Multi-Scale Analysis of Sand Behavior Under Rigid and Flexible Membrane Boundaries in DEM Triaxial Compression

Abstract

Laboratory triaxial tests are essential for studying sandy soil behavior but have limited ability to capture localized deformation and microstructural evolution. The discrete element method (DEM) overcomes these limitations by enabling particle-scale analysis, where boundary conditions can critically affect simulation results. This study employed DEM-based triaxial compression simulations to compare rigid wall and flexible membrane boundaries for sand specimens with initial porosities of 35.5%, 38.2%, 40.8%, and 41.5% under confining pressures of 50, 100, and 150 kPa. The analyses covered macroscopic stress–strain and volumetric responses, shear band morphology, local porosity evolution, and contact force fabric. The results indicate that rigid and flexible boundaries produce similar pre-peak responses, but differ markedly in post-peak behavior and volumetric strain. Rigid boundaries constrain lateral deformation, induce stress concentrations, and underestimate post-peak strength, while flexible membranes apply confining pressure more uniformly and reproduce realistic bulging and porosity evolution. Based on these findings, rigid boundaries are suitable for dense sands when post-peak strength is not a concern, and for loose sands at small strains, whereas flexible membranes are necessary to capture volumetric contraction and realistic post-peak responses. This work provides mechanistic insights into boundary effects and offers a basis for more efficient selection of boundary conditions in DEM triaxial simulations.

1. Introduction

Sand, as a representative non-cohesive granular material, is ubiquitous in geotechnical engineering, and its mechanical behavior plays a decisive role in the stability and safety of engineering structures [1,2,3,4]. The axisymmetric triaxial test remains one of the most fundamental experimental approaches for investigating the constitutive response of soils. By enclosing a cylindrical specimen in an elastic latex membrane and applying a confining pressure, the test enables the characterization of essential properties such as stress–strain relationships, dilatancy behavior, and failure modes [5]. Nevertheless, conventional triaxial testing is inherently constrained to the measurement of macroscopic mechanical responses, offering limited insights into the underlying micromechanical processes. Even advanced techniques, such as surface digital image correlation (DIC) [6] and X-ray computed tomography (CT) [7,8], while capable of capturing deformation patterns or internal fabric changes, are often restricted in spatial resolution and fail to fully establish a three-dimensional correspondence between particle-scale kinematics and bulk-scale responses [9]. Their limitations mainly lie in spatial resolution, the ability to capture dynamic loading, the measurement of mechanical quantities, and the high cost of experiments, making it difficult to systematically quantify inter-particle mechanical mechanisms.

The discrete element method (DEM), pioneered in geotechnical research by Cundall and Strack [10], provides a robust numerical framework for explicitly simulating the motion and interaction of individual particles. In contrast to continuum-based approaches, DEM inherently resolves particle translation, rotation, contact forces, and fabric evolution, thereby enabling direct linkage between macroscopic mechanical behavior and microscopic mechanisms [11,12]. This capability overcomes the observational limitations of traditional experimental techniques and offers unique opportunities for exploring the fundamental mechanisms governing the deformation and failure of granular materials.

In DEM simulations of triaxial compression, the choice and implementation of boundary conditions exert a profound influence on the measured mechanical responses [13,14]. Among the various options, rigid and flexible boundaries have been the most widely adopted. Rigid boundaries are favored for their simplicity and computational efficiency; for example, Kozicki et al. [15] employed fixed-wall loading in triaxial simulations of cylindrical specimens to assess the influence of non-uniform confining stresses on shear band formation, while Mehmet Cil [16] used rigid end platens to investigate the interplay between dilatancy and strain localization. However, such rigid boundary restricts lateral deformation and often induces non-uniform stress fields, leading to discrepancies from laboratory observations.

