Influence of Fines Content and Particle Shape on Limiting Void Ratios of Sand Mixtures: DEM and AI Approaches

Abstract

The mechanical behavior of sand–fines mixtures is governed by their limiting void ratios, which are sensitive to fines content and particle morphology. Conventional empirical correlations often fail to generalize to a wide range of soils, limiting their applicability in engineering design. This study develops an integrated approach combining laboratory calibration, discrete element method (DEM) simulations incorporating realistic particle morphologies and machine learning to predict maximum and minimum void ratios. Glass beads were first tested to validate DEM contact parameters, after which sand particles obtained through 3D scanning were employed to capture morphological effects. Correlation and partial least squares analyses confirmed fines content as the dominant factor, while particle shape also contributed to packing behavior. A fully connected neural network (FCNN) was trained to establish predictive relationships, demonstrating closer agreement with DEM simulations than traditional empirical formulations. The proposed approach provides a reliable and generalizable tool for evaluating packing characteristics and offers new insights into the role of particle morphology in the mechanical response of sand–fines mixtures.

1. Introduction

Granular soils, particularly sand–fines mixtures, are widely encountered in geotechnical engineering, such as foundation design, embankment construction and liquefaction assessment. The strength, compressibility and permeability of these soils primarily depends on their density state [1]. This state is described by the maximum (e_max) and minimum void ratios (e_min), which correspond to the loosest and densest configurations of sandy soils, respectively. In mixtures where fine particles are present, the packing structure becomes more complex due to interactions between coarse and fine particles. Owing to their nonlinear impact on void ratios, the presence of fines significantly complicates the prediction of the mechanical behavior of sand–fines mixtures [2]. Consequently, accurately quantifying the maximum and minimum void ratios of sand–fines mixtures is crucial for engineering practice.

Over the past three decades, extensive experimental studies on maximum and minimum void ratios have been conducted. Triaxial test results from Thevanayagam [3] and Lade [4] showed that the presence of fines significantly alters the packing structure of sand–fines mixtures, resulting in irregular variations in void ratio with fines content. Through oedometer tests on sand–kaolinite mixtures, Kaothon [5] revealed that at the optimum fines content (15–20%), the soil exhibited the maximum density, thereby reducing the settlement potential. To further investigate this relationship, Cheng and Zhang [6] demonstrated that the CRR of sand–fines mixtures decreases and then stabilizes with increasing fines content at constant relative density, but decreases and later increases at constant global void ratio. To unify these opposing trends, they proposed the density index Dr/e. Through a series of undrained triaxial tests on HN31 sand mixed with non-plastic fines, Zhu [7] identified a mean diameter ratio D₅₀/d₅₀ = 14.5 as the critical threshold governing undrained shear strength and the development of excess pore water pressure. With the rapid development of numerical computation in geotechnical engineering, the discrete element method has been widely applied in granular material research, as it provides valuable insights into particle interaction mechanisms that explain the macroscopic behavior under various loading conditions [8,9,10,11,12]. For example, using the discrete element method, Voivret [13] showed that the solid fraction increases with particle size span in a nonlinear manner. Zhong [14] demonstrated that both the minimum and maximum void ratios in DEM specimens first decrease and then increase with increasing fines content, consistent with experimental observations. Moreover, the study revealed that larger particle size ratios enhance the filling effect of fines, although this influence becomes limited beyond a critical threshold. Through DEM simulations, Kodicherla [15] found that particle elongation increases peak strength and promotes strain softening, while the critical state is attained at larger strains with higher shear resistance. A study by Lei [16] showed 3D-printed particles to establish a holistic geometric parameter and showed a clear correlation between shape metrics and the maximum and minimum void ratios. A decrease in packing efficiency and an increase in minimum void ratio associated with greater particle shape irregularity were observed experimentally and further validated through relevant DEM simulations performed by Xu [17].

