The coral sand from a reef in the Nansha Islands was used for testing, which is a unique marine soil formed from the remains of marine organisms through physical, chemical, and biological processes. The coral sand grains are mainly composed of aragonite, high-magnesium calcite, and 90% CaCO₃. It is a calcareous sand with a specific gravity of 2.84. Coral sands are widely distributed in tropical and subtropical waters, and extensively found across China’s Nansha Islands, serving as the primary geotechnical media with yellowish–white grains in the Nansha Islands area. The coral sand used in this study was found in the complex marine environment of the Nansha Islands, which results in several distinctive properties of coral sand grains: they are angular, prone to cementation and breakage, and have rough surfaces and internal pores (Karatza et al., 2017; Deng and Haigh, 2022).

The coral sand was first dried in a constant temperature oven at 103°C. Then, to separate the fine grains of the coral sand from the sand grains, the coral sand was sifted using a standard sieve with a mesh size of 0.075 mm. This process allowed the separation of two types of coral sand grains based on the 0.075-mm threshold (Zhao et al., 2022). Coral sand grains<0.075 mm were considered as fine grains, while those above this size were considered sand grains. Fine particles have no discernible liquid or plastic limit and are classified as non-plastic particles (ASTM International, 2000). Fine and sand grains of different masses were uniformly mixed to obtain coral sands with different FC. The gradation curves of pure sand, pure fine, and coral sand grains with different FCs are shown in Figure 1 . Figure 1 lists the basic physical indicators of the coral sand. The maximum and minimum porosity ratios were measured according to ASTM specifications. To avoid changes in grain size distribution due to breakage during measurement, the minimum porosity ratio was measured using the vibration metho (ASTM D4253, 2016a; and ASTM D4254, 2016b).

A GDS cyclic triaxial apparatus was used for performing stress-controlled drainage cyclic loading tests. This apparatus applies a maximum axial stress of ±5 kN with an accuracy of up to 0.001 kN. The confining pressure chamber can withstand a maximum pressure of 2 MPa. The axial displacement range is 100 mm with a displacement accuracy of 0.07% F.S (full scale). To ensure the uniformity of coral sand with fine grains, samples were prepared using the dry sampling method with a diameter of D = 50 mm and a height of H = 100 mm. According to gradations shown in Figure 1 , the sand required for each specimen was weighed and evenly divided into five layers for filling into film-bearing cylinders. During the filling process, the layer height was strictly controlled, and the surface of each layer was brushed to ensure uniformity. Upon completion of sample preparation, CO₂ was introduced to displace air, followed by presaturation using air-free water. Then, the samples were saturated using the graded back pressure saturation method. When the measured pore pressure coefficient B was ≥0.97, the samples were considered fully saturated. After saturation, an initial effective confining pressure σ _m’ = 100 kPa was uniformly applied for consolidation.

To investigate the development patterns of volumetric strain of coral sands with different D _r, FC, and cyclic stress ratio (CSR) values under cyclic drainage loading conditions, coral sands with FC = 0%, 10%, 20%, and 30% were used to prepare samples with D _r = 30%, 50%, and 70%, respectively, followed by uniform consolidation at σ’ _m = 100 kPa. After consolidation, drainage cyclic loading tests with CSR = 0.20, 0.25, and 0.30, were conducted. The detailed test scheme is shown in Table 1 , with CSR:

where σ _d is the cyclic deviatoric stress. The test was performed using sine waves with a loading frequency of 0.1 Hz.

In this study, the dynamic elastic modulus E _d-1 at the first cycle N = 1 is defined as the initial dynamic elastic modulus. Figure 4 shows the relationship curve between the initial dynamic elastic modulus E _d-1 and FC. With FC ranging from 0 to 30%, the initial dynamic elastic modulus decreases with increasing FC and CSR and increases with increasing D _r. Thus, the magnitude of the initial dynamic elastic modulus is affected by FC, D _r, and CSR. This is because for D _r, the number of sand grains that make up the soil skeleton decreases gradually with the increase of FC and some of them are replaced by fine grains, thus decreasing the stiffness of the sample in the initial state and E _d-1. With the increase of D _r, the degree of compaction of the sample increases and the contact between grains is closer, thus increasing the stiffness of the sample and E _d-1. The above analysis indicates that the impacts of the composition of grains on E _d-1 cannot be reasonably characterized by FC or D _r alone. Thevanayagam (2000) proposed the equivalent skeleton void ratio e _sk ^* considering influence of fines on the contact of sand particles and the interparticle contact force chain. The formula is as follows:

where e is void ratio, b is the proportion of fines involved in the force chain between soil particles on the premise that FC is smaller than the threshold fines content FC _th (FC _th = 30% for the coral sand in this study). The parameter b is calculated using the semi-empirical formula proposed by Rahman et al. (2008) and Mohammadi and Qadimi (2015), and its validity has been verified by Chen et al. (2020).

Therefore, the equivalent skeleton void ratio e _sk ^* is used to characterize E _d−1. It can be found that E _d−1 decreases with the increase of e _sk ^*, indicating a clear relationship between E _d-1 and e _sk ^*; e _sk ^* is an appropriate physical indicator for characterizing E _d−1. As shown in Figure 5 , a clear linear relationship between the CSR-corrected initial dynamic elastic modulus and equivalent porosity ratio of the skeleton is established:

When the volumetric strain (Δϵ _v) development plateaus during drained cyclic loading (Δϵ _v/ΔN< 0.01), the specimen is considered to have reached steady state. The E _d−N in this state is defined as the steady-state dynamic elastic modulus E _d-s. The variation in the steady-state dynamic elastic modulus E _d-s is affected by D_r and FC, as shown in Figure 6 . According to the experimental results, E _d-s decreases with increasing D_r , and increases linearly with decreasing FC. In addition, the experimental results show that E _d-s increases slightly with increasing CSR. This is consistent with the finding of Ma et al. (2024b) that elastic modulus increases slightly with increasing strain rate.

When E _d-N reaches E _d-s at a given D _r, it indicates that the sample has reached the maximum degree of compaction under the applied stress. When FC< FC_th, fine grains fill in between the coarse particle skeleton or wrapped around the surface of the coarse sand, and their lubricating effect reduces the stiffness of the soil. This observation is consistent with the findings Huang et al. (2023) obtained using the three-dimensional discrete element method. The greater the FC, the more the fine grains in the soil skeleton, thus remarkably reducing the stiffness. An increase in the initial D _r would increase the degree of compaction of the consolidated sample, which strengthens the contact interaction between soil grains and manifests itself as an increase in the dynamic elastic modulus.

The equivalent skeleton void ratio e _sk ^* is introduced to characterize the combined impact D _r and FC. As shown in Figure 7 , a clear linear correlation exists between D_r and FC, which allows us to conclude that

The ratio of E _d-N to E _d-s is defined as the relative dynamic elastic modulus E _r (Equation 6). E _r represents the stability development of the coral sand under cyclic drainage loading and is of guidance for the foundation design under long-term loading.

Figure 8 shows the development curve of the relative dynamic elastic modulus E _r with the number of cycles N. It can be seen that the degrees of discreteness of E _r–N curves at different FCs and the same D _r are extremely small, indicating that FC has almost no impact on the development of E _r with N.E _r eliminates the impact of various FC, and during cyclic loading, E _r develops with N in the form of an approximate inverse tangent function.

Equation 7 was used to analyze the development of E _r, namely, parameters A and B. A mathematical analysis indicated that parameter A is related to the final value of the relative dynamic elastic modulus E _r, while parameter B is a variable related to the growth rate of E _r. According to the A–B relationship shown in Figure 9 , parameter A is related to the soil, and remains constant at 0.63 for the saturated coral sand used in this study, unaffected by FC, CSR, or D _r. The growth rate of the E _r is affected by the combined impact of FC, CSR, and D_r. When the impact of FC is eliminated, changes in D _r will substantially affect the fitting parameter B. The value of B shows a linear increase with increasing D _r. This is because an increase in the D _r would increase the degree of compaction of the loaded sample, which strengthens the contact interaction between soil grains and manifests itself as an increase in the dynamic elastic modulus.

Parameter B is corrected by CSR, and in this study, the density correction parameter D _r ^CSR is used in place of the relative density D _r. As shown in Figure 10 , there is a clear linear relationship between the two.

Thus, a prediction model for the dynamic elastic modulus of saturated coral sands can be developed as follows: