# Micro-scale investigation of the role of fabric on internal stability of gap-graded soils

Rohan R Dhamne, Mahdi M Disfani, Antoine Wautier

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Rohan R Dhamne, Mahdi M Disfani, Antoine Wautier. Micro-scale investigation of the role of fabric on internal stability of gap-graded soils. Computers and Geotechnics, 2026, 190, pp.107713. 10.1016/j.compgeo.2025.107713. hal-05343175

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Micro-scale investigation of the role of fabric on internal stability of gap-
graded soils

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Rohan R. Dhamneª, Mahdi M. Disfaniª,*, Antoine Wautierᵇ

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ªDepartment of Infrastructure Engineering, Faculty of Engineering and Information Technology, The University of Melbourne, Australia.

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ᵇINRAE, Aix Marseille University, Aix-en-Provence, France.

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Rohan R. Dhamne, M.Tech. (ORCID: 0000-0002-6520-9986)

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Ph.D. Candidate in Geotechnical Engineering, Department of Infrastructure Engineering, Faculty of Engineering and Information Technology, The University of Melbourne, Australia.

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Mahdi M. Disfani, M.Sc., Ph.D. (ORCID: 0000-0002-9231-8598)

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Professor, Department of Infrastructure Engineering, Faculty of Engineering and Information Technology, The University of Melbourne, Australia.

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Antoine Wautier, M.Sc., Ph.D. (ORCID: 0000-0002-2551-103X)

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Researcher, INRAE, Aix Marseille University, Aix-en-Provence, France.

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*Corresponding Author:

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Mahdi M. Disfani

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Professor in Geotechnical Engineering

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Department of Infrastructure Engineering, Faculty of Engineering and Information Technology

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The University of Melbourne

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Block B, Room 207, Parkville, VIC 3010, Australia.

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Phone: +61 3 8344 5972

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Email: [mahdi.miri@unimelb.edu.au](mailto:mahdi.miri@unimelb.edu.au)
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**Abstract**

Internal stability of gap-graded soils susceptible to suffusion is governed by soil fabric, which evolves through particle rearrangement. Using three-dimensional Discrete Element Method (DEM) simulations, this study examines the impact of fine content (FC), gap ratio (GR), and relative density ($D_R$) on the role of fine particles in fabric evolution and their potential for erodibility. Stress in fine particles is evaluated using particle- and contact-based stress partitioning, together with fabric indicators such as the stress reduction factor ($\alpha$). Particle connectivity is assessed through fine coordination ($Z_{avg}^{fine}$), mechanical coordination ($Z_m$), and the mechanical coordination number of fine particles ($Z_m^{fine}$). To advance beyond conventional binary rattler methods, a novel fabric index, $I_{rattler}$, is introduced as a continuous, coordination-based measure of fine particle inactivity, enabling more accurate detection of suffusion onset and fabric transitions. Contact force transmission across different contact types is also analyzed, with emphasis on the strong contact network and its variation with FC and GR, highlighting their influence on packing stability with respect to $D_R$. The DEM framework is validated against previously published experimental and numerical studies on gap-graded soils. Results demonstrate that for GR = 4–7 with FC between 20–35%, soils occupy a transitional zone where densification promotes a shift from F–C to F–F contact dominance, thereby stabilizing the fine matrix. In contrast, at GR = 20, fines remain largely inactive until FC ≈ 35%, with stability attained abruptly at higher FC, marking a geometric upper bound for fine contribution under extreme particle size disparity.

**Keywords:** Gap-graded soil, Internal erosion, Fine particle contribution, Fabric indicators, Discrete element method (DEM).
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# 1. Introduction

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Water seepage in earthen structures can detach and transport fine particles, leading to internal erosion (Gelet and Marot, 2022), which encompasses concentrated leak, backward erosion, contact erosion, and suffusion (Fell and Fry, 2007). This research primarily focuses on the initial detachment stage of suffusion, which refers to the selective removal of the finest grains from the soil matrix. Materials that are susceptible to suffusion are said to be internally unstable. At the representative elementary volume (REV) scale, this process involves three concurrent mechanisms: detachment, transport, and migration of non-plastic fines through constrictions in the coarse skeleton, progressively altering fabric, porosity, hydraulic conductivity, and strength (Liu et al., 2025; Mehdizadeh et al., 2019; Sterpi, 2003; Yang et al., 2019).

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Internal stability is often evaluated solely from particle size distribution (PSD); however, it fundamentally depends on the contribution of fines to the stress-carrying skeleton. In unstable soils, fines reside loosely in coarse voids and are readily mobilized by seepage. In stable fabrics, they are integrated into force chains and enhance structural stability, while transitional states occur when fines are only weakly stressed and provide lateral support. Thus, fines may act as inactive, semi-active, or active constituents, corresponding to unstable, transitional, and stable zones, respectively (Prasomsri et al., 2021; Shire et al., 2014; Sufian et al., 2021a). These roles are illustrated in **Fig. 1** for a gap-graded assembly under isotropic compression.
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Schematic of a gap-graded assembly under isotropic compression, showing coarse particles (white) and fine particles (blue: inactive, grey: semi-active, red: active). Red lines indicate active fine–coarse contacts forming part of the load-bearing network</figcaption></figure>

Suffusion arises when three conditions are satisfied: (i) material susceptibility, (ii) a critical hydraulic load, and (iii) a critical stress condition (Garner and Fannin, 2010). In gap-graded soils, susceptibility depends on constriction size and continuity, which are controlled by gap ratio (GR), fine content (FC), relative density ($D_R$), particle shape, and effective stress. Although both size ratio and gap ratio describe particle size disparity, their interpretations differ. The size ratio, $D_{p,15}^c/d_{p,85}^f$ (15% of coarse particles smaller than $D_{p,15}^c$, 85% of fines smaller than $d_{p,85}^f$), is a PSD-based index long applied in broadly graded soils (Kezdi, 1969; Sherard JL, 1979) and more recently in binary mixtures (Adesina et al., 2025; Zhu et al., 2020). However, it does not capture the actual geometric separation between coarse and fine fractions. In contrast, the gap ratio (GR), the ratio of the smallest coarse to the largest fine particle (Kwang, 1990), directly measures inter-fraction spacing and better reflects transitional packing states where fines are most susceptible to detachment.

Alongside GR, the fine content (FC, mass percentage of fines) and relative density ($D_R$) strongly govern fabric transitions (Ahmadi et al., 2020; Sufian et al., 2021). Mobilization of fines generally requires a threshold
71 hydraulic gradient or seepage force (Deng et al., 2023; Liu et al., 2023; Nguyen and Indraratna, 2020; Xiong et al., 2021). Depending on the evolving stress state, fines may act as load carriers, provide lateral support, or remain passive fillers, making their role highly dependent on density. The relative density index ($D_R$),

$$D_R = \frac{e_{max} - e}{e_{max} - e_{min}} \qquad (1)$$

74 quantifies compactness by relating the current void ratio ($e$) to the loosest ($e_{max}$) and densest ($e_{min}$) packing states of the soil (Bedja et al., 2022; Igwe et al., 2012; Shi et al., 2021; Standard, 2006). In DEM studies, $D_R$ is often adjusted by varying the interparticle friction coefficient $\mu$ (Shire et al., 2014). For example, when $FC < 24\%$ and $GR$ is large, fines remain unstressed regardless of density; at $FC > 35\%$ they form a fine skeleton; and within 24–35% the effect of $D_R$ is most pronounced. These results highlight the transitional zone as the most density sensitive. However, such analyses often remain qualitative, without explicitly linking $D_R$ to material behaviour.

81 Later DEM studies further explored the combined effects of density and stress-induced anisotropy. Sufian et al. (2021) performed constant-mean-stress triaxial tests with size ratios ($\chi = d_{p,max}/d_{p,min}$, the ratio of maximum to minimum particle diameters) of 5.7, 8.0, and 14.3 ($d_{p,max} = 1$ mm), and fine contents (FC) ranging from 18–45%. Loose assemblies ($\mu = 0.3$) showed little change in the stress-reduction factor ($\alpha_p$), whereas dense assemblies ($\mu = 0.0$) exhibited sharp reductions, particularly in transitional states (25–35% FC). Similar patterns were observed by Adesina et al. (2025). Nonetheless, DEM procedures for defining $e_{min}$ and $e_{max}$ remain inconsistent, with different $\mu$ values adopted for “loosest” states. Ahmadi et al. (2020) addressed this by proposing an algorithm in which $\mu = 0$ and $\mu = 1$ correspond to $e_{min}$ and $e_{max}$, respectively. They showed that soils with $GR \ge 4$ and $FC \le 15\%$ remain unstable, while stability improves at higher density for $FC = 35\%$ and $4 \le GR \le 7$. These findings underline the need for consistent density-based approaches to quantify fine participation in soil fabrics.
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Complementary experimental work also points to the structural role of fines. For example, Cabalar and Alosman (2021) found rock powder enhanced strength in organic soils, Brito et al. (2018) showed coarse fraction variability influenced hydromechanical behaviour in soil-rockfill mixtures, and Xu et al. (2009) reported distinctive compressibility in rock-soil blends. Together, these studies emphasise that fines can either destabilise or reinforce soil skeletons depending on their content and interaction with coarse grains. Building on these insights, the present study employs a relative density-based DEM framework to systematically investigate the influence of fines on fabric evolution and suffusion susceptibility across varying gap ratios (GR), fine contents (FC), and relative densities ($D_R$). Spherical particles are employed for computational efficiency and clear interpretation of micromechanical behaviour, while acknowledging that real soils exhibit angularity and crushing, which enhance interlocking and constraint. This simplification provides a controlled baseline, with future DEM extensions (e.g., incorporating particle shape or breakage; Asadi et al., 2024) capable of capturing more complex natural effects.