Flexible boundaries, by contrast, more faithfully replicate the mechanical role of latex membranes. Their implementation generally falls into two categories: discrete flexible membranes composed of bonded particles [17,18] and continuum–DEM coupled membranes [8,19]. In the discrete approach, a deformable membrane is formed from spherical particles, to which a uniform confining pressure is applied, offering simplicity, high controllability, and ease of implementation. In the continuum–DEM approach, finite element or finite difference methods are employed to capture membrane deformation with high fidelity, albeit at a significantly higher computational cost. Various studies have underscored the advantages of flexible boundaries: Binesh [20] developed a MATLAB–PFC3D coupling framework for stress-controlled flexible membrane loading; Qu Tongming [21] employed multilayer particle membranes to maintain stable and uniform confining pressures; Hua-Xiang Zhu [22] used FDM–DEM coupling to reproduce the deformation of rubber membranes across different loading stages; Ye Lu [23] demonstrated that flexible membranes reduce stress concentrations and promote more uniform deformation; and Tarek Mohamed [11] achieved accurate bulging simulations through finite element–DEM coupling. However, the implementation of flexible membrane boundaries in triaxial simulations is more complex and often entails substantial computational cost, which has limited their broader application.

Comparative investigations between rigid and flexible boundaries have consistently confirmed the superior ability of the latter to reproduce uniform stress states and realistic deformation patterns. For instance, the study by Tingting Yang [24] demonstrated that, compared with rigid boundaries, flexible membrane boundaries more accurately captured strain localization and volumetric changes in triaxial tests, reducing peak strength prediction errors by 15–20% and improving volumetric strain prediction accuracy by approximately 30%. Similarly, Mehmet B. Cil [16] reported that rigid boundaries induced lateral stress gradients of up to 20%, whereas flexible membranes achieved nearly uniform stress fields. Despite these advances, systematic comparative studies remain limited. Most existing work has concentrated on macroscopic responses, with relatively few addressing multi-scale metrics. Moreover, analyses of the differences between rigid and flexible boundaries have largely remained at the level of constraint conditions, without sufficiently probing into the fundamental distinctions in their loading principles. Consequently, the influence of boundary conditions across multiple scales has yet to be comprehensively established [16,25,26].

To address these gaps, the present study employs a unified DEM modeling framework to systematically compare the multi-scale responses of sand specimens in triaxial compression under rigid and flexible boundaries, with identical initial conditions and loading schemes. The analyses encompass macroscopic mechanical behavior, shear band characteristics, contact force distribution, and the evolution of fabric and local porosity. The differences between rigid and flexible boundaries are further examined from the perspective of their fundamental loading principles, with particular attention to the influence of specimen volume change. On this basis, recommendations are proposed for the selection of triaxial loading boundaries under different conditions. Collectively, these efforts not only provide mechanistic insights into the observed differences but also inform the rational selection of boundary conditions in DEM-based triaxial simulations and facilitate the interpretation and extension of laboratory test results in geotechnical engineering practice.

2. Materials and Methods

This study employed the three-dimensional Particle Flow Code (PFC3D, Itasca Consulting Group) to simulate triaxial compression tests on sand specimens using both rigid and flexible membrane boundaries. The flowcharts of the two approaches are presented in Figure 1. The rigid boundary simulation is relatively straightforward, in which servo-controlled confining pressure is achieved by moving rigid walls [27]; in this work, the classical rigid boundary modeling approach proposed by Potyondy [28] was adopted. The flexible membrane boundary was implemented following the bonded-ball membrane approach proposed in previous studies [14,21], in which membrane particles are arranged in the widely used hexagonal packing pattern—a configuration proven to be mechanically stable and reliable. Given the complexity of the loading process in flexible membrane modeling, this method is described in detail in the following section.

2.1. Flexible Membrane Boundary Method

In generating the DEM specimen with a flexible membrane, a stepwise membrane installation method was adopted. As shown in Figure 1, the specimen particles were first loaded to the target confining pressure under the servo-controlled conditions of rigid walls. Once the specimen reached a stable state, the boundary condition was switched and the flexible membrane particles were installed. This approach facilitates model stability and maintains the regular arrangement of the initial membrane particles. Moreover, it allows for direct comparison with the rigid boundary loading group. The overall procedure is as follows:

(1)

Generate the specimen within rigid walls and achieve initial equilibrium.

(2)

Apply servo-controlled loading through rigid walls to reach the target confining pressure.

(3)

Reset particle velocities to zero and install the bonded-ball flexible membrane (detailed in the following section).

(4)

Apply confining pressure to the membrane particles (loading method described below).

(5)

Conduct axial loading via the top platen, with membrane particles at both the top and bottom rigidly bonded to the platens to simulate the actual clamping effect in physical experiments.

Figure 1. Schematic illustration of different boundary loading methods.

Figure 1. Schematic illustration of different boundary loading methods.

2.1.1. Generation of Flexible Membrane Particles

The flexible membrane was constructed using a bonded-ball particle assembly arranged in a hexagonal packing pattern, ensuring that each membrane particle had six contacts forming equilateral triangular micro-elements, thereby enhancing stability under large deformations and facilitating both particle interaction searches and the application of equivalent confining pressure. All membrane particles were assigned an identical radius to maintain regular packing. The relationship between the membrane particle radius and the specimen radius was determined according to Equations (1) and (2), which were derived to prevent particle overlap and thus avoid excessive initial stresses in the membrane.

$$2\pi \left(R+\mathrm{r}\right)=2N\mathrm{r}$$

(1)

$$\mathrm{r}={\displaystyle \frac{\pi R}{N-\pi }}$$

(2)

where R is the radius of the specimen. r is the radius of the flexible membrane particles. N is the number of particles required for each layer of the annular membrane.

Previous studies have shown that the radius of membrane particles can significantly affect computational performance: excessively large particles may reduce calculation accuracy, whereas overly small particles can lead to low computational efficiency. Based on comprehensive testing and relevant research, N was set to 138 in this study. The ratio of the membrane particle radius r to the minimum particle radius was 0.37, satisfying the recommended range of 0.2–0.8 proposed by Yang et al. [24].

For the numerical specimen with a diameter of 40 mm and a height of 80 mm, 100 annular particle layers were generated along the z-axis, with each layer consisting of 138 particles (Figure 2). Taking the specimen center as the coordinate origin, with the z-axis aligned in the axial direction, the position of each particle (x, y, z) can be determined by [29]:

$$\mathrm{x}=(\mathrm{R}+r)\times \cos ({\displaystyle \frac{2\pi \times m}{N}}+{\displaystyle \frac{\pi (n-1)}{N}})$$

(3)

$$\mathrm{y}=(\mathrm{R}+r)\times \sin ({\displaystyle \frac{2\pi \times m}{N}}+{\displaystyle \frac{\pi (n-1)}{N}})$$

(4)

$$\mathrm{z}=-{\displaystyle \frac{H}{2}}+2\times r\times \cos ({\displaystyle \frac{\pi (n-1)}{6}})$$

(5)

where m represents the serial number of the particle in each layer, n represents the layer number where the particle is located, and H represents the height of the specimen, which is taken as 80 mm in this paper.

The terminal membrane particles were rigidly bonded to the loading platens to simulate the constraining effect of O-rings in physical experiments, thereby preventing particle loss during loading. Interactions between membrane particles were modeled using the linear contact bond (linearcbond) model, which transmits only normal and shear forces without moment transfer, thereby effectively replicating the mechanical behavior of rubber membranes in conventional triaxial tests. To ensure numerical stability while preserving physical relevance, the bond strength parameters were deliberately set to high values to prevent artificial fracture, with parametric analyses confirming that this adjustment had a negligible influence on the macroscopic simulation results.

2.1.2. Application of Boundary Confining Pressure

During specimen shearing and deformation, the specimen dimensions and the effective number of membrane particles continuously change. Consequently, the force applied to each membrane particle must be frequently updated. To address this, a dedicated servo mechanism was implemented to track the geometric information of both the specimen and the membrane, and to update the forces applied to each membrane particle at every calculation cycle.