Numerous empirical formulas have been documented in the literature for predicting e_min and e_max of granular soils. Early studies primarily emphasized empirical relationships incorporating gradation parameters. For soils, an empirical method for predicting the maximum packing density based on different particle sizes was proposed by Humphres [18]. Subsequent investigations further established systematic links between limiting void ratios and median grain size. Based on an extensive experimental database, Cubrinovski and Ishihara [1] demonstrated that the void ratio range (e_max − e_min) decreased consistently with increasing median particle size (D₅₀) and proposed empirical equations that use D₅₀ to predict this range. For clean sands, Patra [19] found that relative density could be systematically related to D₅₀ and compaction energy, with the empirical coefficients derived through regression against Proctor test data.

In addition, particle shape has been recognized as an important factor influencing the limiting void ratios of granular soils. Youd [20] demonstrated that the maximum and minimum void ratios of clean sands are primarily controlled by particle roundness in conjunction with gradation, and proposed empirical relationships for estimating e_min and e_max. Riquelme and Dorador [21] found an empirical methodology that correlates gradation parameters and particle angularity with test results, providing reliable equations to estimate the e_min and e_max of coarse granular soils. A predictive model for e_min of sand–fines mixtures was introduced by Chang [22] using the concept of dominant particle network. The model captured the influence of particle shape on packing through the filling coefficient (a) and embedment coefficient (b), which were reversely calculated from experimental data. Chang [23] later extended this framework to predict both e_min and e_max for the full range of fines contents, which successfully captured the V-shaped trend with fines content. Based on this, Polito [24] developed straightforward regression equations to estimate the coefficients a and b directly from the median grain sizes of sand (D₅₀) and silt (d₅₀), allowing void ratios to be predicted accurately without the need for extensive laboratory testing.

Although experimental studies have provided valuable insights into the influence of macroscopic parameters (e.g., fines content and particle size ratio) on e_min and e_max, they are usually restricted to specific materials available to individual researchers. Moreover, particle shape is often simplified or idealized, limiting the generality of these models when applied to sands with complex and irregular morphologies. As a result, the conclusions derived from such tests are highly material-dependent and difficult to generalize, making it challenging to establish a consistent database for engineering practice. At the same time, most empirical formulas proposed in the literature rely on fitting or inverse-calibrated parameters, which reduces their generality and makes practical application challenging. To overcome these limitations, the present study develops an AI-based predictive framework for e_min and e_max of sand–fines mixtures. Glass bead experiments were first conducted to calibrate the contact model parameters, after which DEM simulations were performed on mixtures with actual particle shapes. The results were then processed through partial least squares (PLS) analysis for dimensionality reduction. Finally, a fully connected neural network (FCNN) was trained to establish the prediction model. This integrated approach avoids the need for empirical parameters and combines micromechanical modeling with artificial intelligence, offering a novel tool for evaluating the packing behavior of granular soils.

2. Research Methods

2.1. Experimental Program

To calibrate the DEM contact model parameters, a set of simplified laboratory tests was conducted using spherical glass beads. These tests yielded the maximum and minimum dry densities, from which void ratios were obtained. Glass beads were selected for their uniform shape and stable size distribution. More importantly, their composition is primarily silicon dioxide (SiO₂), which is very similar to that of natural quartz sand [25,26].

In this study, glass beads with two diameters of 0.5 mm and 2.0 mm were adopted to create a mean diameter ratio D/d = 4, as shown in Figure 1. Both the maximum and minimum dry density tests were conducted for six fines contents of F_C = 0.0, 0.2, 0.4, 0.6, 0.8 and 1.0. For the determination of e_max, coarse and fine particles were thoroughly mixed. A 500 g portion of the mixture was taken and divided into three equal parts. The material was then placed into a 250 mL mold in three layers. During placement, a vibrating fork (150–200 vibrations per minute) and a rammer (30–60 blows per minute) were used to compact the specimen, as shown in Figure 2a. After compaction, the top surface was leveled and the mass of the specimen was measured to calculate the corresponding e_min. To determine e_min, a 400 g portion of the mixture was gently poured into a 500 mL graduated cylinder using a long-neck funnel (see Figure 2b). After the deposition was completed, the cylinder was sealed and gently inverted three times to achieve the loosest configuration. The final volume was recorded to derive the corresponding e_min [27]. It should be mentioned that each test was performed in triplicate to confirm experimental repeatability.