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Although seepage flow is not simulated, the study focuses on the mechanical preconditions of suffusion onset, specifically, when non-plastic fines lose kinematic constraint and become detachable. Since suffusion is multi-stage, with transport and migration governed by hydraulics, this study isolates the detachment stage using micromechanical indicators such as fines coordination ($Z_{avg}^{fine}$), the stress-reduction factor ($\alpha = p'_{fine}/p'$, where $p'$ is the total mean effective stress and $p'_{fine}$ is the fines contribution), and a new inactivity index. For instance, during drained shearing of a gap-graded assembly, fines that evolve to $Z_{avg}^{fine} < 1$ with $\alpha \approx 0$ are mechanically disengaged and would require only a modest hydraulic gradient to move, whereas fines maintained at $Z_{avg}^{fine} \ge 2$ and confined by load-bearing constrictions remain immobilized. These flow-independent criteria quantify detachment susceptibility and provide a rigorous precursor to subsequent CFD-DEM modeling of transport and migration. To capture evolving structure, a novel fabric index, $I_{rattler}$, provides a continuous, coordination-based measure of fine particle inactivity, improving detection of suffusion onset and fabric transitions beyond conventional binary rattler methods. Together with parameters such as the stress reduction factor ($\alpha$),
coordination number (Z), and contact type distributions (e.g., α<sup>FF</sup>), these indicators enable a comprehensive
assessment of fabric transitions and fine particle contributions to internal stability under different packing
conditions.

## 2. DEM Simulation

The Discrete Element Method (DEM), originally developed by Cundall and Strack (1979), is widely used to investigate the fabric of granular materials at the particle scale. In this study, DEM simulations were performed using Particle Flow Code PFC3D (Itasca Consulting Group, 2024). This section describes the simulation setup, including the material properties, sample preparation procedure, and the applied loading conditions.

### 2.1 Material

Gap-graded cohesionless soils are highly susceptible to suffusion (Wan and Fell, 2008) and are commonly idealized in numerical studies as binary granular mixtures for simplicity (Ahmadi et al., 2020; Sufian et al., 2021). In this study, cubical assemblies of spherical particles were generated to represent gap-graded soils (Fig. 2). To capture the internally stable, transitional, and unstable regimes of gap-graded soils with non-plastic fines, 88 binary granular assemblies spanning loose and dense states (Table 1) were analyzed, with fine content (FC) and gap ratio (GR) systematically varied. The corresponding particle size distributions modelled in DEM are presented in Fig. 3.

<figure><img src="image_3.png" /></figure>

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<figure><img src="image_7.png" /><figcaption>Fig. 3. Particle size distribution curves at (a) GR = 2, (b) GR = 4, (c) GR = 5, (d) GR = 7, and (e) GR = 20</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 2.** Cubical assemblies of gap-graded soils with varying fine contents (FC) at a gap ratio (GR) of 2</figcaption></figure>
Table 1

<table><caption>Number of particles to generate cubical size assemblies</caption><thead><tr><th>FC</th><th colspan="2">5%</th><th colspan="2">10%</th><th colspan="2">15%</th><th colspan="2">20%</th><th colspan="2">25%</th><th colspan="2">30%</th><th colspan="2">35%</th><th colspan="2">40%</th><th colspan="2">45%</th><th colspan="2">50%</th></tr><tr><th>GR</th><th>C</th><th>F</th><th>C</th><th>F</th><th>C</th><th>F</th><th>C</th><th>F</th><th>C</th><th>F</th><th>C</th><th>F</th><th>C</th><th>F</th><th>C</th><th>F</th><th>C</th><th>F</th><th>C</th><th>F</th></tr></thead><tbody><tr><td>2</td><td>2512</td><td>1612</td><td>2512</td><td>3211</td><td>2512</td><td>5187</td><td>2512</td><td>6476</td><td>2512</td><td>9719</td><td>2512</td><td>12629</td><td>2512</td><td>15730</td><td>2512</td><td>21963</td><td>2512</td><td>24709</td><td>2512</td><td>29070</td></tr><tr><td>4</td><td>2512</td><td>13412</td><td>2512</td><td>26021</td><td>2512</td><td>41085</td><td>2512</td><td>48983</td><td>2512</td><td>77648</td><td>2512</td><td>95526</td><td>2512</td><td>125321</td><td>2512</td><td>167820</td><td>2512</td><td>188798</td><td>2512</td><td>232610</td></tr><tr><td>5</td><td>522</td><td>5272</td><td>522</td><td>10542</td><td>522</td><td>16716</td><td>522</td><td>25130</td><td>522</td><td>31618</td><td>522</td><td>42100</td><td>522</td><td>50970</td><td>522</td><td>56134</td><td>522</td><td>63151</td><td>522</td><td>94692</td></tr><tr><td>7</td><td>522</td><td>14393</td><td>522</td><td>28738</td><td>522</td><td>45703</td><td>522</td><td>68065</td><td>522</td><td>86215</td><td>522</td><td>114030</td><td>522</td><td>139238</td><td>522</td><td>197389</td><td>522</td><td>222062</td><td>522</td><td>259947</td></tr><tr><td>20</td><td>200</td><td>94481</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>200</td><td>598382</td><td>-</td><td>-</td><td>200</td><td>974715</td><td>-</td><td>-</td><td>-</td><td>-</td><td>200</td><td>1795146</td></tr></tbody></table>

where, FC - Fine Contents (%), GR - Gap Ratio, C - Number of Coarse Particles, F - Number of Fine Particles
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## 2.2 Specimen Preparation

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This study considers fine content (FC) values from 5% to 50% by weight, at 5% intervals (i.e., 5%, 10%, 15%, 20%, 25%, 30%, 35%, 40%, 45%, and 50%), combined with gap ratios (GR) ranging from 2 to 7. To further capture the effect of extreme particle size disparity, an additional case with GR = 20 was examined. However, to limit computational cost, this high GR was only simulated for FC = 5%, 25%, 35%, and 50%, representing the three key fabric regimes, underfilled, transitional (near threshold FC), and overfilled, as illustrated in **Fig. 4**. This targeted selection allows meaningful insights into the role of fines at very large gap ratios while avoiding redundant simulations across all FC levels.

<figure><img src="image_9.png" /><figcaption>**Fig. 4.** Various soil fabric states [adapted from Shire et al. (2016)]- (a) Underfilled, (b) Filled, and (c) Overfilled</figcaption></figure>

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To determine the representative elementary volume (REV), the required particle number was calculated using the power-law distribution (Ahmadi et al., 2020). For computational efficiency, the DEM specimen size was simplified, while ensuring accurate REV representation. Previous studies (Head, 2006; Liu et al., 2023; Shi et al., 2024) suggest that in physical compression tests with constrained boundaries, a specimen size to maximum particle size ratio $\ge$ 5 minimizes boundary effects. In this study, periodic boundaries were used with a ratio exceeding 5.
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The DEM timestep (Eq. 2) governs stability and was derived from a one-dimensional mass-spring system (Itasca Consulting Group, 2024), balancing accuracy and cost. A critical timestep of $\Delta t_{cr} = 4.04 \times 10^{-8}$s was obtained (Bathe and Wilson, 1976);

$$t_{crit} = \sqrt{\frac{m_i}{k_i}} \qquad (2)$$

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where $m_i$ and $k_i$ are the mass and stiffness of particle i. The timestep and other DEM input parameters are listed in Table 2.

