The test sample is coral sand from an island reef in the Maldives, and the coral sand samples are sieved to obtain the particle groups of different sizes of <0.075 mm, 0.075–0.25 mm, 0.25–0.5 mm, 0.5–1 mm, 1–2 mm, and 2–5 mm, which are presented in Figure 1. According to the ‘Code for Geotechnical Investigation on Port and Waterway Engineering (Partial Revision)-Geotechnical Investigation of Coral Reefs (General Revision Draft)’ (CCCC Second Navigation Engineering Survey and Design Institute, 2022), coral gravel sand, coral coarse sand, coral medium sand, coral fine sand, and coral powder sand are prepared, respectively. The five types of coral sands have a wide range of particle gradation distribution, which allows for the acquisition of all mechanical parameters of coral sands with typical representativeness.

The control particle size and gradation indexes are summarized in Table 1. The gradation curves of the five types of coral sands are depicted in Figure 2.

The tests are carried out on GZQ-1 type fully automatic pneumatic consolidation apparatus. The equipment is produced by Nanjing Ruitai Geotechnical Testing Instrument Co., Ltd., with an axial load error of ±0.5% FS and a lateral pressure sensor nonlinearity of 0.1%. The whole system consists of a consolidation apparatus, a pneumatic pressure controller, a multi-channel communication converter, and a data acquisition system, as presented in Figure 3. The interior of the pressure chamber is inlaid with a ring-shaped rubber membrane, and the water is filled between the rubber membrane and the outer wall of pressure chamber. Coral sand squeezes the rubber membrane, leading to a varying water pressure on the side wall, which allows for the determination of the lateral pressure of coral sand sample by water pressure. The pressure chamber has a diameter of 61.8 mm and a height of 40 mm, and the maximum axial consolidation pressure it can withstand is 6 kN. It is able to provide a horizontal stress measurement range of 0–1,000 kPa and an axial displacement effective test range of 0–10 mm. The vertical load can be set freely by computer programme, and six vertical load levels of 50 kPa, 100 kPa, 150 kPa, 200 kPa, 400 kPa, and 600 kPa are designed for the tests.

The test procedure includes sample preparation, sample installation, loading, and data acquisition, while the dry and saturated scenarios are configured. For preparation of dry sample, the dried sand samples weighed according to the design density of 1.45 g/cm3 are loaded into the pressure chamber of the lateral pressure gauge for compaction in three times, and the compactness designed for the test is achieved by controlling the mass of sample added each time and the height of sample after compaction. The specific operation is as follows: the sample is filled in three layers, and each layer is manually compacted 10 times using a standard compaction hammer (hammer weight 2.5 kg, drop distance 200 mm). The height error of each layer after compaction is controlled to be ≤0.5 mm. For preparation of saturated sample, according to the design density, the coral sand sample of the corresponding mass is placed into the pressure chamber of the lateral pressure gauge, and then the lateral pressure gauge is put into the vacuum bucket. The vacuum pump is activated to evacuate the sample for 48 h to achieve a complete saturation.

The test adopts the rapid loading method, with a stabilization duration of 1 h for each level of loading (NSAI, 2017). The gas-free distilled water is selected for the test to ensure that no gas exists in water bladder, which results in a more accurate measurement of the horizontal pressure. The saturated coral sand sample is always covered by water during the testing process, and a certain amount of pre-pressure needs to be applied first to make the sample reach the design compactness prior to the test. The lateral pressure after stabilization of each level of load is continuously recorded during the test, and the test is terminated when the deformation under the last level of load is less than 0.005 mm/h. The preloading load is uniformly set at 30 kPa (equivalent to 60% of the minimum formal load of 50 kPa), which has been verified through pre testing as the optimal solution for eliminating sample gaps.

The relationships between the horizontal and vertical stresses of samples in dry and saturated states are obtained, respectively, and then the K ₀ curves of coral sand are determined, as presented in Figure 4.

As indicated in Figure 4, the horizontal stress increases linearly with the increase of vertical stress, which demonstrates a basically constant K ₀ along the coral sand at different depths. The K ₀ values of coral sands in dry and saturated states are statistically obtained, as summarized in Table 2.

The values of the K ₀ of coral powder sand, coral fine and medium sand, as well as coral coarse and gravel sands in dry state are 0.32, 0.25–0.27, and 0.22–0.23, respectively, while those in saturated state are 0.33, 0.30–0.32, and 0.27–0.27, respectively. These values are smaller than the laboratory test results of quartz sand (whose K ₀ is 0.39–0.47) (Xu et al., 2007) (The test conditions for quartz sand are relative density Dr = 70% and dry state). Also, the K ₀ values in saturated state are higher than those in dry state. The increase in K ₀ of saturated coral sand is essentially due to the dual influence of water on the geotechnical system - firstly, pore water forms pressure chambers in the particle gaps, similar to a micro hydraulic system that converts vertical loads into lateral thrust, directly pushing up horizontal stress; Secondly, water molecules wrap around the edges and corners of coral sand to form a lubricating film, weakening the mechanical interlocking friction between particles and making them more prone to rolling adjustment. This lubricating effect further releases the originally locked horizontal stress. The synergistic effect of these two mechanisms results in a significantly stronger horizontal stress response in the saturated state than in the dry state.

Figures 5, 6 illustrate the relationships between particle gradation and K ₀ in dry and saturated states, where the quantitative calculation formulas are also presented. The correlation coefficients are all above 0.9, indicating a good correlation between particle gradation and K ₀ and a high calculation accuracy of the listed formulas. It is observed that the K ₀ tends to decrease gradually with the increase of coefficient of uniformity and coefficient of curvature. This is due to the fact that with the increase in the gradation range from coral powder sand to coral gravel sand, the requirements of Cc = 1-3 and Cu ≥ 5 are gradually fulfilled, and the particle gradation changes from poor to good. Coarse particles form the skeleton, which is filled by fine particles. The occlusion between particles is enhanced while the overall resistance to deformation is increased. Under the vertical load, the horizontal stress is smaller, which leads to a lower K ₀ value measured. Narrow graded sand (such as silt) has uniform particles, concentrated force chains under vertical loads, and high horizontal stress transmission efficiency, resulting in a K0 of 0.32; Coarse particles form the main skeleton in wide graded sand (such as gravel sand), and fine particles fill the pores to enhance the “three-dimensional interlocking effect”, converting more vertical loads into particle interlocking internal forces and reducing horizontal stress release. Therefore, K0 is reduced to 0.22. The decreasing trend of K ₀ suggests that the magnitude of change of K ₀ in saturated state is significantly smaller than that in dry state. This is due to the fact that the lubrication of water in coral sand in saturated state reduces the occlusal friction between the particles, which leads to a decrease in the effect of the gradation on the K ₀.

The existing numerical tests report that the particle size enlargement method has less influence on the mechanical behavior of the material (Evans and Valdes, 2011; Belheine et al., 2009). The simulation in this study uniformly enlarges the particle size by 1.5 times. The numerical test procedure is as follows:

(1) Coral sand particles of the same gradations as those in laboratory tests are generated in a cylinder with a diameter of 61.8 mm and a height of 40 mm. Finally, 7,394 coral gravel sand particles, 10,263 coral coarse sand particles, 14,840 coral medium sand particles, 15,268 coral fine sand particles, and 17,196 coral powder sand particles are generated, respectively.

This paper utilizes a parallel bond model to simulate the bonding force between the coral sand particles. The parallel bond model can describe the contact characteristics between the bonded particles, where spring components with constant normal and tangential stiffness are distributed within the contact surface of the particles. When the displacement or force between particles exceeds the critical parallel bonding strength, the parallel bonding between particles fails and fracture occurs.

Discrete element numerical simulation requires the input of meso-parameters such as contact stiffness and friction coefficient. Referring to its range in previous numerical experiments (Stratton and Wensrich, 2010; Thompson et al., 2009), the particle normal stiffness in this study is set to 100 MN/m, while the particle stiffness ratio is set to 1.0, which meets the recommended range of 1.0–1.5 by Goldenberg and Goldhirsch (2005). By reverse fitting the key characteristics of the stress-strain curve of the indoor triaxial test, it is achieved that the basic microscopic parameters (friction coefficient, stiffness, etc.) are first fixed, and then the cohesion and friction angle are gradually adjusted to make the peak stress, residual strength trend, and elastic segment slope simulated by PFC consistent with the test curve. The iterative process aims to match the overall shape of the curve, and the final parameter values are listed in Table 3 after multiple verifications.

Figure 8 presents the comparison between the numerical and experimental results regarding the relationship between the horizontal and vertical stresses for coral powder sand. It is observed that there is only a minimal discrepancy between the K ₀ values obtained from numerical simulations and laboratory tests, indicating a high overall agreement between the two results. This verifies the feasibility of numerical simulation and the rationality of calibrated meso-parameters. An in-depth analysis of the numerical simulation results can be conducted based on this foundation.

Coral sand particles are angular with irregular shapes, and exhibit relatively strong inter-particle cohesion, which is characterized by the bonding strength in PFC simulation. To study the influence of bonding cohesion on K ₀, K ₀ tests are conducted under six different bonding cohesion cases of 5 × 10⁴, 1 × 10⁵, 5 × 10⁵, 1 × 10⁶, 3 × 10⁶, and 5 × 10⁶ Pa. The particle gradation of coral powder sand is used for all cases, and the other meso-scale parameters are the same as those in Table 3. The obtained K ₀ values for different bonding cohesions are presented in Figure14a. It is evident that as the inter-particle cohesion increases, the K ₀ value of coral sand gradually decreases. Once the cohesion exceeds 1 × 10⁵ Pa, the K ₀ tends to stabilize gradually and finally remain unchanged as cohesion increases. The relationship between them approximately follows an exponential function relationship, as depicted by the red fitted curve.

To explore the influence of bonding friction angle on K ₀, K ₀ tests are conducted under six different bonding friction angle cases of 0°, 10°, 20°, 30°, 40°, and 50°. The particle gradation of coral powder sand is used for all cases, and the other meso-scale parameters are the same as those in Table 3. The obtained K ₀ values for different bonding friction angles are displayed in Figure14b. It is observed that the K ₀ of coral sand gradually decreases as the inter-particle bonding friction angle increases. When the friction angle is greater than 30°, the K ₀ tends to stabilize gradually with minimal change in magnitude as friction angle increases. The relationship between the K ₀ and the bonding friction angle approximately satisfies an exponential function relationship, as indicated by the red fitted curve.