2.2. Simulation of Minimum and Maximum Void Ratios

In this study, a commercial DEM program, PFC 6.0 [11,28], developed by the Itasca group, was used to simulate the packing behavior of sand–fines mixtures. Given that the actual morphology of the coarse particles was directly considered, the linear model [29,30] was applied at interparticle contacts. The angularity of the coarse particles inherently provided rotational resistance, thus eliminating the need for additional rolling-resistance components. Although fine particles in natural soils may exhibit irregular shapes, they were modeled as spheres in this study for both particle generation and contact interaction. This assumption is justified by two considerations [30,31]. At low fines contents, fine particles mainly participate in filling the voids within the coarse-particle skeleton rather than forming a force-transmitting framework, and their detailed shape has a limited influence on packing density. At high fines contents, the number of fine particles exceeds 10⁵, which already imposes a substantial computational burden; explicitly modeling irregular fine particles under such conditions would further increase computational cost without providing commensurate improvement in predictive accuracy. In addition, the linear contact model was widely applied to spherical particles and was demonstrated to adequately capture particle rearrangement and packing characteristics under gravity-dominated and low-stress conditions. Therefore, this simplification has a negligible impact on the simulated packing behavior [32].

To simulate the minimum void ratio, particles with a specified fines content were first generated within a cylindrical container measuring 190.5 mm in height and 100 mm in diameter (Figure 3). The container’s dimensions were selected to be over ten times greater than the maximum particle diameter so as to eliminate boundary effects [33,34]. Following static equilibrium, gravitational acceleration (g = 9.81 m/s²) was applied, and an axial stress of 950 kPa was imposed via a rigid top plate [14]. Under this constant load, the entire particle assembly was subjected to cyclic vibration, promoting gradual rearrangement and densification. The simulation continued until the top plate exhibited no further vertical displacement, referring to the attainment of the densest packing configuration. The corresponding e_min was subsequently calculated from the final specimen volume.

To simulate the maximum void ratio, particles with a specific fines content were first generated within a cylindrical container measuring 270 mm in height and 100 mm in diameter, followed by a static equilibrium process. Afterwards, the bottom plate of the container was replaced with the same rigid diffuser as that used in laboratory experiments [26]. The entire particle assembly was then released under gravity, allowing the particles to fall freely and settle into the loosest packing configuration. Due to interparticle friction, the particles naturally formed a conical-shaped deposition at the top of the specimen (Figure 4). To avoid overestimating the specimen volume, this upper portion was removed, and the remaining volume was used to calculate the corresponding e_max.

2.3. Validation of Contact Model Parameters

A trial-and-error approach was adopted to calibrate the contact model parameters for the linear model until the DEM simulations could accurately match the experimental data, as listed in

Table 1. DEM model parameters with linear model.

Parameter		Particle–Particle	Particle–Facet	Unity
Linear Group
Effective Modulus	E*	1.0 × 108	1.0 × 108	Pa
Normal-to-Shear Stiffness Ratio	κ*	2.0	2.0	-
Friction Coefficient	μ	0.2	0.0	-
Dashpot Group
Normal Critical Damping Ratio	βn	0.2	0.2	-
Shear Critical Damping Ratio	βs	0.2	0.2	-
Dashpot Mode	Md	3.0	3.0	-

. The subplots (a) of Figure 5 and Figure 6 compare the experimental and simulated values of e_min and e_max, respectively. The results show that the fitting lines fall within a narrow band around the data points. The subplots (b) in the two graphs present bar charts comparing e_min and e_max at each fines content, in which only small gaps are consistently observed. Both the above phenomena suggest that the model contact model parameters were calibrated with a high level of accuracy.

2.4. Particle Morphology Parameters

In this study, four types of sand particles with distinct shapes were selected. Each particle was precisely examined using blue-light 3D scanning, which preserved its geometric properties and surface textures. The obtained digital models were exported in STL format and subsequently imported into PFC 6.0, where each particle was reconstructed as a rigid cluster component (see Figure 7). These clusters do not break upon external force, which is in agreement with the high resistance to breakage of natural quartz sand particles [35].

