Development patterns of the dynamic elastic modulus of saturated coral sand under different drainage conditions

Long-term cyclic loading can have a significant effect on the modulus of sand, and the influence on saturated coral sand has yet to be established. In this paper, the significant influence of non-plastic fines content (FC) and relative density (D_r) on dynamic elastic modulus (E) of saturated coral sand has been evaluated by a series of cyclic triaxial drainage tests. The results show that the dynamic elastic modulus increases rapidly at the beginning of loading; then the growth slows down and finally stabilizes. In general, the development of E is influenced collectively by FC, D_r and cyclic stress ratio (CSR). The initial dynamic elastic modulus E_d-1 and steady-state dynamic elastic modulus E_d-s increase with the increase of D_r, and decrease as FC increases. The linear fitting equations are given by introducing the equivalent skeleton void ratio e_sk*. Furthermore, the relative dynamic elastic modulus E_r is defined as the ratio of E_d-N to E_d-s, and the prediction equation for E_r was developed to provide a basis for the engineering mechanical parameters of coral sands under long-term loads.

1 Introduction

The carbonate sand formed by marine biological deposition, whose CaCO₃ content is >90%, is referred to as coral sand. Coral sand is abundant in nature (e.g. in the Persian Gulf and the South China Sea) and is often used as the primary geotechnical material for constructing infrastructures such as coral islands and reefs or harbors (Ding et al., 2021). Coral islands and reefs or harbors are located in marine environments that are susceptible to dynamic loads from long-term waves, storm surges, tsunamis, and earthquakes. Differential subsidence or permanent deformation caused by these marine dynamic loads can have a remarkable impact on the foundations of coral sand (Wen et al., 2021). During analysis of soil dynamics problems, the dynamic elastic modulus of the soil is considered a key parameter for studying the soil’s ability to resist deformation (Seed and Idriss, 1970; Rollins et al., 1998, 2020; Chen et al., 2024; Ma et al., 2024a). This modulus effectively reveals the complex nonlinear relationship between dynamic stress and strain in soils, thereby providing an important theoretical basis for investigating the dynamic properties of soils. The dynamic elastic modulus of a sandy soil is an important parameter when describing its deformation features under dynamic loading. In the fields of earthquake, foundation, and geotechnical engineering, the dynamic elastic modulus is one of the key indicators for assessing the response of a sandy soil under seismic or other dynamic loads.

There are many researches on the dynamic deformation characteristics of soils in different regions, but most of the researches devoted to clay and granular soils (Iwasaki et al., 1978; Kallioglou et al., 2008; Vucetic and Mortezaie, 2015; Kantesaria and Sachan, 2021; Nong. and Park, 2021), studies on the dynamic elastic modulus of coral sand are rather limited. Long et al. (2022) investigated the impact of coral sand grain breakage on its dynamic properties. The result of resonant column tests indicated that the maximum dynamic shear modulus (G_max) of the coral sand first increased and then decreased with the increase in the degree of coral sand grain breakage. Furthermore, Wu et al. (2022) conducted a series of resonant column tests on coral sand with different fines content (FC) and found that the shear modulus ratio G/G_max - shear strain γ curves show FC-dependent development and are characterized by nonlinear enhancement with increasing FC. Catano and Pando (2012) investigated the CaboRojo coral sand and found the shear modulus G value of coral sand was smaller than that of quartz sand. Alternatively, Jafarian and Javdanian (2020) investigated the Bushehr coral sand using the resonant column technique and found that the relative density had a minor impact on the dynamic shear modulus ratio. While, the decay of G/G_max gradually slows down as the effective confining pressure increases. Liang et al. (2023) studied the small-strain shear modulus G₀ of coral sands from Nansha and Xisha Islands. A G₀ prediction model for different types and gradations of sandy soils is proposed by introducing the concept of critical porosity ratio. Wichtmann et al. (2015) focused on the impacts of FC on G_max of quartz sands. At the same porosity ratio e, when FC< 10%, G_max decreases substantially with increasing FC; when FC = 10%–20%, G_max increases with increasing FC; however, when FC is >20%, G_max does not change significantly with FC. These findings provide an important basis for a deeper understanding of the impacts of FC on the mechanical properties of quartz sands.

Previous studies have investigated the development of G of sands within small strain conditions using resonant column tests. In this study, the drained cyclic triaxial tests have been conducted to study the influence of FC and relative density (D_r) on the initial dynamic elastic modulus (E_d-1) and steady-state dynamic elastic modulus (E_d-s). Based on the experimental results, a prediction equation for the relative elasticity modulus ratio (E_r) is given, so as to provide a basis for the engineering mechanical parameters of coral sands with non-plastic fines under long-term loads.

2 Test materials and methods

2.1 Test materials

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).

Figure 1

Figure 1. Gradation curve of coral sand.

2.2 Test apparatus and sample preparation

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.

2.3 Test scheme

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:

$\begin{array}{ll}CSR=\frac{{\sigma }_{\text{d}}}{2{\sigma }_{\text{m}}^{\text{'}}}& (1)\end{array}$

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

Table 1

Table 1. Test scheme.

3 Test results and analysis

There is relatively limited documentation on how to use a dynamic triaxial apparatus to test the coral sand from the South China Sea under drainage cyclic loading conditions to obtain its dynamic elastic modulus. Hence, the analysis of the dynamic modulus of the soil is crucial, which is defined as:

$\begin{array}{ll}{E}_{\text{d}}\text{ = }\frac{{\sigma }_{\text{d}}}{{ϵ}_{\text{d}}}& \text{(2a)}\end{array}$

where, σ_d and ϵ_d are axial dynamic stress and strain, respectively.

Under cyclic loading, the contact force between saturated coral sand grains gradually increases with increasing soil density, thereby increasing the dynamic elastic modulus E_d. When the dynamic elastic modulus stabilizes, the contact force between grains remains the same and E_d reaches a steady state. The average dynamic elastic modulus of saturated coral sand at the N-th cycle id denoted as E_d-N. Figure 2 illustrates the calculation of E_d-N using Equation 2b.

Figure 2

Figure 2. Schematic calculation of E_d-N..

$\begin{array}{ll}{E}_{\text{d-N}}=\frac{{\sigma }_{\text{d},\max }-{\sigma }_{\text{d,}\min }}{{ϵ}_{\text{d},1}-{ϵ}_{\text{d},2}}& \text{(2b)}\end{array}$

Where σ_d,max and σ_d,min are the maximum and minimum cyclic deviatoric stresses at the N^th cycle, and ϵ_d,1 and ϵ_d,2 are the axial strains corresponding to σ_d,max and σ_d,min, respectively. Figure 3 shows the curve of dynamic elastic modulus of the saturated coral sand E_d-N versus N. As shown in this figure, E_d-N accumulates continuously with increasing N. It increases rapidly at the beginning of loading; then, its growth rate slows down until it stabilizes.During loading, the dynamic elastic modulus of the saturated coral sand increases with the increase in the relative density D_r; the growth rate of E_d-N decreases with increasing D_r and increases with increasing FC and CSR.

Figure 3

Figure 3. Development of dynamic elastic modulus E_d-N of saturated coral sand with N. (A) D_r = 30%, (B) D_r = 50%, (C) D_r = 70%.

3.1 Analysis of the initial dynamic elastic modulus

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:

Figure 4

Figure 4. Curve of initial dynamic elastic modulus E_d-1 versus fines content FC.

$\begin{array}{ll}{e}_{\text{sk}}^{\ast }=\frac{e+(1-b)\times FC}{1-(1-b)\times FC}& (3)\end{array}$

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:

Figure 5

Figure 5. Initial dynamic elastic modulus E_d-1 vs. e_sk^*.

$\begin{array}{ll}{E}_{d-1}\times CS{R}^2=-4.3e_{sk}^{*}+6.67& (4)\end{array}$

3.2 Analysis of the steady-state dynamic elastic modulus

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.

Figure 6

Figure 6. Curve of steady-state dynamic elastic modulus E_d-s versus fines content FC.

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

Figure 7

Figure 7. Curve of steady-state dynamic elastic modulus E_d-s vs. e_sk^*.

$\begin{array}{ll}{E}_{\text{d-s}}=-70.3e_{\text{sk}}^{*}+135.6& (5)\end{array}$

3.3 Prediction modeling for the relative dynamic elastic modulus

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.

$\begin{array}{ll}{E}_r=\frac{E_{\text{d}-\text{N}}}{E_{\text{d}-\text{s}}}& (6)\end{array}$

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.

Figure 8

Figure 8. Relationships between the relative dynamic elastic modulus E _r and N. (A) D_r = 30%, (B) D_r = 50%, (C) D_r = 70%.

$\begin{array}{ll}{E}_r=A\arctan [B\times (N+1)]& (7)\end{array}$

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.

Figure 9

Figure 9. Relationships between parameters A, B and relative density D_r.

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.

Figure 10

Figure 10. Relationship between parameter B and D_r..

$\begin{array}{ll}B\times \text{CSR }= 0.036+0.16D_{\text{r}}{}^{\text{CSR}}& (8)\end{array}$

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

$\begin{array}{ll}{E}_r=\frac{2}{\pi }\arctan [\frac{0.036+0.16D_{\text{r}}{}^{\text{CSR}}}{\text{CSR}}\times (N+1)]& (9)\end{array}$

4 Conclusions

In this study, we investigated the impacts of D_r, FC, and CSR on the dynamic elastic modulus E_d-N of saturated coral sands with non-plastic fines and its development patterns, and drew the following conclusions:

(1) E_d-N of saturated coral sands keeps accumulating with increasing N during cyclic loading, and it increases with increasing D_r and decreases with increasing FC during the loading process.

(2) The initial dynamic elastic modulus E_d-1 is affected by FC, D_r, and CSR. Alternatively, the steady-state dynamic elastic modulus E_d-s is affected by the combined impact of FC and D_r and to a lesser extent by the CSR. Linear relationships between E_d-1 and e_sk^* and E_d-s and e_sk^* for FC< FC_th were established respectively by introducing the equivalent skeleton void ratio e_sk^*.

(3) The relative dynamic elastic modulus E_r is defined as the ratio of E_d-N to E_d-s. E_r is not affected by the FC, but increases with increasing N. A prediction model for E_r in the form of an inverse tangent function was developed, and its fitting parameters A and B were analyzed.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material. Further inquiries can be directed to the corresponding author.

Author contributions

RZ: Writing – original draft. ZH: Formal Analysis, Writing – review & editing. QL: Data curation, Investigation, Writing – original draft. QY: Project administration, Supervision, Validation, Writing – review & editing. QW: Writing – review & editing, Methodology.

Funding

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This work is supported by the National Natural Science Foundation of China under grant no 52378346.

Conflict of interest

Author HZ was employed by China Nuclear Power Engineering co.,Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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