Introduction

The analysis of earth pressures induced by the backfill soil behind a retaining wall is considered one of the oldest and most studied problems in geotechnical engineering, dating back to Coulomb1 and Rankine2 who conducted studies on the resultant force acting on retaining walls, from which the conventional design methods assume a linear distribution of earth pressures, contrasting with the actual non-linear earth pressure distribution revealed in later measurements3,4,5,−6. However, the experimental tests mentioned above were conducted at 1g, resulting in stresses significantly lower than those in real-world scenarios, thus failing to reveal the stress-dependent performance accurately. More recently, Deng and Haigh7 conducted a comprehensive analysis of earth pressure distributions mobilised with active rigid wall movements using centrifuge modelling, replicating a stress state in backfill sand to a depth of 10 m. Their study revealed non-linear earth pressure distributions and the corresponding mobilisation mechanism, and provided a valuable database for reassessing the theoretical methods and validating numerical models.

With the advancement of computational capabilities, the application of the discrete element method (DEM) has been expanded from modelling small-scale element-level tests with few particles to large-scale physical model tests and scenarios to study soil deformation, stability mechanisms, and soil-structure interaction in geotechnical engineering. Experimentally validated DEM models can reveal supplementary soil responses, providing insights into underlying microscopic behaviours in physical modelling. Regarding modelling the response of backfill sand to retaining wall movement, several studies have used the DEM approach to simulate 1g or centrifuge tests with either 2D or 3D models. Chang and Chao8 made one of the earliest attempts at using 2D DEM by modelling soil blocks connected by elasto-plastic springs, highlighting the potential of DEM to estimate the distribution of earth pressures behind retaining walls. Jiang et al.9 applied 2D DEM with spherical particles to simulate active and passive earth pressures across various wall movement modes, underscoring the need to incorporate rolling friction for a more realistic representation of sand response. Their later work expanded on this by simulating passive failure of sand with inter-particle bonding to represent structured sand, revealing the relationship between the mobilised earth pressure and bond strength10. While 2D DEM offers insights, it faces inherent limitations in fully capture the spatial behaviour of granular materials, particularly regarding particle movements within adjacent voids in three dimensions. To address this, Widuliński et al.11 utilised 3D DEM to analyse shear zone mechanisms and internal work under different wall movement modes, observing particles within shear zones tended toward a critical state with significant particle rotations. Similarly, Soltanbeigi et al.12 and Gezgin et al.13 simulated small-scale retaining wall movements, investigating active deformation mechanism and illustrating the earth pressure variations through force chains behind the wall. Despite these advancements in understanding microscopic response, current computational limitations still present challenges in simulating large-scale systems with substantial particle counts. As a result, research on analysing earth pressures against rigid retaining walls using 3D DEM, particularly in direct comparison with experimental data, remains relatively limited.

In actual centrifuge tests, sand movements are often observed through the front or back transparent windows using digital image correlation (DIC) or particle image velocimetry (PIV) techniques; however, the interface between sand particles and windows is not perfectly smooth. This friction can be considered as an inherent artefact that violates the plane-strain condition, leading to inaccurate and misleading understanding of sand response to wall movements. The ability to ensure the plane-strain condition is considered an advantage of DEM modelling and the flexibility in applying various boundary conditions enables the assessment of boundary effects.

Taking into account both the plane-strain condition and the interface friction in the centrifuge tests, this study presents a detailed 3D DEM simulation procedure of sand response with three active wall movement models, as shown in Fig. 1 using a setup closely matching the centrifuge tests conducted by Deng & Haigh7. The magnitudes and distributions of earth pressures in the DEM models are compared with those measured in centrifuge tests. The effect of front and back window roughness is analysed, with the influence distance of 9 d50 being then evaluated, beyond which the effect of the friction can be negligible.

Fig. 1
figure 1

Active movement modes of rigid retaining wall: a rotation about the base; b rotation about the top; c translation.