Particle morphology is one of the most important factors governing the packing and mechanical behaviors of sand–fines mixtures. Among the various morphological indices proposed in previous studies, the study adopted four of the most intuitive and commonly adopted parameters (i.e., sphericity, roundness, roughness and aspect ratio) to characterize particle shape [36]. Sphericity describes the closeness of a particle to a perfect sphere, whereas roundness is largely dependent on the sharpness of angular protrusions from the particle [37]. The sphericity (S) and roundness (R) of a given particle can be determined using the following formulas [38]:

$$S=r_{\max -in}/r_{\min -cir}$$

(1)

$$R=({\displaystyle \sum r_i}/N)/r_{\max -in}$$

(2)

where r_max-in and r_min-cir are the radius of the largest inscribed and smallest circumscribed spheres, and ${\displaystyle \sum r_{\mathrm{i}}}/N$ is the average radius of particle surface features. Roughness (RG) describes surface irregularities based on local deviations between the real and benchmark surfaces of a particle with the following equation:

$$RG=\sqrt[3]{{\displaystyle \frac{4\pi }{3V}}}{\displaystyle \frac{1}{S}}{\displaystyle {\sum }_{i=1}^m{d}_i{S}_i}$$

(3)

where S is the area of the benchmark surface, S_i is the area of the ith triangular mesh and V is the particle volume [39]. The aspect ratio (AR) is defined as the ratio between the Feret minimum and Feret maximum diameters [40]:

$$AR=d^{F\min }/d^{F\max }$$

(4)

A Python 3.8 script was built in this study to automatically compute these parameters. They are ranked by magnitude in Figure 8, while their specific values are listed in

Table 2. Shape parameters of different particles.

	Particle Shape
Particle Type	S	R	RG	AR
a	1	1	0	1
b	0.636	0.574	0.119	0.923
c	0.537	0.674	0.051	0.648
d	0.468	0.540	0.130	0.537
e	0.373	0.648	0.111	0.442

, respectively.

3. Results

3.1. Minimum and Maximum Void Ratios

Figure 9 depicts the variation in e_min of spherical particles under different fines content and particle size ratios. The Figure shows that for spherical particles, under different particle size ratios, the minimum void ratio initially decreases with increasing fines content until it reaches a critical threshold (F_C = 0.4). Beyond this point, the minimum void ratio increases as fines content approaches 1.0. This trend reflects a transition in the dominant skeletal structure of the sand–fines mixtures, shifting from a coarse-grained skeleton to a fines-dominated skeleton. Initially, the mixture comprises solely coarse grains, where interparticle voids are relatively large. As fines gradually increase, they fill the voids between coarse particles, making the packing denser and thus leading to a reduction in the minimum void ratio. However, once fines exceed the critical threshold, they begin to replace coarse particles and form their own structural framework. This replacement isolates coarse grains and restricts efficient particle rearrangement, resulting in an increase in void ratios.

Figure 10 illustrates the variations in the minimum void ratio for five different particle shapes under varying fines contents and particle size ratios. The selected shapes range from ideal spheres (type a) to increasingly irregular natural sand particles (types b–e). For non-spherical particles at a size ratio of D/d = 2, the minimum void ratio increases monotonically with increasing fines content. This behavior differs from that of spherical particles and is attributed to the geometric complexity of non-spherical grains. At low size ratios, fines are insufficiently small to occupy the angular or elongated voids between coarse particles. Instead of enhancing packing, they disrupt the original structure by forcing irregular particles apart, leading to a steady increase in void ratio. However, when D/d > 2, fines become sufficiently small relative to non-spherical coarse particles, enabling them to infiltrate and fill the complex voids. Therefore, the minimum void ratio of non-spherical particles exhibits a similar trend to that of spherical particles. Notably, the critical FC for non-spherical particles (F_C = 0.2) occurs earlier than for spherical ones (F_C = 0.4). This shift is attributed to the enhanced interlocking and contact heterogeneity of non-spherical particles, which reduces mobility and accelerates the transition to a fines-dominated skeleton.

