓 𝑐 > 𝑓 𝑐 𝑒𝑞 ; 𝑓 𝑐 𝑒𝑞 ∈ [48% ; 55%] depending on density and stress state).

(iii) We found that the threshold stress which determines whether a fine particle transmits a marginal or a significant stress with respect to the contribution of the fines to the total mean stress is a fraction of the average particle stress, 𝑃 𝑝 ̅̅̅̅ , within an assembly; where the threshold stress values are 0.1 × 𝑃 𝑝 ̅̅̅̅ for 𝑓 𝑐 𝑡ℎ ≤ 𝑓 𝑐 ≤ 45%, and 0.25 × 𝑃 𝑝 ̅̅̅̅ for 𝑓 𝑐 >45%. The threshold stress concept proposed in this study can be useful in determining the particles that constitute important mesostructures (i.e.

important force-chain networks) in gap-graded granular assemblies. The skeleton void ratio determined based on the threshold stress correlates with the critical state strength of the assemblies, while no correlation was found between the critical state strength and alternative void ratio indexes such as the void ratio and the mechanical void ratio at the critical state. The marginal stress transmitting fines (marginally active fines) determined based on the threshold stress may constitute a part of the particles susceptible to internal erosion, in addition to the non-inactive fines in a gap-graded assembly (particles with zero or only one contact).

In addition to the standard underfilled and overfilled regimes, future DEM studies could focus on the two transitional regimes identified here, and seek to understand the role played by marginally active fines i) in the susceptibility of gap-graded assemblies to internal erosion and ii) in the triggering of mechanical instabilities. As an additional perspective, the distinct characteristics of the regimes shown in this study will prove useful in the development of micromechanical models for gap-graded materials in which the typical mesoscale grain arrangements can be tailored for each regime.

Fig. 1 .

 1 Fig. 1. Dense specimens with different percentages of fines content: (a) fc = 10 % (b) fc = 30

Fig. 2 .

 2 Fig. 2. Effect of fines content, 𝑓 𝑐 , on (a) the initial void ratio, 𝑒, (b) the range of attainable void ratios, 𝑒 𝑚𝑎𝑥 -𝑒 𝑚𝑖𝑛 for the assemblies of binary mixture studied

Fig. 3 .

 3 Fig. 3. Effect of fines content, 𝑓 𝑐 , on the stress-strain and the volumetric strain responses for dense assemblies (a,d) 𝑓 𝑐 = 10%-25% (b,e) 𝑓 𝑐 = 30%-40% (c,f) 𝑓 𝑐 = 45%-70%. Inset graphs correspond to the start of the loading.

Fig. 4 .

 4 Fig. 4. Effect of fines content, 𝑓 𝑐 , on the stress-strain and the volumetric strain responses for loose assemblies (a,d) 𝑓 𝑐 = 10%-25% (b,e) 𝑓 𝑐 = 30%-40% (c,f) 𝑓 𝑐 = 45%-70%. Inset graphs correspond to the start of the loading.

Fig. 5

 5 Fig. 5 Effect of fines content on (a) stress ratio at peak, (𝑞 𝑝) ⁄ 𝑝𝑒𝑎𝑘 , for the dense assemblies, and peak void ratio, 𝑒 𝑝𝑒𝑎𝑘 (b) critical state stress ratio, (𝑞 𝑝) ⁄ 𝑐𝑟𝑖𝑡 , (unique value for dense and loose assemblies), and critical state void ratio, 𝑒 𝑐𝑟𝑖𝑡 .

Fig. 7

 7 Fig. 7 Effect of fines content on the (a) mechanical coordination number, 𝑍 𝑚 , for all particles (b) 𝑍 𝑚𝑐 for coarse particles (c) 𝑍 𝑚𝑓 for fine particles (d) number of fines-rattlers (e) number of coarse-rattlers within assemblies of binary mixture.

Fig.

  Fig.8ashows the contact lost or gained during shearing, ∆𝑍, determined here as 𝑍 𝑖𝑛𝑖 -𝑍 𝑐𝑟𝑖𝑡 ,

Fig. 8

 8 Fig. 8 (a) Change in coordination number during shearing, ∆𝑍 (filled circle for dense assemblies and open circles for loose assemblies). ∆𝑍>0 means contact lost and ∆𝑍<0 means contact gained during shearing. (b) excess friction angle (strain softening), 𝜙 𝑝 -𝜙 𝑐 for dense assemblies of binary mixture.

Fig. 9

 9 Fig. 9 Evolution of the proportion of mean stress transmitted by the fines during shearing for (a) dense assemblies (b) loose assemblies 11

Fig. 10

 10 Fig. 10 Proportion of mean stress transmitted by the fines and coarse particles at (a) the initial state for dense assemblies (b) the initial state for the loose assemblies (c) the peak (d) the critical state (unique trends for both dense and loose assemblies)

Fig. 12

 12 Fig. 12 Cumulative distribution of normalised particle mean stress for coarser fraction (a) dense; initial state (b) loose; initial state (c) dense; peak state (d) critical state for both dense (solid line) and loose (dashed line) assemblies

Fig. 15

 15 Fig. 15 Evolution of the proportion of mean stress transmitted by contact types during shearing for (a) c-c contacts; dense (b) c-f contacts; dense (c) f-f contacts; dense (d) c-c contacts; loose (b) c-f contacts; loose (c) f-f contacts; loose

Fig. 16

 16 Fig. 16 Proportion of mean stress transmitted by contact type at (a) the initial state for dense assemblies b) the initial state for the loose assemblies (c) at the peak (d) the critical state (unique trends for both dense and loose assemblies). Void ratio data represented with dashed grey lines are plotted on the right side of each subplot to help identify the different regimes.

Fig. 17

 17 Fig. 17 Proportion of the total contacts belonging to each category of contact type at (a) the initial state for dense assemblies (b) the initial state for the loose assemblies (c) the peak (d) the critical state (unique trends for both dense and loose assemblies). Void ratio data represented with dashed grey lines are plotted on the right side of each subplot to help identify the different regimes.

Fig. 18

 18 Fig. 18 Estimation of the proportion of the mean stress transmitted by the finer fraction when particles with particle stress lower than a threshold are excluded (a) dense; initial state (b) loose; initial state (c) dense; peak state (d) critical state (unique trends for both dense and loose assemblies)

Fig. 21

 21 Fig. 21 Regime identification by macroscale, particle scale and contact scale characteristics

Table 1

 1 Simulation parameters

	125
	Parameter	Value
	Diameter of coarse particles, 𝐷 𝑐 , mm	3.075
	Diameter of fine particles, 𝐷 𝑓 , mm	0.375
	Particle size ratio 𝐷 𝑐 / 𝐷 𝑓	8.2
	Particle density, 𝜌, 𝑘𝑔 𝑚 3 ⁄	2700 × 10 3
	Contact law	Elasto-frictional
	Inter-particle friction coefficient during isotropic compression	0.03 (dense), 0.5 (loose)
	Inter-particle friction coefficient during shearing	0.5
	Wall-particle friction coefficient	0
	Particle normal stiffness, 𝑘 𝑛 , 𝑁 𝑚 2 ⁄	3.56 x 10 8
	Stiffness ratio	1.0

Table 2

 2 Number of particles for each assembly

	127
	Specimens			Percentage of fine particles by weight
		10%	15%	20%	25%	30%	35%	40%	45%	50%	60%	70%
	Number	Coarse 1,612 1,021	723	543	423	337	272	222	182	121	78
	of
	particles

Fine 98,388 98,979 99,277 99,457 99,577 99,663 99,728 99,778 99,818 99,879 99,922 Total 100,000 128 129

Table 3 :

 3 Summary of regime based on macromechanical and micromechanical characteristics

	∆ in parameters	Regime 1	Regime 2	Regime 3
	as the 𝑓 𝑐 ↑	(𝑓 𝑐 < 𝑓 𝑐 𝑡ℎ )	(𝑓 𝑐 𝑡ℎ < 𝑓 𝑐 <45%)	(𝑓 𝑐 >45%)
	𝑒 𝑖𝑛𝑖 ; 𝑒 𝑐𝑟𝑖𝑡	Decreases		Increases
	(𝑞/𝑝) 𝑚𝑎𝑥	Increases		Decreases
	(-𝑑𝜀 𝑣 𝑑𝜀 𝑎 ) ⁄	𝑚𝑎𝑥	Increases		Decreases
	𝑍 𝑚		Decreases	Increases	Increases marginally to a plateau
	𝑍 𝑚𝑐 ; 𝑍 𝑚𝑓	No change	Increases	Increases marginally to a plateau
	∆𝑍		No change	Increases	Decreases marginally to a plateau
	𝜙 𝑝 -𝜙 𝑐		Increases	Plateau	Decreases

  𝑓 𝑐 𝑡ℎ <𝑓 𝑐 <𝑓 𝑐 𝑡𝑟 , where 𝑓 𝑐 𝑡𝑟 is referred here as transitional fines content where 𝛼 𝑐-𝑐 =𝛼 𝑓-𝑓 ), the condition 𝛼 𝑐-𝑓 > 𝛼 𝑐-𝑐 > 𝛼 𝑓-𝑓 generally holds. In the third regime where 𝑓 𝑐 𝑡𝑟 <𝑓 𝑐 <𝑓 𝑐 𝑒𝑞 (𝑓 𝑐 𝑒𝑞

  Proportion of the total number contacts belonging to the coarser fraction Ratio of the mean size of the coarse grains to the mean size of the fine grains

	q	Deviatoric stress
	𝑞 𝑓	Deviatoric stress transmitted by the finer fraction
	𝑞 𝑐	Deviatoric stress transmitted by the coarser fraction
	𝑞 𝑓	Proportion of the deviatoric stress transmitted by the finer fraction
	R-squared	Goodness of fit
	𝑅 𝑑
	𝑅 𝑝	Radius of particle
	𝑅 1	Radius of first particle in contact
	𝑑 50 𝑅 2 𝐷 50 𝑅 𝑓 emax 𝑒 𝑠𝑘𝑒𝑙 emin 𝑒 𝑠𝑘𝑒𝑙 ;𝑥 𝑒 𝑐𝑟𝑖𝑡 𝑆𝑘𝑙 𝑓 𝑒 ℎ𝑐 𝑥 𝑒 ℎ𝑓 𝜇 𝑝𝑟𝑒𝑝 εa 𝜀 𝑞 µ εvol 𝜆 𝑓 𝑐 𝑉 𝑓 𝑡ℎ 𝑉 𝑝 𝑓 𝑐 𝑡ℎ 𝜎 𝑧𝑧 𝑓 𝑐 𝐿 𝜎 𝑦𝑦 𝑓 𝑗 𝑐 𝜎 𝑥𝑥 𝑙 𝑁 𝑝 𝑁 𝑐 𝜎 𝑖𝑗 𝑐 𝜎 𝑖𝑗 𝜎 𝑖𝑗 𝑝 𝑁 𝑝 𝑐 𝜎 𝑝 𝑐 𝑁 𝑝 𝑓 𝜎 𝑝 𝑓 𝑁 𝑐 𝑝 𝜙 𝑁 0 𝜙 𝑝 𝑁 1 𝜙 𝑐 𝑁 𝑐≥1 𝜓 𝑚𝑎𝑥 𝑁 𝑓≥1 𝑍 𝑁𝐶 𝑓 𝑍 𝑖𝑛𝑖 𝑁𝐶 𝑓 𝑍 𝑐𝑟𝑖𝑡 𝑛 𝑖 𝑍 𝑔 p 𝑍 𝑔𝑐 𝑝 𝑐 𝑍 𝑔𝑓 𝑝 𝑓 𝑍 𝑚 𝑝 𝑘 𝑍 𝑚𝑐 𝑃 𝑝 ̅̅̅̅ 𝑃 𝑝 𝑍 𝑚𝑓	Radius of second particle in contact Mean size of the fine grains Proportion of non-active fines Mean size of the coarse grains Skeleton void ratio Maximum void ratio Skeleton void ratio determined using a threshold coefficient Minimum void ratio Proportion of fines included in skeleton Void ratio at the critical state Threshold coefficient Void ratio of the host sand Inter-particle friction coefficient during assembly preparation (isotropic Void ratio of the silt compression) Axial strain (in y-direction) Inter-particle friction coefficient Deviatoric strain Size fraction Volumetric strain Volume of system Fines content Volume of particle Threshold fines content Normal stress in z-direction Threshold fines content Normal stress in y-direction Limiting fines content Normal stress in x-direction Contact force vector Unit branch vector Number of particles in the granular system Second order stress tensor Second order stress tensor for all contacts Second order stress tensor for a particle Number of contacts in a granular system Average mean stress transmitted by the coarser fraction Number of coarse particles in the system Average mean stress transmitted by the finer fraction Number of fine particles in the system Friction angle Number of contacts involving a particle Peak friction angle Number of particles with no contact Critical state friction angle Number of particles with one contact Maximum dilatancy angle Number of coarse particles with one or more contacts Mean coordination number Number of fine particles with one or more contacts Mean coordination number at the initial state (after isotropic compression) Normalised total contacts for the coarser fraction Mean coordination number at the critical state Normalised total contacts for the finer fraction Geometrical coordination number Unit branch vector Geometrical coordination number for coarse particles Mean effective stress Geometrical coordination number for fine particles Mean stress transmitted by the coarser fraction Mechanical coordination number Mean stress transmitted by the finer fraction Mechanical coordination number for coarse particles Mean stress transmitted by a contact type Average particle mean stress Particle mean stress Mechanical coordination number for fine particles
	𝛼 𝑐	Proportion of the total mean stress transmitted by the coarser fraction
	𝛼 𝑓	Proportion of the total mean stress transmitted by the finer fraction
	𝛼 𝑘	Proportion of the total mean stress transmitted a contact type
	(𝑞/𝑝) 𝑚𝑎𝑥	Stress ratio at the peak
	(𝑞/𝑝) 𝑐𝑟𝑖𝑡	Stress ratio at the critical state