Confining pressure was applied through the flexible membrane by exerting uniform stresses on the triangular micro-elements formed between bonded membrane particles. These stresses were subsequently converted into equivalent nodal forces for the three constituent particles based on the finite element method formulation. As illustrated in Figure 3, a planar flexible membrane composed of bonded triangular micro-elements can be generated, where each micro-element consists of three membrane particles of identical radius bonded together. According to the finite element principle for equivalent nodal force calculation, the concentrated force applied to each particle can be expressed as

$${\overrightarrow{{\displaystyle F}}}_{1A}={\overrightarrow{{\displaystyle F}}}_{2A}={\overrightarrow{{\displaystyle F}}}_{3A}={\displaystyle \frac{1}{3}}\times ‖F‖\times {\overrightarrow{n}}_{123}$$

(6)

$$\overrightarrow{{\displaystyle F}}={\sigma }_3\times S_{123}\times {\overrightarrow{n}}_{123}$$

(7)

where ${\sigma }_3$ is the confining pressure applied by the flexible membrane; $\overrightarrow{{\displaystyle F}}$ represents the equivalent force acting on triangle 123; ${\overrightarrow{{\displaystyle F}}}_{1A}$, ${\overrightarrow{{\displaystyle F}}}_{2A}$, ${\overrightarrow{{\displaystyle F}}}_{3A}$ is the equivalent concentrated forces borne by particles 1, 2, and 3, respectively.

2.2. Specimen Preparation

The triaxial shear tests on sand specimens were conducted in accordance with ASTM standards. The specimens were prepared by layered placement and compaction to control the relative density [30,31], achieving a target dry unit weight of 18.5 kN/m³. The resulting relative density was approximately 0.72, classifying the material as dense sand. Tests were performed under confining pressures of 50 kPa, 100 kPa, and 150 kPa, with a constant axial loading rate of 0.1 mm/min. In the numerical simulations, directly using the fine particle size from the laboratory tests would significantly reduce computational efficiency. To overcome this limitation while maintaining simulation accuracy, the particle size was uniformly scaled up by a factor of ten [32,33]. The particle size distribution adopted in the simulation is illustrated in Figure 4. The sand specimen was modeled as a cylindrical specimen with a diameter of 40 mm and a height of 80 mm, consisting of 6878 particles—sufficient to ensure numerical stability. The initial porosity of the numerical specimen was set to 0.37 to match the laboratory test conditions.

To account for the rolling resistance between sand particles during shearing, the anti-rolling contact model (rrlinear model) was chosen for the particle contacts. For the flexible membrane boundary particles, the linear contact bond (linearcbond) model was adopted, ensuring consistency with the mechanical behavior of real flexible membranes by allowing the transmission of normal and shear forces while prohibiting moment transfer. Additionally, to prevent fracture at the particle boundary due to excessive concentrated forces, the bonding strength was set to a very high value [17]. Based on the study of flexible membrane deformation characteristics by Qu et al. [21], the deformation parameters for the membrane particle contacts were selected as shown in

Table 1. Meso-parameters of the particle contact.

Microscopic Parameters	Kn/Ks	Friction	cb_Tenf (N/m2)	cb_Shearf (N/m2)	rr_Fric
flexible membrane boundary	1	0	1 × 10300	1 × 10300	0
Soil particles	3	0.4	/	/	0.5

. A total of 13,800 membrane particles were generated to simulate the boundary behavior of the flexible membrane during the loading process.

Based on laboratory test results and studies of the rrlinear contact model and mesoscopic parameter calibration methods [27], the micromechanical parameters at the particle scale were determined, as detailed in Table 1. Figure 5 presents a comparison between the triaxial test and flexible membrane boundary DEM specimens. As shown in the figure, the stress–strain curve from the numerical simulation closely matches the experimental data, which is consistent with the mechanical behavior of medium-density sand observed in the experiments, confirming the reliability of the selected micromechanical parameters.

3. Results and Analyses

The numerical simulations were conducted under varying confining pressures of 50 kPa, 100 kPa, and 150 kPa, combined with four initial porosity levels (35.5%, 38.2%, 40.8%, and 41.5%). This matrix of conditions enabled a systematic evaluation of how boundary type—rigid or flexible—affects both the macroscopic and microscopic responses of sandy soils. The subsequent analysis examines stress–strain and volumetric strain–axial strain relationships, deformation patterns, porosity evolution, and contact force distributions, highlighting the influence of boundary constraints under different stress states and initial fabric conditions.

3.1. Macroscopic Mechanical Response

The stress–axial strain and volumetric strain–axial strain curves for each group of loaded specimens are presented in Figure 6. With increasing confining pressure levels, the specimens exhibited significantly enhanced shear strength and greater initial volumetric compression, which aligns well with theoretical expectations. Notably, dense specimens demonstrated more strain-softening characteristics under high confining pressures. Experimental data revealed that specimens with different initial porosities ultimately tended to reach stable states with comparable strengths under constant confining pressure conditions.

Comparative analysis of volumetric strain simulation results showed that under relatively dense conditions (initial porosities of 35.5% and 38.2%), the results from flexible membrane boundaries and rigid boundaries were similar. Specimens exhibited dilatant behavior after slight contraction, and high confining pressure significantly suppressed dilatancy. However, when specimens underwent substantial contraction, high confining pressure resulted in greater volumetric compression. Analysis of Figure 6c,d reveals inherent limitations of rigid boundaries in capturing specimen contraction behavior. This is attributed to the superior local deformation adaptability of flexible boundaries, which conform more closely to specimen deformation. Under rigid boundary conditions, mutual constraints among particles hindered the overall servo motion of the walls, making it difficult to accurately reflect certain volumetric changes. In contrast, this hindrance effect was less pronounced during dilatancy, leading to closer agreement between the two boundary conditions.

3.2. Failure Modes and Shear Band

To investigate the failure characteristics of specimens under different mechanical states, this study selected two representative groups for analysis: specimens exhibiting significant strain-softening behavior (150 kPa confining pressure, 35.5% initial porosity) and those demonstrating continuous hardening characteristics (50 kPa confining pressure, 41.5% initial porosity). Through comparative analysis of their failure mode differences and shear band evolution processes under rigid and flexible membrane boundary conditions, we systematically examined the critical influence of boundary constraints on specimen failure mechanisms.

Figure 7 presents the particle rotation diagrams during loading for each specimen group. In the initial loading stage up to 8% axial strain, particle rotation developed gradually, forming only localized rotation zones. With increasing axial strain, localized rotation zones in strain-softening specimens progressively expanded and interconnected, ultimately forming macroscopically continuous shear bands. The magnitude of particle rotation within shear bands was significantly higher than in other regions. Upon further loading to 20% axial strain, non-uniform dilatant deformation occurred in the specimen’s central region, which intensified local particle rotation and induced remarkable changes in the geometric characteristics of shear bands (including distribution patterns, width, and morphology). In contrast, continuously hardening specimens failed to develop distinct shear bands, with their localized rotation zones maintaining a disordered distribution throughout the specimen.

Comparative results under different boundary conditions revealed that specimens constrained by rigid boundaries consistently maintained cylindrical geometry, as the rigid wall motion enforced coordinated deformation of internal particles. Under flexible membrane boundary conditions, peripheral particles exhibited greater freedom of movement. Shear bands in rigid-boundary specimens displayed fully penetrating characteristics, with more pronounced particle rotation observed at contact regions with both top and lateral boundaries. This phenomenon likely originates from the constraint effect of rigid boundaries, where enforced wall displacement not only drives rotation of contact particles but also dominates the specimen’s overall deformation pattern. In comparison, flexible-boundary specimens demonstrated enhanced load-bearing capacity in end regions due to the fixed constraint effect of peripheral particle membranes, which provided additional circumferential confinement.

Figure 8 illustrates the displacement field evolution characteristics of specimens under different failure modes. For strain-softening specimens, distinct inclined shear bands are observed, exhibiting the following key features: the internal particle displacements within shear bands are significantly smaller than those in external regions [15]. Further analysis reveals that displacements within shear blocks are predominantly vertical, while particle displacements in dilatant zones are mainly governed by horizontal components.

Particularly noteworthy is that under rigid boundary conditions, the inability of boundary walls to accommodate local deformation results in simultaneous development of significant vertical and horizontal displacements for particles near specimen ends. This phenomenon correlates well with the penetrating characteristics of shear bands. In contrast, flexible membrane boundary specimens demonstrate different deformation patterns, showing excellent consistency with the aforementioned particle rotation analysis results.

For strain-hardening specimens, both rigid and flexible membrane boundary conditions ultimately produce displacement fields with similar distribution characteristics: vertical displacements dominate in end regions, while the central portion forms a turbulent deformation zone featuring smaller displacements with orientations deviating from the specimen axis.

3.3. Local Porosity Evolution

The variation in porosity during specimen loading is closely associated with material failure behavior. However, macroscopic porosity indicators often fail to accurately capture localized failure characteristics. To address this limitation, this study employs a mesoscale porosity monitoring method [24], utilizing measurement spheres distributed across the xoz plane to obtain porosity distribution contour maps at failure under different loading conditions, thereby providing deeper insights into the mesoscopic failure process.

Figure 9 presents the porosity distribution contours of each specimen group at the end of loading. The results demonstrate that the local porosity distribution at failure is strongly dependent on the initial porosity state [22]. For densely packed specimens (initial porosity = 35.5%), high-porosity zones progressively expanded and interconnected during loading, ultimately forming a macroscopically continuous shear band. In contrast, loose specimens exhibited discontinuous high-porosity regions that remained locally clustered without developing into a fully connected failure band.

The influence of boundary conditions on porosity distribution was also significant. Under rigid boundary constraints, high-porosity zones predominantly aligned along diagonal directions, fully penetrating the specimen—consistent with the observed shear band formation pattern. Conversely, flexible membrane boundary conditions led to high-porosity concentrations in the bulging central region, attributable to the membrane’s adaptability to non-uniform particle displacements, which facilitated greater lateral expansion.

To elucidate the evolution of localized porosity under different failure modes and boundary conditions, this study conducted comparative analyses on two representative specimen types: strain-softening specimens (confining pressure: 150 kPa, initial porosity: 35.5%) and strain-hardening specimens (confining pressure: 50 kPa, initial porosity: 41.5%). The experiments were performed under both rigid and flexible membrane boundary conditions to evaluate boundary effects. As illustrated in Figure 10, fifteen measurement zones were strategically distributed within each specimen for porosity monitoring. The inclination angle of shear band measurement zones relative to the xoy plane was set at 51° based on classical soil mechanics theory. All measurement zones had a radius of 10 times the mean particle diameter (approximately 3.2 mm), ensuring statistically representative results.

Figure 11 presents the evolution curves of localized porosity during specimen loading. As shown in Figure 11, all specimens exhibited porosity reduction during the initial loading stage, primarily attributed to particle rearrangement and densification induced by the compression from loading plates.

For dense specimens, when axial strain reached 5%, the porosity range decreased from (0.33–0.38) to (0.36–0.39), followed by an overall increasing trend in specimen porosity. The porosity differences among measurement points gradually diminished, indicating enhanced homogeneity of the internal structure. Notably, when axial strain exceeded 7.5%, specimens under different boundary conditions demonstrated significantly distinct porosity evolution characteristics. Under flexible membrane boundary conditions, the porosity growth rates in top (Point 2) and bottom (Point 3) regions decelerated markedly and stabilized. At axial strains exceeding 15%, non-shear-band regions (Points 5, 7, 8, 9, 15) exhibited porosity reduction, while shear-band regions (Points 1, 4, 6, 10) maintained continuous porosity increase, forming pronounced localized features. In contrast, under rigid boundary conditions, all measurement points except Point 8 (showing slight porosity decrease) maintained increasing trends, with shear-band regions (Points 1, 9, 11, 12, 14) demonstrating significantly higher porosity growth rates than other regions. Porosity variations stabilized when axial strain surpassed 15%. The above analysis is consistent with the results reported by Lu et al. [26] regarding the evolution of localized porosity in dense sand under triaxial loading with different boundaries. However, Lu et al. did not conduct a comparative analysis on loose specimens.

Figure 11c,d reveal distinct porosity evolution patterns in loose specimens compared to dense specimens. During initial loading (0–10% axial strain), porosity ranges at all measurement points converged under both boundary conditions, indicating overall specimen compression and progressive homogenization under loading. When axial strain exceeded 10%, mid-region points (Points 1, 4, 5, 6, 7) in flexible-boundary specimens showed sustained porosity growth, while edge regions exhibited markedly reduced variation rates, with top and bottom regions even displaying porosity reduction. Under rigid boundary conditions, only the bulging region (Point 4) demonstrated rapid porosity increase, with other regions showing relatively stable variations.

These analyses demonstrate that porosity evolution under flexible membrane boundary conditions exhibits stronger spatial heterogeneity, consistent with its accommodation of free specimen deformation. Conversely, the enforced constraints of rigid boundaries resulted in more uniform porosity variation trends across specimen regions.

3.4. Fabric and Force Chain Analyses

In discrete element method (DEM) simulations, statistical analysis methods are employed to systematically extract information on the magnitude, direction, and type of contact forces between particles [34,35]. This allows the evolution of fabric tensors to be analyzed, revealing the material’s response mechanisms at the microscopic scale in relation to its macroscopic mechanical behavior. During the triaxial shear process of sand, particle contacts are continuously formed and broken, with significant evolution in the distribution of normal contact directions, which intuitively reflects changes in the internal fabric structure of the specimen.

To explore the fabric evolution patterns of specimens with different deformation behaviors under various boundary conditions, this study selected typical strain-softening (150 kPa confining pressure, 35.5% initial porosity) and strain-hardening (50 kPa confining pressure, 41.5% initial porosity) specimens, plotting their three-dimensional polar distribution of average normal contact forces. As shown in Figure 12, the figure illustrates the evolution of the direction distribution of normal contact force branch vectors in three-dimensional space, extracted from the numerical simulations.

For strain-softening specimens, during the early loading stages, the vertical average contact force between particles gradually increases, showing a slight polarity. Once the axial strain exceeds 8%, the average normal contact force decreases and stabilizes. This evolution corresponds to the macroscopic behavior of strain-softening specimens, transitioning from peak strength to residual strength after failure. Notably, under rigid boundary conditions, the contact force distribution in these specimens exhibits more pronounced polarity after the peak, which is further analyzed by plotting the force chain [35,36] distribution at the end of loading for four different conditions (Figure 13).

From Figure 13, it is observed that under flexible membrane boundary conditions, the force chain distribution in the strain-softening specimen is relatively uniform, without significant concentration of strong force chains. In contrast, under rigid boundary conditions, a pronounced, thick force chain region forms along the specimen’s diagonal. This phenomenon supports the previous analysis from a microscopic perspective: under rigid boundary conditions, particles on either side of the shear band tend to compress the rigid walls, leading to an overestimation of the confining pressure by the rigid walls and, consequently, an underestimation of the residual strength in the simulation. In contrast, the flexible membrane boundary maintains a constant confining pressure through the particle membrane, effectively preventing the polarization of the contact force distribution caused by boundary effects.

For strain-hardening specimens, the distribution of normal contact forces remains relatively uniform throughout the loading process, with no significant directional shift, indicating that these specimens primarily undergo uniform compaction without significant shear localization. Further analysis of the force chain structure reveals that under flexible membrane boundary conditions, the force chain distribution is dense and evenly spread, with fewer and shorter strong force chains, lacking distinct long-range force chains. Under rigid boundary conditions, however, the overall contraction of the rigid walls leads to the formation of more robust primary force chain pathways during loading. Although this locally enhanced force chain structure improves overall shear resistance, it limits the specimen’s ability to express volumetric changes, making it less effective at capturing volumetric contraction under rigid boundary conditions.