To describe the relationship between e_min and F_C, Zhong [14] employed a quadratic function (y = ax² + bx + c) to fit the numerical results under different particle size ratios. The reason for choosing a quadratic function was that it could capture the observed trend and it requires only three fitting parameters for the fitting. Therefore, in this study, a quadratic function is used to fit the variation in the minimum void ratio of irregular particles with fines content and particle size ratios. The fitted functions and corresponding coefficients of determination are presented in

Table 3. The fitting function for the minimum void ratio.

D/d	Particle	Fitting Function (y = ax2 + bx + c)	R2
2	a	y = 0.367x2 − 0.373x + 0.579	0.798
	b	y = 0.083x2 + 0.178x + 0.302	0.998
	c	y = 0.075x2 + 0.227x + 0.260	0.999
	d	y = 0.078x2 + 0.293x + 0.191	0.993
	e	y = 0.067x2 + 0.245x + 0.251	0.999
3	a	y = 0.687x2 − 0.695x + 0.576	0.809
	b	y = 0.250x2 + 0.005x + 0.299	0.956
	c	y = 0.236x2 + 0.060x + 0.256	0.971
	d	y = 0.234x2 + 0.127x + 0.193	0.977
	e	y = 0.234x2 + 0.075x + 0.244	0.973
4	a	y = 0.942x2 − 0.949x + 0.584	0.796
	b	y = 0.425x2 − 0.156x + 0.293	0.911
	c	y = 0.437x2 − 0.133x + 0.260	0.907
	d	y = 0.395x2 − 0.019x + 0.187	0.938
	e	y = 0.402x2 − 0.080x + 0.241	0.925
5	a	y = 1.020x2 − 1.034x + 0.598	0.784
	b	y = 0.543x2 − 0.264x + 0.289	0.858
	c	y = 0.526x2 − 0.213x + 0.255	0.873
	d	y = 0.505x2 − 0.126x + 0.188	0.908
	e	y = 0.532x2 − 0.208x + 0.244	0.876
6	a	y = 1.273x2 − 1.316x + 0.632	0.790
	b	y = 0.645x2 − 0.362x + 0.290	0.840
	c	y = 0.608x2 − 0.294x + 0.256	0.856
	d	y = 0.577x2 − 0.204x + 0.196	0.903
	e	y = 0.622x2 − 0.299x + 0.249	0.855

.

Figure 11 and Figure 12 show the maximum void ratio as a function of particle size ratios and fines contents. Although the trend mirrors that of e_min, the underlying mechanisms are associated with particle deposition and rearrangement. At low fines contents, coarse particles dominate the packing and form a loose configuration during free deposition, resulting in relatively large void spaces and a high e_max. When the fines content exceeds a critical value, the packing structure transitions to a fines-dominated skeleton. In this regime, the large number of fine particles, combined with increased interparticle contacts and friction, restricts particle mobility during deposition, causing e_max to increase again. This evolution reflects a transition in the dominant packing structure and explains the observed non-monotonic variation in e_max with fines content. Based on this trend, a quadratic function is still adopted to fit the variation in e_max. The fitted quadratic functions for e_max and their coefficient of determination are presented in

Table 4. The fitting function for the maximum void ratio.

D/d	Particle	Fitting Function (y = ax2 + bx + c)	R2
2	a	y = 0.374x2 − 0.386x + 0.713	0.831
	b	y = 0.099x2 + 0.124x + 0.472	0.989
	c	y = 0.123x2 + 0.160x + 0.416	0.987
	d	y = 0.091x2 + 0.264x + 0.341	0.993
	e	y = 0.121x2 + 0.157x + 0.420	0.987
3	a	y = 0.799x2 − 0.800x + 0.702	0.812
	b	y = 0.485x2 − 0.257x + 0.463	0.929
	c	y = 0.393x2 − 0.095x + 0.391	0.960
	d	y = 0.503x2 − 0.181x + 0.371	0.927
	e	y = 0.400x2 − 0.104x + 0.390	0.969
4	a	y = 0.912x2 − 0.911x + 0.691	0.846
	b	y = 0.656x2 − 0.403x + 0.439	0.888
	c	y = 0.625x2 − 0.323x + 0.390	0.888
	d	y = 0.606x2 − 0.325x + 0.317	0.938
	e	y = 0.679x2 − 0.385x + 0.398	0.891
5	a	y = 1.099x2 − 1.055x + 0.652	0.793
	b	y = 0.852x2 − 0.608x + 0.462	0.874
	c	y = 0.843x2 − 0.544x + 0.406	0.923
	d	y = 0.802x2 − 0.428x + 0.330	0.92
	e	y = 0.821x2 − 0.508x + 0.393	0.911
6	a	y = 1.15x2 − 1.066x + 0.617	0.772
	b	y = 0.875x2 − 0.615x + 0.441	0.830
	c	y = 0.838x2 − 0.520x + 0.379	0.872
	d	y = 0.835x2 − 0.450x + 0.309	0.901
	e	y = 0.816x2 − 0.503x + 0.381	0.815