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3D DEM simulations

DEM simulations were conducted in accordance with the centrifuge tests conducted by Deng & Haigh7, consisting of three active rigid wall movement modes in both loose and dense specimens, with relative densities of 35% and 85%, respectively. The simulations were performed using the open-source code LIGGGHTS, and high-performance computing was employed with 4 nodes and 128 CPUs. As shown in Fig. 2, sand with dimensions of 550 mm × 200 mm × 250 mm (width × thickness × height) was modelled and enclosed by a container measuring 550 mm × 200 mm × 300 mm.

Fig. 2
figure 2

Schematic diagram of the DEM model (unit: mm).

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Modelling real particle sizes and shapes is computationally expensive, therefore, simplifications were made in the DEM simulations to enhance computational efficiency while maintaining the representative behaviour of the granular material similar to actual sand. In this study, spherical particles were used in the DEM models, with a particle size distribution (PSD) based on Hostun sand14 scaled up by a factor of 10, following approaches adopted in previous research on soil–structure interaction using DEM9,15,16. Approximately 500,000 particles were generated to fill the desired specimen volume. It should be noted that due to the simplification of using spherical particles, the actual maximum and minimum void ratios, \({e_{{\text{max}}}}\) and \({e_{{\text{min}}}}\), of Hostun sand used in centrifuge tests cannot be replicated. Instead, the \({e_{{\text{max}}}}\) and \({e_{{\text{min}}}}\) of the simulated spherical particles were calculated as 0.739 and 0.577, respectively. Consequently, the relative density, \(RD\), was adopted as an index to assess the soil density and to ensure that the specimens were comparable to those in the centrifuge tests. Specifically, \(RD\) values of 35% and 85% were used to represent loose and dense specimens, respectively.

The nonlinear Hertz contact model available in LIGGGHTS was used to calculate normal and tangential forces between particles, with a detailed implementation described by Lin & Ng17. Simulation input parameters were listed in Table 1 which were determined through direct shear test simulations validated by the experimental measurement from Da Silva14, following the procedure outlined in ASTM-D308018. The particle density, \({\rho _{\text{s}}}\), of 2650 kg/m3 was used, based on the specific gravity of quartz reported by Deng & Haigh19. Parametric studies were conducted to determine the Young’s modulus, E, Poisson’s ratio, v, and the coefficient of restitution, \(COR\), of Hostun sand. The simulation results were found to be insensitive to Young’s modulus of particles, \({E_{\text{p}}}\), for values greater than 2 GPa; therefore, 2 GPa was chosen to achieve a larger time-step for improving computational efficiency while maintaining small interparticle overlaps. The Young’s modulus for the retaining wall and the container, \({E_{\text{r}}}\) was set to be 200 GPa, approximately equalling to the modulus of steel, to ensure rigidity during the simulation. A v of 0.3 and a \(COR\) of 0.5 were applied, as they showed insensitivity to the macroscopic results based on parametric studies.

Table 1 Parameters used for sand and wall simulation.
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The stress-strain response is governed by the interparticle friction coefficient, \({\mu _{\text{p}}}\), as discussed by Thornton20, while the particle-wall interface friction, \({\mu _{{\text{pw}}}}\), also plays a crucial role in determining the magnitude of earth pressures induced in the backfill sand, as noted by Grabowski et al.21, both of which were therefore validated through simulating direct shear tests. By applying the material properties mentioned earlier, a series of direct shear tests were simulated with different values of \({\mu _{\text{p}}}\) as shown in Fig. 3, and the peak and critical state shear friction angles, \(\phi _{{\text{p}}}^{\prime }\) and \(\phi _{{{\text{crit}}}}^{\prime }\), 46.2° and 31.4°, of Hostun sand, measured by Da Silva14 and Lan (personal communication, 2023) at the confining pressure of 100 kPa, were closely approximated using an \({\mu _{\text{p}}}\) of 0.7, with the values of 46.5° and 31.7° being obtained. Similarly, employing the same \({\mu _{\text{p}}}\) of 0.7 and setting the particle-wall interface friction, \({\mu _{{\text{pw}}}}\), to 0.7 in the direct shear tests of the particle-wall interface resulted in interface angles, \({\delta _{{\text{pw}}}}\), of 21.1° and 24.5° for loose and dense specimens, closely matching the interface friction angles measured by Deng & Haigh22 at 21.7° and 24.7°, respectively. The particle-window interface friction, \({\mu _{{\text{sw}}}}\), of 0 was applied to elucidate the earth pressure distribution under the plane-strain condition, and a value of \({\mu _{{\text{rw}}}}\) of 0.7 was used to evaluate the boundary effect of rough PMMA windows according to the friction measurements by Widisinghe and Sivakugan23.