The word "regime" in this paper refers to zones of unique behaviour within the range of fines content in gapgraded assemblies.

Throughout this paper, soil mechanics conventions are used with compression being positive and extension being negative.

The maximum resultant force on the different grains divided by the mean contact force.

Note that the coordination number of particles with more than two contacts accounts for the contacts from particles with only one contact in the standard definition of 𝑍 𝑚 .

The contacts made by the coarse particles include both the coarse-coarse and the coarse-fine contacts while the contacts made by the fine particles include both the fine-fine and the coarse-fine contacts.

Note that 𝜎 𝑖𝑗 𝑐 and 𝜎 𝑖𝑗 𝑓 correspond to the contribution of coarse and fine particles to the total stress and not to the mean stress in coarse and fine particles. To compute the stress in the coarse particles, we consider both the coarsecoarse and the coarse-fine contacts. Similarly, to compute the stress in the fine particles, we consider both the fine-fine and the coarse-fine contacts.

 10 In Appendix B and Fig.A2, we show that the evolution of the deviatoric stress transmitted by the fines is generally similar to the evolution of the mean stress presented in Fig.8.

By comparing the 𝑞/𝑝 data(Figs. 3 & 4) with the 𝛼 𝑓 data (Fig.9), it is observed that the fluctuation in the 𝑞/𝑝 data for the assemblies having 𝑓 𝑐 ∈ [30% ; 40%] is not reflected in the relatively smooth 𝛼 𝑓 data. This is because the fluctuations evident in the 𝑞/𝑝 data is masked behind the apparently smooth 𝛼 𝑓 data as a result of the normalisation of the stress transmitted by the fines with the total mean stress (𝑝 𝑓 𝑝 ⁄ ).

The contacts made by the coarse particles include both the coarse-coarse and the coarse-fine contacts while the contacts made by the fine particles include both the fine-fine and the coarse-fine contacts. The proportion of the total number of contacts shared by the fines, 𝐶 𝑛 𝑓 , is calculated as: 𝐶 𝑛 𝑓 = 2𝑁 𝑓-𝑓 +𝑁 𝑐-𝑓 2𝑁 𝑓-𝑓 +2𝑁 𝑐-𝑓 +2𝑁 𝑐-𝑐 .

Note that, depending on the sample preparation procedure, we could imagine having fewer non-active fines at initial state. This is for instance the case when samples are prepared in the lab using the moist tamping method that gather fines around contacts between coarse grains because of capillary effects.