.

3.2. Pearson Correlation Coefficient Analysis and PLS Analysis

To quantify the dominant factors influencing the evolution of e_min and e_max, Pearson correlation analysis was first conducted, followed by partial least squares regression. This two-step approach addresses multicollinearity among variables while extracting latent variables that optimally explain variance in void ratios.

3.2.1. Pearson Correlation Analysis

Pearson correlation analysis was used to evaluate the strength and direction of linear relationships between variables. Compared with Spearman or Kendall correlation, Pearson correlation is more suitable for detecting linear dependencies and is computationally efficient, making it appropriate for initial variable screening. The correlation coefficient ranges from −1 to 1, with values closer to 1 indicating stronger positive correlations [41,42].

Figure 13 presents the Pearson correlation matrix generated using Python. The fines content exhibits the strongest positive correlation with both e_min (r = 0.6958) and e_max (r = 0.6441), underscoring its dominant role in packing behavior. This finding aligns with Section 3.1, where fines content was shown to markedly influence void ratio trends. Other particle shape parameters, such as sphericity (S), roundness (R), and roughness (RG), also display moderate correlation with void ratios. Notably, roughness (RG) demonstrates a negative correlation (−0.3035 with e_min and −0.2515 with e_max), indicating that higher surface texture enhances interparticle interlocking. This enhanced interlocking restricts particle movement and rearrangement, thereby leading to lower limiting void ratios. However, the magnitude of these correlations reveals that no single variable can adequately account for void ratio variation. Thus, while fines content emerges as the most influential factor, changes in the void ratio are still governed by the combined effects of multiple parameters. Partial least squares (PLS) regression is adopted in the subsequent analysis to address this multivariate nature.

3.2.2. Partial Least Squares Analysis (PLS)

To address the multicollinearity among variables and reduce input dimensionality for neural network modeling, PLS regression was applied. PLS is a dimensionality reduction technique that identifies latent variable components by projecting the original predictors onto a new space that maximizes the covariance with the target response [43]. Compared with standard regression or principal component analysis, PLS is particularly effective for small-sample, high-dimensional, and multicollinear datasets, which are conditions often encountered in mechanics simulations [44].

Using Python’s scikit-learn library, PLS was performed on the six variables (S, R, RG, AR, D/d, and F_C), resulting in two latent variables. LV1 and LV2 represent the combined effects of the original particle shape parameters, size ratio, and fines content that govern the variation in void ratios [45]. These latent components are linear combinations of the original predictors, with weights representing each variable’s contribution to explaining the variance in e_min and e_max.

The weight distribution (Figure 14) indicates that fines content has the largest weight in the first component, confirming its dominant influence, while shape parameters such as sphericity, roughness, and aspect ratio also contribute substantially. By condensing six correlated predictors into two orthogonal latent variables, PLS clarifies the combined effects of particle shape, size ratio, and fines content, while simplifying the input space. These two components are subsequently used as inputs for the fully connected neural network, promoting faster convergence and better generalization by reducing redundancy and retaining the essential information from all original variables.

It should be noted that the DEM simulation results were not directly used as the training dataset for the neural network. In the DEM simulations, each particle size ratio produced a set of 30 values for the maximum and minimum void ratios, which is insufficient in size for training a deep learning model. Therefore, the fines content was discretized from 0 to 1000 into 1001 intervals, resulting in 5005 data points for each of the maximum and minimum void ratios. This procedure enabled the construction of an expanded and continuous dataset while preserving the intrinsic correlations identified by the DEM-PLS framework, thereby providing a sufficient and structured dataset for neural network training.

3.3. Fully Connected Neural Network

Based upon the dimensionality reduction and feature extraction achieved through PLS, a fully connected neural network (FCNN) is constructed to predict the minimum and maximum void ratios. As a deep learning architecture, the FCNN is well-suited for capturing complex, nonlinear relationships between inputs and outputs through multiple layers of interconnected neurons [46]. Compared with conventional regression models, FCNNs offer greater flexibility and stronger predictive capability, especially when dealing with multivariate, interdependent features such as particle shape parameters [47].

As shown in Figure 15, the architecture of FCNN consists of an input layer, two hidden layers, and an output layer. The input layer receives the two latent variables (LV₁ and LV₂). The first hidden layer contains 16 neurons to capture the features of variables, while the second hidden layer has 8 neurons to further refine these representations and reduce overfitting. All neurons are fully connected to those in adjacent layers, enabling dense signal propagation and transformation throughout the network. The output layer contains two neurons that generate the predicted values of e_min and e_max. To improve the learning capacity and adaptability of the network, an Adaptive Activation Function (AAF) was applied in both hidden layers. Unlike fixed functions such as ReLU or Tanh, the AAF adjusts functional form during training, providing greater flexibility to fit both convex and non-convex patterns in the data [48]. This adaptability is particularly beneficial for small datasets or features with complex nonlinear interactions, as is often the case for particle morphology parameters [49]. The network was implemented in Python using the PyTorch library and trained for 300 epochs with a batch size of 16 and a learning rate of 0.002. The dataset was randomly split into 80% for training and 20% for validation to ensure reliable generalization assessment. The mean squared error (MSE) loss function was used to measure deviations between predicted and observed void ratio values. Early stopping based on validation loss was also employed to prevent overfitting and enhance training stability.

Based on the loss curves, the performance of the fully connected neural network can be effectively assessed. As shown in Figure 16, mean squared errors for both training and validation datasets drop sharply within the first 30 epochs, indicating rapid learning and efficient parameter optimization in the early stage. The training loss then continues to decrease and stabilizes at a low value, reflecting a strong fit to the training data. The validation loss remains slightly higher than the training loss but is consistently low, demonstrating that the model achieves good generalization without overfitting. Similarly, Figure 17 presents the evolution of the coefficient of determination (R²) over 300 epochs for both the training and validation sets. As training progresses, R² for both datasets converges and remains steadily above 0.85, confirming the model’s robustness and precision in capturing the relationships between input features and target variables.

4. Discussion

4.1. Comparison with Classical and Recent Studies

To further evaluate the predictive performance of the proposed framework, its results are compared with a representative empirical approach from the literature, namely the parameterized formulation of Chang’s binary packing theory [23] as refined by Polito. Polito [50] relies on empirically calibrated coefficients (the filling coefficient a and the embedment coefficient b) inferred from regression analyses based on the median particle sizes of sand and fines. This accounts for the variation in the limiting void ratios of sand–fines mixtures across sand-dominated and fines-dominated regimes.

The trained FCNN is then applied to predict e_min and e_max. As shown in Figure 18, the predicted values agree closely with the DEM results, while the empirical correlation of Polito [50] captures only the general tendency and shows clear deviations at intermediate fines content. This indicates that the FCNN provides a more accurate description of the influence of fines content and particle morphology.

The root mean square error (RMSE) is defined as the square root of the mean of the squared differences between the predicted values and the corresponding DEM results [51]. RMSE is used as an indicator to evaluate the overall accuracy of the proposed model, with a smaller RMSE indicating better agreement between predictions and DEM data. A more detailed comparison is given in Figure 19, which shows the RMSE of both e_min and e_max for different particle shapes. Overall, the FCNN outperforms the empirical model, with the advantage becoming more evident as the aspect ratio or size disparity increases. The neural network is a feasible approach for predicting the maximum and minimum void ratios of sand particles.