Fig. 3
figure 3

Peak and critical state friction angles mobilised with interparticle friction coefficient.

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In the centrifuge tests, the earth pressures were measured using a Tekscan pressure mapping system attached to the wall, as reported by Deng24. In the DEM models, particle forces were calculated and output at frequent time step, along with forces at the interface between particles and the retaining wall. The sand deformation in centrifuge tests were observed through transparent poly methyl methacrylate (PMMA) windows using the PIV technique, with only the sand behind the window being measured. Conversely, in DEM simulations, data regarding particle kinematics can be obtained not only behind the window but also across any section along the thickness.

The DEM simulation procedure consists of three stages: sample preparation, gravity increase and wall movement. Initially, approximately 500,000 particles were generated as a non-contacting cloud within the container, with an \({\mu _{\text{p}}}\) of 0.7. This was followed by a gravity-settling stage, in which particles were stabilised under the earth’s gravity over several time steps. Since the specimen did not experience any disturbances during this stage, it was considered to be in its loosest state. The \({\mu _{\text{p}}}\) was reduced from 0.7 to 0.22 for the loose and from 0.7 to 0.04 for the dense specimen to further settle the particles and achieve the desired \(RD\) of 35% and 85%, as attained through parametric studies. Once the specimens reached target \(RD\), \({\mu _{\text{p}}}\) was returned to 0.7, which is the value for simulating interparticle contacts of Hostun sand, as established in the parametric studies as mentioned earlier. After sample preparation, the gravity was increased from 1g to 40g in increments of 10g and then different movement modes were applied to the wall using displacement-controlled approaches. According to Perez et al.25, the simulation results are not sensitive to the shear rate if the quasi-static conditions are maintained, with the inertia number, I, shown in Eq. 1 being lower than 2.5 × 10− 3. A horizontal wall movement velocity of 1 mm/s was used in the DEM simulations, applied to either the top or the base of the wall, giving an I expressed in Eq. 1 of approximately 6.64 × 10− 6.

$$I=\dot {\varepsilon }\bar {d}\sqrt {\frac{{{\rho _{\text{s}}}}}{{p^{\prime}}}}$$
(1)

Where \(\dot {\varepsilon }\) is the shear strain rate; \(\bar {d}\)\(\bar {d}\)\(p^{\prime}\) is the average particle diameter; and \(p^{\prime}\) is the mean effective stress.

Earth pressure results

Figures 4, 5 and 6 depict the comparison of DEM simulations with the centrifuge tests conducted by Deng & Haigh7 regarding the mobilisation of earth pressure coefficients, \(K\), with three different modes of wall movements in loose and dense specimens. The values of \(K\) are shown for wall movements of 10% and 100% of the maximum displacements, corresponding to the points at which the reduction in earth pressure terminated and the active state was considered to be fully mobilized. The earth pressure coefficient at-rest, \({K}_{0}\), corresponding to the “DEM 0” in the figures, is consistent for all the loose DEM specimens, as shown in Figs. 4a, 5a, and 6a, and likewise for the dense specimens shown in Figs. 4b, 5b, and 6b, given the identical loose or dense specimens at the beginning of wall movements. This can be considered as an enhancement of DEM simulations compared to the experimental counterpart, as it eliminates any initial variations during sand pouring. Another improvement of the DEM simulations is that the retaining wall can maintain totally stationary during the acceleration increasing from 1g to 40g, while the wall in the centrifuge tests might move slightly due to insufficient prop stiffness as mentioned by Deng24, potentially disturbing the at rest state of the specimens.