Appendix A. REV scale analysis

In order to confirm that the sample size of 100,000 particles used for all 𝑓 𝑐 in this study is a representative element volume (REV), in Fig. A.1, we show that increasing the assembly size to 200,000 particles for both the dense and loose assemblies having 𝑓 𝑐 = 30% did not significantly influence the stress-strain responses. The slight sensitivity to assembly size observed in the volumetric strains was also reported in the DEM study on REV for granular materials by [START_REF] Adesina | Determining a representative element volume for DEM simulations of samples with non-circular particles[END_REF] and can be attributed to the small variation in the initial void ratio of the assemblies. The REV test is shown here for 𝑓 𝑐 = 30% because it represents the threshold fines content and is the fines content with the highest coarse-fine particle interaction (Fig. 17) which indicates the largest REV size. that all stress tensors are diagonal in the frame (x, y, z), the deviatoric stress for the entire system is calculated as:

Similar to the mean stress (see Section 4), the proportion of the total deviatoric stress transmitted by the finer fraction, 𝑞 𝑓 , is calculated as 𝑞 𝑓 𝑞 ⁄ , while the proportion of the total deviatoric stress transmitted by the coarser fraction, 𝑞 𝑓 , is calculated as 𝑞 𝑐 𝑞 ⁄ , where 𝑞 𝑓 and 𝑞 𝑐 are the deviatoric stress transmitted by the finer and the coarser populations, respectively.

As expected, the 𝑞 𝑓 data is generally similar to the 𝛼 𝑓 data presented in Fig. 9. It is worthy of note that the 𝑞 𝑓 data presented in Fig. A2 starts after the initial state, at 𝜖 𝑎 = 0.02, since 𝑞 = 0 at the initial state. We observed that for 𝑓 𝑐 ≥ 30%, the magnitude of 𝑞 𝑓 is lower than the magnitude of 𝛼 𝑓 during shearing. This indicates that the stress is more isotropic in the finer fraction than in the coarser fraction. In Table A. 1 and Fig. A.3,we show that at the critical state, 𝑒 𝑠𝑘𝑒𝑙 ; 𝑥 = 0.1 exhibited a negative correlation with (𝑞/𝑝) 𝑐𝑟𝑖𝑡 and yielded the highest R-squared value (𝑅 2 =0.84), while both the 𝑒 and 𝑒 𝑚𝑒𝑐ℎ do not correlate with (𝑞/𝑝) 𝑐𝑟𝑖𝑡 . 𝑒 𝑠𝑘𝑒𝑙 ; 𝑥 = 0.05 and 𝑒 𝑠𝑘𝑒𝑙 ; 𝑥 = 0.25 also yielded a negative correlation with (𝑞/𝑝) 𝑐𝑟𝑖𝑡 , with 𝑅 2 =0.73 and 𝑅 2 =0.76, respectively. 𝑒 𝑠𝑘𝑒𝑙 ; 𝑥 = 1 yielded a poor corelation (𝑅 2 =0.08) with (𝑞/𝑝) 𝑐𝑟𝑖𝑡 , indicating that using a threshold coefficient, 𝑥 = 1, for the skeleton void ratio calculation for the gap-graded assemblies, resulted in the exclusion of important stress-transmitting fines, especially from the overfilled assemblies. 𝑥 = 1 is the standard for defining force-chains for narrowly-graded assemblies [START_REF] Peters | Characterization of force chains in granular material[END_REF]. In agreement with earlier studies [START_REF] Adesina | DEM study on the effect of particle shape on the shear behaviour of granular materials[END_REF][START_REF] Holubec | Effect of particle shape on the Engineering Properties of Granular Soils. Evaluation of Relative Density and Its Role in Geotechnical Projects Involving Cohesionless Soils[END_REF][START_REF] Ng | Triaxial Test Simulations with Discrete Element Method and Hydrostatic Boundaries[END_REF], at the peak, 𝑒, correlates negatively with (𝑞/𝑝) 𝑚𝑎𝑥 , and yielded the highest R-squared value (𝑅 2 =0.97), followed by 𝑒 𝑚𝑒𝑐ℎ (𝑅 2 =0.66). 𝑒 𝑠𝑘𝑒𝑙 correlated poorly with (𝑞/𝑝) 𝑚𝑎𝑥 .

Compliance with ethical standards

Conflict of interest The authors declare that there is no conflict of interests regarding the publication of this article. Publication has been approved by all authors. None of the material presented in the paper is submitted or published elsewhere.