Recent advances in artificial intelligence have been widely applied to the prediction of soil mechanical properties, with most studies focusing on strength- and deformation-related parameters such as shear strength [52], moisture [53], thickness and distribution [54]. In contrast, fundamental packing descriptors, particularly the minimum and maximum void ratios, have received relatively limited attention and are commonly treated as prescribed inputs rather than prediction targets. The present study addresses this limitation by developing a DEM-AI framework that directly predicts the e_min and e_max of sand–fines mixtures, thereby extending recent AI-based approaches toward more fundamental and physically meaningful soil parameters.

More recent numerical studies have attempted to overcome these limitations using the discrete element method. For example, Zhong [14] investigated the evolution of minimum and maximum void ratios by systematically varying fines content and mean particle size ratio, employing spherical particles combined with a rolling-resistance contact model to represent particle angularity. While their results reproduce similar global trends to those observed in the present study, particle shape effects in Zhong are incorporated implicitly through contact-level rotational constraints.

In contrast, the present study captures particle morphology effects explicitly by introducing non-spherical sand geometries while adopting a classical linear contact model. This modeling strategy allows particle shape to be treated as an independent physical variable rather than being embedded within empirical contact parameters. Consequently, the present framework provides a clearer physical interpretation of how fines interact with a non-spherical sand skeleton to govern packing limits, while avoiding the need for empirical fitting coefficients.

4.2. Practical Applicability and Relation to Standard Methods

The minimum and maximum void ratios are fundamental parameters in geotechnical engineering practice, as they define the density state of granular soils and directly affect liquefaction susceptibility, shear strength, and compressibility. In this context, the FCNN-based predictions of e_min and e_max proposed in this study should be regarded as a numerical research and analysis tool, rather than a direct substitute for laboratory testing or design standards. More importantly, the proposed framework is not intended to function as a purely numerical regression model. By integrating FCNN predictions with DEM observations, it provides a systematic and efficient means to identify and quantify the nonlinear coupling among fines content, particle size ratio, and particle morphology that governs void ratio limits under controlled conditions. As such, the framework supports parametric studies, sensitivity analyses, and preliminary investigations, offering physical insights into packing behavior and guidance for experimental program design.

In current practice, e_min and e_max are commonly determined using standardized laboratory procedures [27]. The DEM-based procedures adopted in this study are conceptually consistent with these standards, as both aim to reproduce the loosest and densest achievable packing states under controlled boundary conditions. However, the DEM framework enables systematic control of fines content, particle size ratio, and particle morphology, which are difficult to isolate in laboratory testing.

5. Conclusions

This study comprehensively investigated the influence of particle morphology and particle size ratio on the minimum and maximum void ratios of sand–fines mixtures using the discrete element method (DEM). Authentic sand particle models were reconstructed from STL files obtained via blue-light scanning and implemented in PFC to explore the role of shape parameters and fines content. Correlation analysis and machine learning were further used to quantify and predict void ratios. The main conclusions are as follows:

• For mixtures with D/d > 2, both e_min and e_max exhibit a clear non-monotonic evolution with increasing fines content, characterized by an initial decrease, a minimum at a transitional fines content, and a subsequent increase. Furthermore, the results reveal that the critical transitional fines content varies with morphology, specifically with particle shape and size ratio. This finding provides new insight into the variability of packing transitions in sand–fines mixtures and explains the inconsistencies observed in traditional empirical correlations.
• Through partial least squares (PLS) analysis, these coupled effects were distilled into a reduced set of latent variables, clarifying the relative contributions of grading and morphology while preserving the essential micromechanical trends. This provides a more physically interpretable framework for understanding void ratio limits beyond traditional empirical correlations.
• By integrating DEM-derived data with a fully connected neural network (FCNN), the study demonstrates that the limiting void ratios of sand–fines mixtures can be reliably predicted as target variables rather than prescribed constants. Compared with conventional empirical models, the proposed DEM-AI approach exhibits improved accuracy and robustness across a wide range of fines contents, particle size ratios, and particle shapes. This highlights a shift from empirical estimation toward data-driven, morphology-informed prediction of fundamental packing parameters.