Fig. 4
figure 4

Comparison of DEM simulating and centrifuge testing earth pressure coefficients for rotation about the base: a in loose sand; b in dense sand.

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Fig. 5
figure 5

Comparison of DEM simulating and centrifuge testing earth pressure coefficients for rotation about the top: a in loose sand; b in dense sand.

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Fig. 6
figure 6

Comparison of DEM simulating and centrifuge testing earth pressure coefficients for translation: a in loose sand; b in dense sand.

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The theoretical earth pressure coefficient at-rest, \({K_0}\), calculated using Eq. (2)26, and that at the active state, \({K_{\text{a}}}\), calculated using Eq. (3)1 are plotted in Figs. 4, 5 and 6 along with the numerical and experimental results for comparison. It should be noted that Eq. 2 only predicts the \({K_0}\) for loose specimens.

$${K_0}=1 - \sin \phi$$
(2)
$${K_{\text{a}}}=\frac{{{{(\cos \phi )}^2}}}{{\cos \delta {{\left[ {1+\sqrt {\left( {\sin (\delta +\phi } \right)\sin \phi )/\cos \delta } } \right]}^2}}}$$
(3)

.

As shown in Figs. 4, 5 and 6, prior to wall movement, the DEM simulations effectively capture the overall non-linear distribution of K across both loose and dense specimens, showing larger values at relatively shallow depths that gradually decrease and stabilise with depth. This distribution of earth pressure coefficients with depths is comparable to the observed relationship between K and the applied vertical stress, \({\sigma _{\text{v}}}\), in consolidation experiments. The relatively large K values at shallow depths behind the wall resemble the initial isotropic state under low effective vertical stress noted in several consolidation studies, such as those by Lo and Chu27, Watabe et al.28 and Chu and Gan29. For the loose specimens in particular, the predicted \({K_0}\) by Jáky26 is closely approached by both DEM simulations and the experimental results at greater depths.

Force chains from the simulations are presented in Fig. 7 to provide a microscopic perspective on earth pressure development at the point of maximum wall displacement, corresponding to the conditions analysed in Figs. 4, 5 and 6. The interparticle contacts are presented by solid lines, with the line thickness proportional to the magnitude of the contact force. A key modification in this study, compared to conventional approaches, is the normalisation of line thickness by the vertical stress of the specimen. This adjustment ensures the preservation of microscopic details, particularly in scenarios with significant force discrepancies between particles in shallow and deep layers. Notable differences can be observed across the wall movement modes. In the rotation about the top mode, strong force chains develop in the shallow layers, forming an arch shape. These chains are orientated horizontally near the wall and gradually transition to a vertical orientation with increasing distance from the wall. In contrast, the rotation about the base and translation modes predominantly generate the force chains aligned along the vertical direction. Overall, different wall movement modes result in distinct patterns of earth pressure mobilisation and force chain formation, observed from both macroscopic and microscopic viewpoints. These patterns are analysed in detail in the following section.

Fig. 7
figure 7

Force chains in six simulations captured at the points of maximum wall displacement: a, b rotation about the base mode; c, d rotation about the top mode; e, f translation mode.

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K for rotation about the base

By comparing the “DEM 0” and “Test 0” lines with the other soil pressure coefficient plots presented in Fig. 4a and b, it is evident that K for both loose and dense specimens in the DEM simulations reduces with active wall rotation about the base, with the value of reduction being approximately uniform along the full depth behind the retaining wall, consistent with the observed trends in the centrifuge tests. At the depth of 2.5 m, where the retaining wall had reached a stable position after displacement, the earth pressure coefficients for the loose sand samples and dense sand samples were 0.35 and 0.31, showing reductions of 43.5% and 34.0%, respectively. Larger K values at shallow layers can be observed for both DEM and experiments at 2%H, compared to the analytical values by Coulomb1, attributed to the isotropic state under low effective vertical stress as previously discussed. The synchronous reduction of earth pressures is the result of a uniform shearing wedge behind the retaining wall, in which uniform shear strain is mobilised as discussed by Bolton & Powrie30 and Deng & Haigh22. It can also be seen from Fig. 4 that DEM simulations exhibit faster reduction in K than that in the centrifuge tests, due to greater initial stiffness of the specimen. Figure 4a shows similar distributions of K calculated numerically, measured experimentally, and derived analytically at the wall top displacement of 2%H for the loose specimen, showing the satisfactory performance of DEM models. While it is worth noting in Fig. 4b that the measured K arrived at the active state earlier than that simulated, which might be caused by the lower K values at the beginning, possibly due to the slight wall movement during the centrifugal acceleration increasing as mentioned by Deng & Haigh7. Overall, DEM simulations have captured the mechanism of earth pressure mobilisation in the rotation about the base mode.