In conclusion, boundary conditions have a significant impact on the evolution of contact force fabric and force chain structures. Flexible boundaries are more conducive to the natural rearrangement of particles and the reconstruction of the true contact structure, thereby providing a more accurate representation of the specimens.

4. Discussion

4.1. Explanation for the Differences Between the Two Boundary Loading Results

As indicated by the preceding analyses, the differences in specimen deformation characteristics and loading results under the two boundary conditions essentially stem from their distinct loading principles. The flexible membrane boundary applies confining pressure equivalently through membrane particles, leading to a more uniform stress distribution. Consequently, specimen volume changes do not cause local stress concentrations, making it more consistent with the boundary conditions of physical triaxial tests. In contrast, the rigid boundary controls confining pressure through the servo-driven motion of lateral walls. When the specimen undergoes significant volumetric change, the overall lateral movement of the rigid walls is constrained by the particle assembly, resulting in localized stress concentrations. This tends to overestimate confining pressure and leads to a lower post-peak strength. Moreover, such effects also contribute to discrepancies in volumetric strain. In particular, for contractive specimens, the movement of rigid walls is hindered, making it difficult to capture the volume contraction behavior accurately.

4.2. Appropriate Selection of Loading Boundaries

Based on the simulation results of this study, several observations can be made regarding the appropriate use of boundary conditions. For strain-softening specimens (dense sand), if post-peak strength is not a research focus, rigid boundaries can provide reasonable numerical results without incurring excessive computational costs. For strain-hardening specimens (loose sand), when the axial strain is relatively small (less than 8%), rigid boundaries are also capable of producing stress–strain curves comparable to those obtained with flexible boundaries. However, due to the limitations of rigid boundaries in capturing volumetric contraction behavior, they are not recommended for studies involving contractive specimens.

5. Conclusions

This study employed DEM-based triaxial compression simulations to investigate the influence of rigid and flexible membrane boundaries on the macro–micro mechanical behavior of sandy soils with initial porosities of 35.5%, 38.2%, 40.8%, and 41.5% under confining pressures of 50, 100, and 150 kPa. The main findings can be summarized as follows:

(1)

Confining pressure and initial porosity jointly influence boundary effects. At low confining pressures, the differences between rigid and flexible boundaries are mainly reflected in deformation patterns; at high confining pressures, macro-mechanical disparities become more pronounced. Dense specimens under flexible boundaries form sharper shear bands and undergo stronger stress redistribution, while loose specimens exhibit broader deformation zones and gentler post-peak softening.

(2)

Rigid and flexible boundaries produce similar stress–strain responses before peak strength, but rigid boundaries generally underestimate post-peak deviatoric stress—particularly for dense specimens under high confining pressures—while flexible boundaries better capture localized volumetric contraction and heterogeneous deformation.

(3)

Rigid boundaries constrain lateral deformation, producing regular, diagonally oriented shear bands and more uniform porosity changes. Flexible boundaries permit central bulging, heterogeneous shear localization, and greater spatial variability in local porosity, with increases in shear zones and reductions outside them.

(4)

In terms of contact force networks, flexible boundaries generate more uniform force distributions and dispersed force chains, facilitating natural particle rearrangement; rigid boundaries concentrate force chains along shear directions, increasing local anisotropy and affecting residual strength.

(5)

Rigid boundaries apply confinement through servo-controlled walls, while flexible membranes distribute pressure uniformly via membrane particles. Rigid boundaries are applicable for dense sands when post-peak behavior is not critical and for loose sands at small strains, whereas flexible boundaries are necessary for contractive sands and for capturing realistic post-peak responses.

Overall, rigid boundaries are applicable for dense sands when post-peak behavior is not critical and for loose sands at small strains, whereas flexible boundaries are necessary for contractive sands and for capturing realistic post-peak responses. This work enhances the mechanistic understanding of boundary effects and offers a basis for the more efficient selection of boundary conditions in DEM triaxial simulations.