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Table 2

<table><caption>Simulation Parameters</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Poisson's ratio, v</td><td>0.3</td></tr><tr><td>Shear modulus, G, GPa</td><td>27.0</td></tr><tr><td>Particle density, ρ, kg/m³</td><td>2670</td></tr><tr><td>Particle shape</td><td>Perfect sphere</td></tr><tr><td>Contact law</td><td>Simplified Hertz-Mindlin</td></tr><tr><td>Boundary conditions</td><td>Periodic</td></tr><tr><td>Loosest state particle-particle friction coefficient, μ<sub>max</sub></td><td>1.0</td></tr><tr><td>Densest state particle-particle friction coefficient, μ<sub>min</sub></td><td>0.0</td></tr><tr><td>Particle-wall friction coefficient, μ</td><td>0.0</td></tr><tr><td>Friction coefficient to achieve min. practical relative density (D<sub>R,min</sub>)</td><td>0.3</td></tr></tbody></table>

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In this study, the Hertz-Mindlin contact model was used to simulate inter-particle interactions in gap-graded soil assemblies. This nonlinear elastic model, widely applied in DEM for spherical particles, accounts for normal and tangential forces, capturing elastic-frictional behaviour relevant to internal stability (Haq and Shah, 2025; Shire et al., 2014). Rolling resistance and particle shape effects were not included; however, idealized spheres are sufficient to examine fabric evolution and contact-force redistribution governing instability in gap-graded soils.
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## 2.3 Loading Conditions

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Samples were generated by isotropically compressing a non-interacting particle cloud to a confining pressure of $p' = 140$ kPa (Ahmadi et al., 2020). Gravitational effects were omitted to minimize segregation and anisotropy. Relative densities were controlled using the procedure of Ahmadi et al. (2020), where $e_{min}$ and $e_{max}$ were quantified with friction coefficients of 0 and 1, respectively, regardless of actual soil friction. This methodology was directly adopted without replicating variations in $e$ due to changes in $\mu$ (Ahmadi et al., 2020). The simulations were executed for 100,000 timesteps, ensuring stabilization of creep.

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Absolute limits of $D_R = 0\%$ and 100% are rarely achievable in practice; thus, the minimum relative density ($D_{R,min}$) often exceeds zero. In DEM, this was obtained with $\mu = 0.3$, typical for glass beads (Barreto, 2009), under $p' = 140$ kPa. Higher densities were generated by gradually reducing $\mu$ from 0.3 to 0.0 in steps of 0.01, with equilibration at each step (Ahmadi et al., 2020). Dense and loose states were defined using $\mu = 0.01$ and 0.3, respectively (Adesina et al., 2025).

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As shown in Fig. 5, the minimum practical relative density ($D_{R,min}$), measured at $\mu = 0.3$, increases systematically with gap ratio ($GR = 2–7$ and 20) at a constant fine content of 25%, reaffirming the dominant role of the coarse skeleton in packing efficiency. At low GR ($< 4$), fines efficiently occupy small voids but disrupt compact arrangements, resulting in lower $D_R$. For higher GR ($\ge 4$), the skeleton dictates the packing structure: large voids cannot be effectively filled by fines, leading to inherently denser fabrics.

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A notable outcome is observed at GR = 20, where the skeleton alone achieves exceptionally high densities, representing an extreme case of skeleton control. Moreover, the impact of interparticle friction ($\mu$) diminishes with increasing GR, assemblies with GR = 7 and especially GR = 20 retain high $D_R$ even under high friction. These findings demonstrate that gap ratio not only governs void-filling efficiency but also moderates the frictional sensitivity of soil fabric, offering new mechanistic insight into the internal stability of gap-graded soils.
<figure><img src="image_10.png" /><figcaption>**Fig. 5.** Relationship between relative density ($D_R$) and interparticle friction coefficient (μ) for assemblies with different gap ratios (GR), at a constant fine content (FC = 25%)</figcaption></figure>

# 3. Examination of Packing Characteristics

## 3.1 Variations in Void ratio

The variation in void ratio ($e$) with fine content (FC) reflects the interplay between particle sizes and packing in gap-graded soils. The global void ratio, defined as the ratio of void volume to solid volume, is categorized into $e_{max}$, $e_{min}$, and $e_{ref}$, where $e_{ref}$ denotes the reference void ratio, and depends on both FC and GR (Ahmadi et al., 2020).

As FC increases, $e$ decreases until reaching a local minimum, after which it rises again. This “V-shaped” trend (Fig. 6) reflects a transition from underfilled fabrics, where fines occupy voids between coarse particles, to overfilled fabrics, where fines separate the coarse grains and form a fine-dominated matrix. This behaviour aligns with earlier experimental (Lade et al., 1998; Lade and Yamamuro, 1997; Thevanayagam et al., 2002) and numerical studies (Shire et al., 2016).

Void ratio decreases with increasing gap ratio (GR), underscoring the growing dominance of the coarse skeleton in controlling packing. For GR < 4 (Fig. 6(a)), variations are minor and resemble uniform soils,
whereas GR = 4–7 (Figs. 6(b-d)) exhibit marked reductions up to FC ≈ 25–30%, reflecting the transition from underfilled to load-sharing fabrics. At GR = 20 (Fig. 6(e)), void ratios are exceptionally low and largely independent of FC, representing an extreme case of skeleton dominance. These results demonstrate that GR not only governs void-filling efficiency but also dictates whether fines actively reorganize the fabric, providing new insight into fabric transitions and internal stability.

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>
<figure><img src="image_15.png" /><figcaption>**Fig. 6.** Variations in void ratio with changing FC across different GR: (a) GR = 2, (b) GR = 4, (c) GR = 5, (d) GR = 7, and (e) GR = 20</figcaption></figure>

**3.2 Experimental and Numerical Data Validation**

The threshold fine content ($FC_{th}$), defined as the minimum fine content required for fine particles to interact and share contacts with the coarse particles, was found to vary with both initial packing density and particle size disparity, expressed by the gap ratio (GR). In loose assemblies, the reference void ratio ($e_{ref}$) at $\mu = 0.3$ consistently indicated a threshold fine content ($FC_{th}$) of ≈ 30% for GR values of 4–7. For GR = 20, the threshold was observed at $FC ≈ 35\%$, while no distinct threshold was evident for GR = 2 (Fig. 7). This finding aligns well with the DEM results of Sufian et al. (2021a), which reported a comparable threshold value of $FC_{th} ≈ 30\%$ for assemblies prepared using similar methods with size ratios $\lambda \in [5.7; 14.3]$.

It is important to note that while the particle size ratio ($\lambda = d_{max}/d_{min}$, defined as the ratio of maximum to minimum particle diameters) is a distribution-based index derived from particle size distribution (PSD), the gap ratio (GR) used in this study directly quantifies the geometric separation between the coarse and fine fractions. This makes it particularly suited for analyzing the fabric structure and instability potential of soils with distinct bimodal gradation. In the present study, $FC_{th} ≈ 30\%$ was observed across GR values ranging from 4 to 7 (Fig. 7), and this threshold was subsequently used to validate the results against previous experimental and numerical studies. For GR = 20, however, the threshold shifted to about 35%, reflecting
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an extreme case of skeleton dominance; while this variation is highlighted in **Fig. 7**, it is treated here as a
novel extension rather than validated in subsequent comparisons.

<figure><img src="image_16.png" /><figcaption>**Fig. 7.** Effect of fine content (FC) on the reference void ratio ($e_{ref}$) for assemblies with different gap ratios (GR = 2-7 and 20)</figcaption></figure>

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Validation was conducted using studies with similar micromechanical and modeling assumptions. All datasets involved gap-graded assemblies of spherical particles modeled with the Hertz–Mindlin contact law, particle density of 2670 kg/m³, Poisson’s ratio of 0.3, and shear modulus of 27 GPa. Periodic boundary conditions were applied to cubical assemblies to reduce boundary effects, and the reference void ratio at the threshold condition ($e_{ref,threshold}$) was used as the validation metric.

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As expected, the void ratio decreases with increasing particle size disparity. While previous experimental and numerical studies (Lade et al., 1998; Shire et al., 2016; Adesina et al., 2025) characterized this disparity using the particle size ratio ($\lambda = d_{max}/d_{min}$), the present study adopts the gap ratio (GR) to more explicitly capture void-filling behaviour between coarse and fine particles. Despite the conceptual difference between these two metrics, **Fig. 8** demonstrates a comparable trend: a steep reduction in $e_{ref,threshold}$ up to $\lambda \approx 7$ and GR $\approx 4$ followed by a more gradual decline at higher values. The present 3D-
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DEM results (GR ∈ [2; 7]) show strong agreement with prior findings (Fig. 8), thereby validating the GR-based framework and confirming its consistency with established size ratio-based trends.

<figure><img src="image_17.png" /><figcaption>**Fig. 8.** Comparison of threshold reference void ratios ($e_{ref,threshold}$) from prior studies using particle size ratio ($\lambda = d_{max}/d_{min}$) and the present study using Gap ratio (GR). Despite differing metrics, all results show a similar decreasing trend with increasing particle size disparity, validating the GR-based approach</figcaption></figure>

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**3.3 Variations in Coordination Number**

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The average coordination number ($Z_{avg}$), defined as the average number of contacts per particle, is commonly used to quantify the packing density of bimodal granular mixtures at the microscale (Eq. 3). Several researchers (Cao et al., 2021; Li et al., 2020; Minh et al., 2014; Shire et al., 2014) have employed this index to evaluate the internal stability of gap-graded soils by examining the interactions between fine-fine (F-F), fine-coarse (F-C), and coarse-coarse (C-C) contacts within bimodal mixtures of coarse and fine particles.
$$Z_{avg} = \frac{2N_c}{N_p} \qquad (3)$$

where $Z_{avg}$ is the average coordination number; $N_c$ defines the number of contacts shared by coarse and fine particles both; and $N_p$ denotes the number of particles involved in the particle assemblies.

Particle stresses are directly governed by their contact points (Skempton and Brogan, 1994). To evaluate the role of fine particles, classified as inactive, semi-active, or active, Minh and Cheng (2013) employed the coordination number of fines (Eq. 4).

$$Z_{avg}^{fine} = \frac{\sum_{i=1}^{N_{p,fine}} (C_i^{fine-fine} + C_i^{fine-coarse})}{N_{p,fine}} \qquad (4)$$

where $Z_{avg}^{fine}$ denotes the fine coordination number; $N_{p,fine}$ denotes the total number of fine particles present in the particle assemblies; $C_i^{fine-fine}$ denotes the number of contacts shared by fine particle i and other fine particles; and $C_i^{fine-coarse}$ denotes the number of contacts shared by fine particle i and other coarse particles in the particle assemblies.

**Fig. 9** presents the evolution of the average coordination number of all particles ($Z_{avg}$), fine particles ($Z_{avg}^{fine}$), and the percentage of fine rattlers under dense conditions across FC and GR values. For GR < 4 (Fig. 9(a)), $Z_{avg}$ and $Z_{avg}^{fine}$ remain nearly identical, showing that fines are fully integrated into the contact network, behaving like a uniformly graded soil. In contrast, for GR = 4–7 (Figs. 9(b–d)), a clear divergence between $Z_{avg}$ and $Z_{avg}^{fine}$ at low FC (≤ 20%) reveals a coarse-dominated regime where fines are largely inactive. This is corroborated by the high percentage of fine rattlers observed at low FC for GR ≥ 4, consistent with classical suffusion concepts (Kezdi, 1969), which emphasize that a minimum gap ratio is required for fines to become structurally inactive. A distinct threshold fine content ($FC_{th}$) is observed at ≈ 25–30%, beyond which fines engage in the contact network, reducing rattlers and marking a transition from unstable to stable fabrics.

For GR = 20 (Fig. 9(e)), this trend intensifies: fines remain inactive until higher FC, with the threshold shifting to ≈ 35%. Once this threshold is reached, rattlers drop sharply as fines begin to provide secondary
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support within the soil skeleton. This demonstrates that increasing GR not only delays the engagement of fines but also elevates the threshold required for stability, reinforcing the critical role of gap ratio in controlling internal stability mechanisms.

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<figure><img src="image_23.png" /><figcaption>**Fig. 9.** Evolution of the average coordination numbers for all particles ($Z_{avg}$) and fine particles ($Z_{avg}^{fine}$) under dense conditions with increasing fine content across different gap ratios: (a) GR = 2, (b) GR = 4, (c) GR = 5, (d) GR = 7, and (e) GR = 20</figcaption></figure>
Following Thornton (2000), the mean mechanical coordination number ($Z_m$) was introduced as an index to assess only the particles with two or more contacts, excluding inactive ones (Eq. 5).

$$Z_m = \frac{2N_c - N_1}{N_p - N_0 - N_1} \qquad (5)$$

where $Z_m$ is the mechanical coordination number, $N_c$ defines the total number of contacts in the particle assembly, $N_1$ and $N_0$ are the number of particles carrying single or no contacts, and $N_p$ defines the number of particles in the assembly.

**Fig. 10** compares the average coordination number ($Z_{avg}$) and the mechanical coordination number ($Z_m$) across different gap ratios (GR). In all cases, $Z_m$ remains above 4, confirming that the load-bearing skeleton is always stabilized by well-connected particles. For GR < 4 (Fig. 10(a)), the difference between $Z_{avg}$ and $Z_m$ is small, showing that fine particles are fully engaged in load transmission. As GR increases (4–7), $Z_{avg}$ drops sharply at low FC (Figs. 10(b-d)), revealing that most of the fine particles become inactive until a threshold FC of about 25–35%, beyond which fines rejoin the skeleton and $Z_{avg}$ rises.

A novel finding emerges for GR = 20 (Fig. 10(e)), fines remain excluded until FC ≈ 35%, but once this threshold is crossed, they provide significant secondary support, rapidly narrowing the gap between $Z_{avg}$ and $Z_m$. This highlights that extreme gap ratios not only delay fine participation but also amplify their contribution once engaged, underscoring the critical role of GR in fabric transitions and stability.

The divergence between $Z_{avg}$ and $Z_m$ reveals a heterogeneous contact network in gap-graded soils, where most of the fine particles remain inactive or weakly integrated, especially at low FC and high GR. This disparity is key to understanding fabric transitions, load transfer efficiency, and the role of fines in suffusion potential and strength development across different densities.
<figure><img src="image_24.png" /><figcaption>**Fig. 10.** Evolution of the average coordination ($Z_{avg}$) and mechanical coordination number ($Z_{m}$) for particles under dense conditions with increasing fine content across different gap ratios: (a) GR = 2, (b) GR = 4, (c) GR = 5, (d) GR = 7, and (e) GR = 20</figcaption></figure>

The mechanical coordination number of fine particles ($Z_{m}^{fine}$), as defined by Liu et al. (2023), provides further insight into the transmission of forces between the coarse and fine fractions (Eq. 6). This index,
originally conceptualized in the work of Thornton (2000), is illustrated in Fig. 11 and serves to characterize
the connectivity of fine particles within the contact network, including both semi-active and active states.

$$Z_m^{\text{fine}} = \frac{2C_{f-f} + C_{c-f} - N_{p,f1}}{N_{p,f} - N_{p,fo} - N_{p,f1}} \qquad (6)$$

where $C_{f-f}$ and $C_{c-f}$ denote the number of force-transmitting fine-fine and fine-coarse contacts, respectively; $N_{p,f}$ is the total number of fine particles; and $N_{p,fo}$ and $N_{p,f1}$ represent the numbers of fine particles with single contacts and no contacts, respectively.

To clarify the role of non-rattler fines (either provides a primary or secondary role in the overall contact network) in assembly stability, Fig. 11 compares the overall mechanical coordination number ($Z_m$) with that of fines ($Z_m^{\text{fine}}$). For GR < 4 (Fig. 11a), both fine and coarse particles achieve $Z_m > 4$, indicating a strong contact network consistent with the contribution of non-rattler fines. At intermediate GR (4–5; Figs. 11b–c), fines undergo a clear transition from semi-active to active states within FC = 20–25%, reflected in $Z_m^{\text{fine}}$ and aligning with overall assembly stability. For GR = 7 (Fig. 11d), a higher fine content of about 25–35% is needed for fines to develop a stable contact network and contribute to overall structural stability.

A novel outcome is observed for GR = 20 (Fig. 11(e)), where fines remain unable to provide meaningful secondary support until FC ≈ 35%. Beyond this threshold, however, $Z_m^{\text{fine}}$ rises sharply, signifying a sudden transition from inactive to structurally engaged states. This finding shows that at extreme particle size disparities; fines exhibit a delayed but intensified contribution to the load-bearing skeleton. Practically, this implies that soils with very large particle size disparities may remain unstable until a higher fines content is reached, after which their stability improves abruptly, a critical insight for assessing suffusion potential and designing internally stable filters.
<figure><img src="image_25.png" /><figcaption>**Fig. 11.** Variation in the mechanical coordination number ($Z_m$) and the coordination number of fines ($Z_m^{fine}$) across varying FC and GR: (a) GR = 2, (b) GR = 4, (c) GR = 5, (d) GR = 7, and (e) GR = 20</figcaption></figure>
To highlight the role of structurally engaged fines, **Fig. 12** compares the average coordination number of fines ($Z_{avg}^{fine}$) with the mechanical coordination number of fines ($Z_{m}^{fine}$). For GR < 4 (**Fig. 12a**), both indices follow a similar trend with minimal variation, since nearly all fines contribute to the contact network. As particle size disparity increases (GR > 4; **Figs. 12b–d**), a pronounced difference emerges between $Z_{avg}^{fine}$ and $Z_{m}^{fine}$ at low fine contents, reflecting the presence of fines that remain poorly connected. This divergence provides a clear indicator of the threshold fine content, beyond which fines begin to actively participate in the load-bearing skeleton. For instance, at GR = 5 (**Fig. 12(c)**), both indices converge at FC ≈ 35% signifying the point where fines attain $Z > 4$ and transition into a fully active state. For GR = 20 (**Fig. 12(e)**), this divergence is even more pronounced, with fines requiring a higher FC (≈ 35%) before their coordination rises sharply, underscoring a delayed but stronger transition into structural engagement.

The novelty of this approach lies in its ability to use coordination-based measures to pinpoint the onset of fine particle engagement, linking microscale contact mechanics to bulk stability. From a practical standpoint, this offers a rational, mechanics-based criterion for identifying threshold fine contents, thereby improving the assessment of internal stability and supporting the design of soil filters that are more resistant to suffusion and internal erosion.

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<figure><img src="image_31.png" /><figcaption>**Fig. 12.** Variation of avg. coordination number ($Z_{avg}^{fine}$) and mechanical coordination number ($Z_{m}^{fine}$) of fines across different fine contents and gap ratios: (a) GR = 2, (b) GR = 4, (c) GR = 5, (d) GR = 7, and (e) GR = 20</figcaption></figure>

Previous studies (Minh et al., 2014; Taha et al., 2019) characterized soil fabric primarily through contact network indicators, focusing on overall connectivity. Yet, the role of fine particles as active load carriers within this network has been less explored. **Fig. 13** addresses this gap by showing the variation in active fine particles across different fine contents (FC), gap ratios (GR), and density conditions.

For low gap ratio (GR < 4), most fines consistently participate in the contact network through either fine-fine (F-F) or fine-coarse (F-C) contacts, largely independent of density. At intermediate GR values (4-7), the structural activation of fines becomes highly dependent on both FC and density: in dense packings (Fig. 13(b)), a sharp increase in active fines occurs once FC exceeds ~15%, while in loose packings (Fig. 13(a)) a similar trend emerges only beyond ~30%.

The extreme case of GR = 20 reveals a distinct fabric response. At such large size disparity, fines remain structurally disengaged across a wide range of FC, with activation only occurring once FC exceeds ~ 35%.
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Beyond this threshold, their contribution rises abruptly, indicating that excessively large coarse voids prevent fines from forming stable contacts until they volumetrically dominate the assembly. Thus, GR not only governs void-filling efficiency but also raises the critical FC required for fines to transition from passive fillers to active load carriers.

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Through the quantification of fines activation, this study advances a new micromechanical criterion that links particle-scale disengagement to fabric transitions, offering improved predictive capability for internal stability in gap-graded soils.

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<figure><img src="image_32.png" /><figcaption>**Fig. 13.** Percentages of active fine particles in terms of distribution of contact networks, across different FC and GR, for (a) Loose, and (b) Dense soil conditions</figcaption></figure>

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**3.4 Variation Index ($I_{rattler}$)**

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Understanding the role of structurally inactive fine particles, commonly referred to as "rattlers", is essential for predicting the erosion potential and fabric transitions in gap-graded soils. Previous studies (Adesina et al., 2024; Christoffersen et al., 1981) have identified rattlers using a binary criterion, typically classifying particles with zero or one contact as inactive. However, such approaches lack a scalable,
assembly-level perspective. To address this limitation, the present study introduces a novel coordination-based fabric index, $I_{rattler}$, defined as:

$$I_{rattler} = \frac{Z_{m}^{fine}}{Z_{avg}^{fine}} - 1 \qquad (7)$$

where $Z_{m}^{fine}$ represents the mechanical coordination number of fine particles and $Z_{avg}^{fine}$ represents the average coordination number of fine particles. This variation index is derived by normalizing $Z_{m}^{fine}$, which helps in identifying semi-active and active fines, with respect to $Z_{avg}^{fine}$, which identifies fines as inactive, semi-active and active. The relative deviation is accounted for by subtracting 1.

Unlike conventional rattler detection methods that rely on a binary classification, the proposed index $I_{rattler}$ offers a continuous, normalized measure of fine particle inactivity. This innovation captures the varying degrees of disengagement among fines, providing a more accurate representation of their structural role within the granular skeleton. Crucially, $I_{rattler}$ advances understanding of internal erosion (suffusion) by directly quantifying the fraction of fines that are mechanically disengaged and therefore more susceptible to detachment and migration under seepage or cyclic loading.

Beyond offering clear physical meaning, the index can be seamlessly integrated into equivalent granular void ratio frameworks (Rahman et al., 2008), improving predictions of key engineering behaviours such as steady-state strength and cyclic resistance. By explicitly linking microscale fabric evolution with macroscale instability, $I_{rattler}$ delivers a novel and practically applicable improvement over existing criteria, representing a robust step forward in the micromechanical modelling of internal instability in gap-graded soils.

**Fig. 14** illustrates the variation in the proportion of inactive fines ($I_{rattler}$) across different fine contents (FC) and gap ratios (GR). For GR = 2 (Fig. 14(a)), most fines are engaged in the contact network, resulting in low $I_{rattler}$ values with little sensitivity to density. As GR increases, inactive fines become more prominent: at GR = 4 (Fig. 14(b)), nearly half the fines remain structurally disengaged when FC < 20%, while density effects become pronounced around the transitional range (25–35%).
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For GR = 5 and 7 (Figs. 14(c–d)), the transitional regime is even clearer, with $I_{rattler}$ dropping sharply as FC exceeds ~30–35%, marking the point where fines begin to laterally support the coarse skeleton and eventually form their own stress-bearing fabric.

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At GR = 20 (Fig. 14(e)), an important trend is observed. At low–moderate FC (5–25%), $I_{rattler}$ remains high, indicating many fines are poorly connected. Density plays a key role at FC = 35%, where dense packings show lower $I_{rattler}$ than loose ones, reflecting added fine–coarse and fine-fine contacts under compaction. At FC = 50%, inactivity drops further as fines cluster and provide secondary support by laterally confining coarse particles. While they cannot form continuous load-bearing chains, this reduction in $I_{rattler}$ demonstrates their stabilising role through confinement rather than direct stress transmission.

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The proposed index provides a practical tool to identify when fines shift from merely occupying voids to actively stabilising the soil skeleton, offering engineers a more reliable basis for assessing internal stability and designing filters against suffusion in hydraulic structures.

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Loose

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Dense

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<figure><img src="image_39.png" /><figcaption>Fig. 14. Proportion of inactive fine particles ($I_{rattler}$) across varying FC and GR, with relative density effects shown for (a) GR = 2, (b) GR = 4, (c) GR = 5, (d) GR = 7, and (e) GR = 20</figcaption></figure>

## 3.5 Stress-Based Partition

### 3.5.1 Particle-Based Stress Partition

The effect of fine particles on soil fabric and their potential erodibility within granular assemblies can be evaluated by calculating the stress tensor at the particle scale. This requires examining the individual contribution of each fine particle to the overall stress transmission. The stress tensor for a given particle $m$, $\sigma^m$, is represented as (Potyondy and Cundall, 2004)

$$ \sigma^m = \frac{1}{V^m} \sum_{c=1}^{N_c^m} f^c (x^c - x^m) \qquad (8) $$

where $V^m$ is particle volume; $N_c^m$ defines the number of contacts involving the particle; $f^c$ is the contact force vector; $x^c$ defines the position of the contact point; and $x^m$ represents the position of the particle centroid. The stress tensor for the entire assembly, $Tr(\sigma)$, can be evaluated as (Eq. 9)

$$ Tr(\sigma) = \frac{1}{V} \sum_{m=1}^{N_p} V^m Tr(\sigma^m) \qquad (9) $$

where $V$ is the total volume of the particle assembly, $N_p$ denotes the number of particles in the assembly, $V^m$ is the volume associated with particle $m$, and $Tr(\sigma^m)$ is the trace of the stress tensor for the individual particle $m$. The stress tensor for the entire assembly can be further decomposed into fine $Tr(\sigma^f)$ and coarse $Tr(\sigma^c)$ fractions, as evaluated by Sufian et al. (2021a) (Eq. 10).
transition to Type-3 ($\alpha > 0.8$) emerges at $FC \ge 35\%$ (Fig. 15(g)), particularly evident for $GR = 7$, where assemblies evolve from unstable at low densities to stable at higher densities.

For $GR = 20$, soils remain in Type-1 until $FC \approx 35\%$ (Figs. 15(a-g)); only at higher densities do fines become structurally engaged, reflecting delayed but abrupt activation. At $FC = 50\%$ across $GR = 2-7$ (Fig. 15(j)), overfilling occurs, with fines dominating the fabric and suspending coarse particles. These findings, consistent with Indraratna et al. (2015) and Ahmadi et al. (2020), provide a micromechanical basis for fabric classification. Practically, this framework serves as a predictive tool for assessing suffusion potential and filter design, particularly in cases where high gap ratios lead to stability only at elevated densities.

<figure><img src="image_40.png" /><figcaption>**Fig. 15.** Variation of the stress reduction factor ($\alpha$) with fine content (FC) and gap ratio (GR) at different relative densities ($D_R$). The initial $D_R$ values correspond to the minimum practical density obtained at $\mu = 0.3$</figcaption></figure>

While the stress reduction factor ($\alpha$) is often applied as a bulk average across all fines, this study advances the approach by computing $\alpha$ at the particle scale. By linking $\alpha$ with the local coordination number ($z$), defined as the number of contacts an individual fine shares with its neighbours, fines were categorised into three structural zones: [0–2], [3–5], and [6–8]. This framework enables a nuanced assessment of fine particle activity based on connectivity rather than relying solely on global averages.
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**Fig. 16** shows the correlation between coordination number (z) and stress reduction factor (α) at GR = 4. For FC < 25% (Figs. 16(a-c)), fines mainly occupy the low-coordination zone (z = 0–2) with α ≈ 0, indicating mechanical inactivity. By contrast, fines with higher coordination (z = 6–8) maintain consistently high α values, highlighting their persistent structural role regardless of FC.

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At intermediate fines content (25–35%; Figs. 16(d–e)), the majority of fines fall within the semi-active zone (z = [3–5]), providing lateral support to the coarse skeleton and strengthening inter-particle contacts. At FC = 50% (Fig. 16(f)), the fractions of fines with z = [3–5] and [6–8] become comparable, with α exceeding 0.5 in both groups, signalling a transition toward fines-dominated, borderline stable structures.

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This particle-scale disaggregation of α not only clarifies how fines progressively evolve from inactive fillers to active load carriers but also introduces a novel micromechanical pathway for linking local connectivity with global internal stability.

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<figure><img src="image_45.png" /><figcaption>**Fig. 16.** Correlation between local coordination number (z) and stress reduction factor (average α) at a fixed gap ratio of 4, across different fine contents: (a) FC = 10%, (b) FC = 15%, (c) FC = 20%, (d) FC = 25%, (e) FC = 35%, and (f) FC = 50%</figcaption></figure>

<figure><img src="image_46.png" /><figcaption>**Fig. 17** illustrates how the gap ratio (GR) governs the structural role of fines when fine content is fixed at 25%. As GR increases (Figs. 17(a–d)), the proportion of fines within the semi-active coordination zone (z = 3–5) shows a marked decline, while a growing fraction of fines shifts into the low-coordination zone (z = 0–2), where their stress reduction factor (α) becomes negligibly small. This transition reflects the gradual suppression of fines from contributing to load transfer as particle size disparity widens. Most notably, at GR = 20 (Fig. 17(e)), fines are rendered almost entirely inactive, revealing a critical state where extreme particle size disparity nullifies their stabilising influence.</figcaption></figure>

The novelty of this analysis lies in its particle-scale resolution. By differentiating fines according to coordination zones, their micromechanical contribution can be explicitly linked to GR. This coordination-based framework demonstrates that wider particle size gaps progressively erode the semi-active role of fines, ultimately pushing them toward structural redundancy. From a practical perspective, these findings provide a mechanistic basis for identifying conditions under which gap-graded soils are prone to instability and suffusion, offering engineers a physically grounded criterion for filter design and long-term soil-structure performance assessment.
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<figure><img src="image_51.png" /><figcaption>**Fig. 17.** Influence of gap ratio (GR) on the distribution of fines across local coordination zones (z) and the corresponding variation in average stress reduction factor (α) at a fixed fine content of 25%: (a) GR = 2, (b) GR = 4, (c) GR = 5, (d) GR = 7, and (e) GR = 20</figcaption></figure>

### 3.5.2 Contact-Based Stress Partition

Analyzing the interparticle contact network provides deeper insight into the interaction between fine and coarse particles, complementing the stress reduction factor (α), which serves as a useful fabric indicator but offers limited information on such interactions. Since stress in a granular assembly is derived from interparticle forces, it is feasible to obtain a contact-based stress partition. As demonstrated in previous
studies (Christoffersen et al., 1981; Gong et al., 2024), the contact force vector and branch length vector are
key elements in determining the overall stress transmitted through contacts (Eq. 12).

$$
\sigma = \frac{1}{V} \sum_{c=1}^{N_c} f^c l^c \qquad (12)
$$

where $f^c$ denotes the contact force vector applied at contact point $c$ of the particles, $l^c$ denotes the branch length vector connecting the centroid of the particle to contact point $c$ of another connected particle, $N_c$ denotes the number of contacts in the domain, $V$ is the total volume of the particle assembly and $\sigma$ is overall stress developed in the particle assembly. $\sigma$ can be further decomposed to understand the contributions of each contact type, that is, contact between (i) coarse-coarse; (ii) coarse-fine; and (iii) fine-fine particles, labelled as cc, cf, and ff, respectively. The stress developed for each contact types, $\sigma^m$, is determined as (Eq. 13)

$$
\sigma^m = \frac{1}{V} \sum_{c=1}^{N_c^m} f^c l^c \qquad (13)
$$

where $N_c^m$ denotes the number of contacts within the specified contact types $m \in \{cc, cf, ff\}$ and $\sigma = \sum \sigma^m$.

$$
\sigma^{cc} = \frac{1}{V} \sum_{c=1}^{N_c^{cc}} f^c l^c; \qquad \sigma^{cf} = \frac{1}{V} \sum_{c=1}^{N_c^{cf}} f^c l^c; \qquad \sigma^{ff} = \frac{1}{V} \sum_{c=1}^{N_c^{ff}} f^c l^c; \qquad (14)
$$

The stress developed from strong contact types ($\sigma^{strong,m}$) can also be evaluated by comparing the stress values of each contact type with the average value ($\sigma^{average,m}$) (Eq. 15) (Radjai and Wolf, 1998; Thornton, 2000). This will further help in differentiating those contact types which carry the lower amount of stress values and are not involved in the force chain formed by the solid skeleton.

$$
\sigma^{strong,m} = \frac{1}{V} \sum_{c=1}^{N_c^{strong,m}} f^c l^c \ge \sigma^{average,m} = \frac{1}{V} \sum_{c=1}^{N_c^{average,m}} f^c l^c \quad (15)
$$

where $N_c^{strong,m}$ refers to the number of strong contacts (if contact forces greater than 1.2 times the average
contact force ($F_{avg}$) values (Minh et al., 2014)) and $N_c^{average,m}$ refers to the average number of contacts
within the specified contact types $m$, representing all (both strong and weak). In similar manner to Eq. 11,
the proportion of stress quantity carried by different contact types ($\beta^m$), when the stress quantity of interest is the mean effective stress developed on the particles, is expressed as follows: (Eq. 16)

$$ \beta^m = \frac{p'^m}{p'} \qquad (16) $$

where $p'^m$ is the mean effective stress for a partition/contact type $m$, $p'$ is the mean effective stress developed on the entire assembly, and $\beta^m$ represents the proportion of stress quantity carried by contact type $m$. For the entire assembly, the following condition holds (Eq. 17):

$$ \beta^{cc} + \beta^{cf} + \beta^{ff} = 1; \quad m \in \{cc, cf, ff\} \qquad (17) $$

While $\alpha$ parameter only distinguishes whether an assembly is dominated by fine or coarse particles, $\beta^m$ allows for a more detailed classification into four categories of material:

(i) $\beta^{cc} \ge \beta^{cf} \ge \beta^{ff}$: coarse-dominated regime (C) (the assembly is primarily governing by coarse-coarse contacts)

(ii) $\beta^{cf} \ge \beta^{cc} \ge \beta^{ff}$: transitional coarse-dominated regime (TC) (coarse-fine contacts play a significant role, with a decreasing influence from coarse-coarse contacts)

(iii) $\beta^{cf} \ge \beta^{ff} \ge \beta^{cc}$: transitional fine-dominated regime (TF) (coarse-fine contacts dominate, followed by fine-fine contacts, with minimal influence from coarse-coarse contacts)

(iv) $\beta^{ff} \ge \beta^{cf} \ge \beta^{cc}$: fine-dominated regime (F) (fine-fine contacts dominate the assembly, with a secondary role from coarse-fine contacts)

Using this concept, Fig. 18 illustrates how the proportion of stress quantity carried by different contact types varies across different FC and GR as the fabric evolves.

For GR < 4 (Fig. 18(a)), the stress matrix is shared almost equally by coarse-coarse (C-C) and fine-related contacts, as the size disparity is minimal. When GR increases to 4 (Fig. 18(b)), C-C contacts dominate at low fines content (FC < 25%), reflecting an underfilled state where fines act mainly as void fillers. As FC rises to 25-35%, C-F contacts progressively assume a controlling role, marking a transitional regime in which fines provide secondary structural support. At GR = 7 (Fig. 18(d)), this transitional phase
emerges only beyond FC ≈ 30%, since a greater volume of fines is required to bridge the wider size gap. Ultimately, once FC exceeds ~35% for GR = 4–5 (Figs. 18(b-c)), or ~40% for GR = 7 (Fig. 18(d)), F–F contacts dominate the stress matrix, indicating a fines-controlled soil fabric.

At GR = 20 (Fig. 18(e)), the contact fabric behaves distinctly from lower GRs. Fines are largely excluded from the stress matrix up to FC ≈ 35%, leaving stress transmission dominated by C–C contacts. Beyond this point, the system shifts into a C–F controlled regime, where fines act as secondary load carriers. This shows that extreme particle size disparity delays stress redistribution, leading to an abrupt onset of C–F dominated support.

The novelty of this framework lies in identifying threshold fines content for fabric transitions under varying GR, directly linking particle-scale contact types to macro-scale instability regimes. This provides a mechanistic basis for anticipating shifts from coarse- to fine-dominated behaviour, offering practical guidance for evaluating internal stability and suffusion resistance.

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<figure><img src="image_57.png" /><figcaption>**Fig. 18.** Proportion of stress transmitted through different contact types as a function of fine content (FC) under dense conditions, for various gap ratios: (a) GR = 2, (b) GR = 4, (c) GR = 5, (d) GR = 7, and (e) GR = 20</figcaption></figure>

The contact network can be divided into weak and strong force networks based on whether the normal contact force ($f_c$) is less than or greater than the average normal contact force values ($f_{avg}$) (Radjai and Wolf, 1998). **Fig. 19** shows the variation in the percentages of strong contact types at various fine contents and gap ratios provides a clear understanding of the role of interparticle contacts. In case of GR = 2 (**Fig. 19(a)**), a similar variation in the stress contribution from strong contact types was noted across different fine contents. Moreover, for GR values of 4 (**Figs. 19(b) & (c)**), the dominance shifts to C-C contact types up to 15% of FC, indicating that fines up to this threshold do not contribute to the stress-bearing matrix and instead occupy void spaces. As GR increases to 7 (**Fig. 19(d)**), the void spaces within the material expand.

This expansion widens the transitional state of the fabric, necessitating ~35% fines to fully fill the void space in the skeleton. At GR = 7 (**Fig. 19(d)**), with FC ranging from 20–35%, the behaviour is governed by C–F interactions, forming columnar structures that effectively sustain external loads. This aligns with the particle-based stress partition in **Fig. 15**, which identifies the threshold FC for different GR values. Notably, at GR = 20 (**Fig. 19(e)**), the stress matrix remains coarse-dominated up to FC ≈ 35%. Beyond this threshold, a sudden rise in C–F contacts marks a delayed yet abrupt transitional regime, where fines provide only secondary support.
<figure><img src="image_58.png" /><figcaption>**Fig. 19.** Proportion of stress transmitted through strong contact types as a function of fine content (FC) under dense conditions, for different gap ratios: (a) GR = 2, (b) GR = 4, (c) GR = 5, (d) GR = 7, and (e) GR = 20</figcaption></figure>

### 3.5.3 Stress Transmission through Fine Particles

To evaluate the contribution of fine particles to the mechanical behavior of gap-graded soils, Otsubo et al. (2022) applied a contact-based stress partitioning approach to examine deeper insights into stress

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transmission through contacts involving fine particles only. Within this framework, stresses transmitted via fine particles were expressed using two indices: (i) $\alpha^{FF}$, which quantifies the proportion of stress carried exclusively by fine-fine contacts (Eq. 18), and (ii) $\alpha_p$, as defined by Shire et al. (2014), which accounts for the stress transmitted by fines through both coarse-fine and fine-fine contacts (Eq. 19), as outlined below:

$$ \alpha^{FF} = \frac{V_s \sum_{m=1}^{N_{p,f}} (Tr(\sigma_{ff}^m) V_s^m)}{V_s^f \sum_{m=1}^{N_p} (Tr(\sigma^m) V_s^m)} \quad (18) $$

$$ \alpha_p = \frac{V_s \sum_{m=1}^{N_{p,f}} (Tr(\sigma^m) V_s^m)}{V_s^f \sum_{m=1}^{N_p} (Tr(\sigma^m) V_s^m)} \quad (19) $$

where $V_s = (1 - n)V = V_s^f + V_s^c$ is the total volume of solids, including both fine ($V_s^f$) and coarse particles ($V_s^c$), n represents the porosity of the sample, and V is the total volume of the particle assembly. The term $V_s^m$ refers to the solid volume of particle m, respectively. Additionally, $Tr(\sigma_{ff}^m)$ and $Tr(\sigma^m)$ represent the trace of stress tensor corresponding to the mean stresses arising from fine-fine interactions and all contacts involving particle m, respectively. These two indices distinguish stresses carried solely by fine-fine contacts from those transmitted through all fine-related contacts, offering deeper insight into the evolving role of fines across varying FC and GR.

**Fig. 20** shows the variation of the particle-based stress reduction factor ($\alpha_p$) and the stress fraction carried solely by fine-fine contacts ($\alpha^{FF}$) with fines content (FC) across different gap ratios (GR) in dense assemblies. For GR < 4 (indicated by solid circles), $\alpha_p$ rises steadily with FC due to combined fine-coarse and fine-fine contacts, while $\alpha^{FF}$ stays negligible until ~20% FC, showing that fines are well integrated into the skeleton and enhance stability.

At GR = 4–5, $\alpha_p$ shows a transitional regime near FC ≈ 25%, reflecting the emergence of semi-active fines through coarse-fine contacts, while $\alpha^{FF}$ becomes significant only at higher FC (~ 50%) once fines form load-bearing networks. At GR = 20 (star markers in Fig. 20), the large voids of the coarse skeleton suppress both $\alpha_p$ and $\alpha^{FF}$ across the FC range, preventing fines from acting as primary load carriers. Instead, with increasing FC, fines cluster within voids and act as lateral constraints, restricting coarse particle rotation
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and rearrangement. This confinement effect stiffens the skeleton indirectly, revealing that under extreme
disparity fines shift from load carriers to lateral stabilisers, a previously overlooked mechanism controlling
fabric stability.

<figure><img src="image_59.png" /><figcaption>**Fig. 20.** Proportion of stress carried by different contact types with varying FC and GR for dense assemblies</figcaption></figure>

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The combined use of α and the mechanical coordination number of fines ($Z_m^{fine}$) provides a robust framework for evaluating the role of fines in fabric evolution and stress transmission. **Fig. 21** illustrates the α-$Z_m^{fine}$ relationships for GR values ranging from 2 to 20, with FC between 5% and 50%. For GR < 4 (**Fig. 21(a)**), assemblies consistently fall in the stable zone (α > 0.8, $Z_m^{fine} > 3.2$), reflecting Type-3 behavior where fines are structurally active. In contrast, for GR ≥ 4 (**Figs. 21(b–d)**), both α and $Z_m^{fine}$ remain low, indicating Type-1 instability; stability improves only with increasing relative density, which drives a Type-2 transitional regime. Across these regimes, $Z_m^{fine}$ values of ~ 2.4–3.5 correspond to underfilled states, 3.4–4.7 to transitional states, and > 4 to stable, overfilled states.

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At GR = 20 (Fig. 21(e)), both α and $Z_m^{fine}$ remain suppressed across most FC values, with fines concentrated in the inactive to semi-active zones. Even at higher FC, fines fail to form a continuous load-bearing network and instead act primarily as lateral stabilisers, providing confinement to the coarse skeleton rather than direct stress transfer. This highlights a unique regime at extreme size disparity, where fines
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undergo a functional shift from load carriers to passive stabilisers, preventing a full transition to Type-3 stability.

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The novelty of this study lies in integrating two complementary fabric indicators, $\alpha$, quantifying stress partitioning, and $Z_m^{fine}$, capturing micromechanical connectivity, into a unified framework. This dual-parameter approach classifies fines as inactive, semi-active, or active, offering a mechanistic understanding of fabric transitions in gap-graded soils. Practically, it provides design engineers with a predictive tool to determine critical thresholds of fine content and gap ratio, improving the assessment of suffusion potential, filter performance, and long-term soil-structure stability.

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<figure><img src="image_60.png" /></figure>

<figure><img src="image_61.png" /></figure>

<figure><img src="image_62.png" /></figure>

Type-1 ($\alpha$ < 0.4) [inactive]

<figure><img src="image_63.png" /></figure>

Type-2 (0.4 ≤ $\alpha$ ≤ 0.8) [semi-active]

<figure><img src="image_64.png" /></figure>

Type-3 ($\alpha$ > 0.8) [active]

<figure><img src="image_65.png" /></figure>

<figure><img src="image_66.png" /></figure>
<figure><img src="image_67.png" /><figcaption>**Fig. 21.** Variation of $\alpha$ with $Z_m^{fine}$: (a) GR = 2, (b) GR = 4, (c) GR = 5, (d) GR = 7, and (e) GR = 20</figcaption></figure>

**3.6 Combined Analysis of Numerical Trends and Statistical Significance**

The influence of relative density on soil fabric evolution is most apparent in the behaviour of the stress tensor ($\alpha$) across varying gap ratios and fine contents. The particle-based stress tensor $\alpha$, which is comparable to the contact-based stress partition $\alpha_p$ derived from fine-fine and fine-coarse contacts, shows strong sensitivity to relative density when GR < 4 and FC ≤ 30% (Fig. 22(a)). As GR increases, this sensitivity shifts to higher fine contents, becoming evident beyond FC ≈ 30%. In the case of GR < 4 and FC ≤ 30% (Fig. 22(b)), the findings align well with the sensitivity of relative density observed in the average coordination number of fine particles ($Z_{avg}^{fine}$).

Furthermore, distinct thresholds of fine content (Fig. 22(b)) were identified within the transitional fabric state, where the soil structure progressively shifts from a coarse-dominated to a fine-dominated matrix. This study uniquely demonstrates that these thresholds are strongly dependent on relative density, highlighting its controlling role in fabric transitions, an influence that has been largely underexplored in previous research.
<figure><img src="image_68.png" /></figure>

<figure><img src="image_69.png" /><figcaption>**Fig. 22.** Variation of (a) the stress reduction factor (α) with fine content (FC) and gap ratio (GR) across relative densities ($D_R$), and (b) the average fine coordination number ($Z_{avg}^{fine}$) with FC and GR, illustrating their role in characterizing soil fabric states</figcaption></figure>

To statistically evaluate the influence of gap ratio (GR) and fine content (FC) on the fabric characteristics of gap-graded soils, a two-way ANOVA was conducted for both the stress reduction factor (α) and the
average fine coordination number ($Z_{avg}^{fine}$), following the approach outlined by Liu et al. (2024). To ensure consistency across all GR values, including the additional case of GR = 20 (which was simulated only at FC = 5, 25, 35, and 50%), the analysis was restricted to these four representative FC levels. This provided a balanced dataset across the full GR range while still capturing the role of fines at low, intermediate, and high contents.

The results (Tables 3–4) indicate that GR, FC, and their interaction all have highly significant effects ($p < 0.0001$) on both $\alpha$ and $Z_{avg}^{fine}$. The large F-values highlight the strong influence of both GR and FC, while the significant interaction confirms that their combined effect is critical in controlling fabric evolution. These findings are consistent with the micromechanical trends illustrated in Fig. 22 and demonstrate that soil fabric is governed not only by GR and FC individually, but also by their interaction, which dictates particle packing, contact networks, and ultimately the internal stability of gap-graded soils (Liu et al., 2024).

**Table 3**

Statistical significance of GR, FC, and their interaction on the stress reduction factor ($\alpha$), based on two-way ANOVA

<table><caption>Response Variable: α</caption><thead><tr><th>Source</th><th>Degree of Freedom (DF)</th><th>Sum of Squares</th><th>Mean Square</th><th>F value</th><th>p-value</th></tr></thead><tbody><tr><td>Gap ratio (GR)</td><td>4</td><td>5.5242</td><td>1.38105</td><td>114.9120</td><td>&lt;0.0001</td></tr><tr><td>Fine content (FC)</td><td>3</td><td>10.4705</td><td>3.49019</td><td>290.4059</td><td>&lt;0.0001</td></tr><tr><td>Interaction (GR &amp; FC)</td><td>12</td><td>1.7832</td><td>0.1486</td><td>12.3647</td><td>&lt;0.0001</td></tr><tr><td>Error</td><td>81</td><td>0.9734</td><td>0.01202</td><td>-</td><td>-</td></tr></tbody></table>

**Table 4**

Statistical significance of GR, FC, and their interaction on the average fine coordination number ($Z_{avg}^{fine}$), based on two-way ANOVA

<table><caption>Response Variable: Z<sub>avg</sub><sup>fine</sup></caption><thead><tr><th>Source</th><th>Degree of Freedom (DF)</th><th>Sum of Squares</th><th>Mean Square</th><th>F value</th><th>p-value</th></tr></thead><tbody></tbody></table>
<table><tr><td>Gap ratio (GR)</td><td>4</td><td>84.1905</td><td>21.0476</td><td>540.7462</td><td>&lt;0.0001</td></tr><tr><td>Fine content (FC)</td><td>3</td><td>95.0085</td><td>31.6695</td><td>813.6384</td><td>&lt;0.0001</td></tr><tr><td>Interaction (GR &amp; FC)</td><td>12</td><td>17.2448</td><td>1.4370</td><td>36.9205</td><td>&lt;0.0001</td></tr><tr><td>Error</td><td>40</td><td>1.5569</td><td>0.03892</td><td>-</td><td>-</td></tr></table>

## 4. Conclusions

This study develops a DEM-based framework to quantify the role of relative density in fabric evolution and internal stability of gap-graded soils, which are key factors influencing erodibility. Simulations were conducted for gap ratios (GR) of 2–7 and fine contents (FC) ranging from 5–50%, with additional cases at GR = 20 to examine extreme particle size disparity. Fabric behaviour was assessed using the stress reduction factor (α), coordination numbers, strong contact networks, and the proposed variation index $I_{rattler}$. The results demonstrate that increasing relative density markedly reduces $I_{rattler}$, particularly near transitional states, thereby enhancing the contribution of fines to the load-bearing skeleton. Importantly, even at GR = 20, relative density remains a decisive factor in activating fines and stabilizing the granular fabric.

The following are the key points that can be derived from the present study:

(a) For GR < 4, soils remain structurally stable and do not represent truly gap-graded behaviour, whereas for GR ≥ 4 with FC ≤ 15%, assemblies are internally unstable as the coarse skeleton alone carries the load and fines act merely as void fillers.

(b) For 4 < GR < 7, the transitional state appears at FC ≈ 20–25%, shifting to ≈ 30% for GR = 7. At higher FC (30–35%), assemblies are transitional at low density but gain stability with densification. In particular, GR = 7 with FC = 35% demonstrates the critical influence of relative density near threshold fines, a trend also reported by Ahmadi et al. (2020) and Indraratna et al. (2015).

(c) At FC = 50%, α remains close to unity across all GR values and densities, indicating a fully stabilized fabric dominated by fines.
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(d) A combined analysis of $\alpha$ and $Z_m^{fine}$ reveals that fines become fully active when $\alpha > 0.8$ and $Z_m^{fine} > 4$, marking their transition into a self-sustained load-bearing skeleton.

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(e) For GR = 20, soils remain coarse-dominated up to FC ≈ 35%, with stability attained abruptly at FC = 50%. This demonstrates that under extreme particle size disparity, fines cannot progressively assume load-bearing roles but instead activate suddenly, defining a geometric upper bound for their contribution.

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The use of spherical particles is a limitation, as natural soils exhibit angularity, surface roughness, and breakage that influence contact mechanics and fabric evolution. Nevertheless, this simplification enables a controlled assessment of how gradation and packing regulate fine particle behaviour and internal stability. The findings establish a micromechanical framework linking relative density to fabric evolution and instability thresholds, providing a basis for future studies on particle shape and breakage while enhancing the prediction of erosion onset and the design of resilient hydraulic filters.
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CRediT authorship contribution statement

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**Rohan Dhamne**: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing – Original draft, Visualization. **Mahdi Disfani**: Conceptualization, Supervision, Methodology, Writing – review & editing, Project administration, Resources, Funding Acquisition. **Antoine Wautier**: Conceptualization, Supervision, Writing – review & editing.

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Declaration of Competing Interest

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The authors affirm that they do not have any known conflicting financial interests or personal relationships that could have influenced the work presented in this paper.

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Acknowledgments

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The authors acknowledge the support of the Australian Research Council through the Discovery Program (Grant Number: DP210106663). The first author is the recipient of the Melbourne Research Scholarship (MRS) from the University of Melbourne.

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Data Availability

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Data is provided within the manuscript.
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