K for rotation about the top

Regarding the rotation about the top mode, DEM simulations well capture the notable trends observed in the centrifuge tests as shown in Fig. 5, in which earth pressures increase in shallow layers but decrease at depth with active wall rotation, indicating that the sand behind the upper wall enters the intermediate passive state. Rotation of the wall about the top restricts sand deformation in shallow layer, resulting in the formation of the arching effect behind the upper part of the wall, causing sand particles there to be compressed. At the depths of 0.5 m, 1.5 m, and 2.5 m, where the retaining wall had moved from 0 to 2%H, the earth pressure reduction for the loose sand samples was 24.7%, 22.7%, and 43.5%, respectively. For the dense sand samples, the corresponding reductions were 29.6%, 38.2%, and 34.0%, respectively. As seen in Fig. 5a that, for the loose DEM specimen, K increases and the sand enters the intermediate passive state behind the top 1/4 H of the wall at a wall base displacement of 0.2%H. The passive zone propagates with the wall movement and develops to the mid-depth behind the wall at the wall base displacement of 2%H. This finding provides valuable insight into the research of sand arching, which influences stress distribution on retaining structures31.

The development of soil arching is analysed using the loose specimen as an example, as shown in Fig. 8. The analysis is illustrated through force chains in Fig. 8a and b, the variation of earth pressure with the wall movements in Fig. 8c, and displacement vectors of the specimen in Fig. 8d, corresponding to wall movements of 0.2%H and 2%H, and through displacement vectors of the specimen in Fig. 8c. At a wall movement of 0.2%H, larger horizontal forces are observed in the upper 1/4 H of the wall, aligning with the earth pressure distribution shown in Fig. 8a, c. As wall movement increases, the arch-shaped force chains become more pronounced at 2%H, concentrating horizontal interparticle forces in the upper half of the wall and transmitting stress to particles at greater depths away from the wall. Figure 8d shows particle displacements in a curved trend, where particles near the wall top are restricted from horizontal movements due to top constraints, leading to compression between particles in the upper part of the wall and resulting in the arching effect. Overall, Fig. 8 clearly illustrates the concept of arching, showing how stress is distributed within the soil body as the wall moves, with some particles bearing a greater load.

Fig. 8
figure 8

Microscopic response in the loose specimen for rotation about the top mode: a soil arching indicated by force chains at 0.2%H; b soil arching indicated by force chains at 2%H; c variation of earth pressure; d relative displacement vectors between 0%H and 2%H.

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K for translation

Figure 6 shows the comparison of K distributions simulated using DEM with those measured in the centrifuge tests, showing that the earth pressure increase in shallow layers is well captured. The earth pressure increase induced by the translation mode is less pronounced compared to that caused by the rotation about the top mode. This difference is due to the tendency for sand collapse to occur more easily in the translation mode. The local increase of earth pressures at the upper section of the wall is attributed to sand arching as noted by previous studies conducting 1g tests6,32, and analytical approaches33,34, wherein such a mechanism, the arching is caused by the friction of the retaining wall providing upwards support to the sand body. One of the possible reasons for the smaller arching area and lower earth pressure increase in DEM simulations could be the enlarged particle sizes with greater particle mass compared to the actual sand. Consequently, the support from the interface friction of the wall might not be able to hold as many particles as in the experiments. Another possible reason could be the spherical shape of the particles, so that the interlock between particles and the wall was reduced, resulting in sand collapse at lower wall movement, reflected by the smaller earth pressure increase in a shallower depth.

Analysis of the boundary effects

Although the PMMA window applied in the centrifuge tests is polished after each test, the interface friction still exists and can be considered as an inherent artefact to simulate a plane-strain condition. Therefore, it is crucial to investigate whether the boundary friction could influence the results and potentially compromise the integrity of the plane-strain condition. One of the advantages of performing 3D DEM simulations in this study is the ability to capture both the earth pressure distributions and particle kinematics across the specimen, facilitating a comprehensive analysis of the boundary effects associated with the front and back windows.

The effects of friction at the interface of the PMMA window are analysed, where a DEM model with a coefficient of friction, \({\mu _{{\text{rw}}}}\), of 0.7 at the particle-window interface is applied, following the same sample preparation procedure as the original models with the smooth windows with a \({\mu _{{\text{sw}}}}\) of 0 as described earlier in Sect. 2. The loose specimen with the rotation about the base mode is chosen for comparison, considering the relatively simple sand response behind the wall rotating about the base, and it is a common scenario in the excavations retained by cantilever walls in the field. Moreover, DEM exhibits effective performance in capturing both the trend and magnitude of earth pressures, as validated by the centrifuge tests shown in Fig. 4a.

Figure 9 illustrates the earth pressures behind the retaining wall in loose specimen at rest as well as when the wall moved by 0.2%H and 2%H. Similar values of earth pressures are shown for both models along the full depth of the retaining wall before the wall moves, indicating that the boundary effects are insignificant during sand pouring. However, the differences in K become pronounced with the wall movement, with slightly smaller magnitudes of K observed in the model with rough windows. The reason for this phenomenon is attributed to the friction from the windows, which balances a portion of the self-weight of the soil wedge, causing the supportive force from the retaining wall to be lower than that in the plane-strain condition. As the reaction to the reducing supportive force from the retaining wall, the earth pressures applied to the wall reduce, resulting in smaller K distributions. This discrepancy warrants careful consideration during geotechnical exploration using physical modelling.

Fig. 9
figure 9

Comparison of K in the models using smooth and rough boundaries for rotation about the base mode.

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Due to the 3D models used in this study, the effects of the rough windows can also be analysed by comparing the earth pressure distributions along the model thickness. Figure 10 demonstrates the earth pressures acting along the thickness of the model at the depth of 0.5 m, 1.5 m, 3.5 m, 6.5 m and 8.5 m at the wall displacement of 2%H for both models. It can be seen from Fig. 10a that, for the model with the smooth windows, the earth pressure distributions are relatively uniform at each depth with only slight fluctuations along the thickness. While the effects of the rough windows are pronounced as shown in Fig. 10b, in which larger earth pressures distribute in the central zone and smaller pressures are observed close to the windows, forming arch-shaped earth pressure distributions along the thickness of the model, reducing the total earth pressure at certain depths as shown earlier in Fig. 9. Comparing Fig. 10a and b, similar magnitudes of earth pressures are observed in the central area along the thickness, showing the weakening of boundary effects, which can be further evaluated through particle kinematics.

Fig. 10
figure 10

Distribution of horizontal earth pressures along the thickness for the loose specimens for rotation about the base mode using a smooth window; b rough window.

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The application of rough windows also results in different particle kinematics compared to those in the model using smooth windows, which are illustrated in Fig. 11. Figure 11 shows the displacement vector diagram for particles at the wall displacement of 2%H, where the displacement components in the x and y directions are combined to illustrate the resultant displacement. The diagram also shows the rotational angle vectors for the particles at the wall displacement of 2%H, representing the resultant rotational angle obtained from the combination of the rotational angle components in the x, y, and z directions. Similar particle displacements and rotations are observed along the window and in the central section of the specimen with smooth windows as shown in Fig. 11a–d, as well as those in the central section of the model using rough windows as shown in Fig. 11g and h, validating the earth pressure distributions shown in Fig. 10. However, comparing Fig. 11a–e, the particle displacements behind the rough window are generally smaller than those behind the smooth window, indicating the restriction of displacements due to window friction. The frictional effects are also reflected by the cumulative particle rotations as shown in Fig. 11b and f, where larger particle rotations are observed behind the rough window due to larger interface friction between the window surface and particles. It is noteworthy that although the simplified spherical particle shape used in DEM simulations does not fully capture the behavior of real particles, particularly with respect to frictional effects between particles, since non-spherical particles are also subjected to mechanical interlocking effects11, Fig. 11b and f clearly show the impact of rough glass surfaces on particle rotations in the boundary region. Therefore, despite using spherical particles, the results from this study still provide valuable insights for analyzing boundary effects. Moving away from the rough window with diminishing boundary effects, increasing particle displacements and decreasing particle rotations can be observed, achieving the similar kinematics as those observed in the model with smooth windows.

Fig. 11
figure 11

Particle displacement and rotation at the wall displacement of 2%H: a and c particle displacements for smooth boundary (unit: m); b and d particle rotations for smooth boundary (unit: m); e and g particle displacements for rough boundary (unit: degree); f and h particle rotations for the rough boundary (unit: degree).

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The region of effects of the rough windows is studied as shown in Fig. 12 by considering the particle rotations at different distances away from the front window. It can be seen from Fig. 12 that the specimen away from the rough window tends to approach a state with particle rotations similar to those adjacent to the smooth windows as shown in Fig. 11b. The median particle rotation within the triangular shearing wedge was calculated to be 15.3°at half the specimen thickness. A significant value of 22.3° was obtained for the specimen behind the rough window. As the distance from the rough window increased, the median rotation decreased to 17.8° in Figs. 12b and 16.0° in Fig. 12c, eventually stabilizing at 15.3° at a distance of 9 d50 from the rough window, which is approximately 1.26 m. This indicates the elimination of the rough window’s influence. It should be noted that this distance may be influenced by particle sizes and window friction coefficients, which deserves further investigations.

Fig. 12
figure 12

Particle rotations at different distances from the rough front window (unit: degree).

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Conclusion

A comprehensive set of 3D DEM simulations of active wall movement was conducted, including wall rotation about the base, rotation about the top, and translation, considering both loose and dense specimens.

  1. 1.

    The DEM models effectively captured the trends in earth pressure coefficient distribution and yielded magnitudes similar to those predicted by the analytical solutions and observed in the centrifuge tests. The simulations accurately predicted the non-linear distribution of earth pressure coefficients at rest, with higher values at shallow depths and values approaching Jáky’s analytical solutions for \({K_0}\) at greater depths, consistent with centrifuge results.

  2. 2.

    As wall movement increased, a reduction in earth pressure coefficients was observed, with the dense specimens exhibiting lower values than the loose specimens. At the larger wall movements of 2%H, where the earth pressure was fully mobilised, the DEM models, regardless of movement mode, show earth pressure coefficients in the lower half of the wall comparable to those from centrifuge tests and Coulomb’s analytical solutions for \({K_{\text{a}}}\). In the rotation about the top and translation modes, DEM simulations also displayed an increase in earth pressure coefficients at shallow depths, effectively capturing the trends observed in centrifuge tests.

  3. 3.

    In rotation about the top mode, greater earth pressure coefficients at shallow depths were attributed to soil arching due to the wall constraints, resulting in localised particle compression and enhanced earth pressure. The arching effect was evident from both the arch-shaped force chains and particle displacement vectors. In translation mode, arching was caused by friction from the retaining wall providing upwards support to the sand body. However, the increase of earth pressure coefficients was less pronounced in DEM simulations due to large particle sizes and reduced interlocking from spherical particle shape.

  4. 4.

    The effects of friction on the front and back windows were assessed through 3D DEM analyses, revealing that a rough window reduced particle displacements and increased particle rotations, resulting in lower earth pressures on the wall. Frictional effects diminished with distance away from the boundaries, impacting both earth pressure distribution and particle kinematics. The analysis of boundary effects offers valuable insights for future researchers aiming to achieve precise measurements of earth pressures and sand deformation mechanisms in physical modelling.