Experimental Study of the Dynamic Shear Modulus of Saturated Coral Sand under Complex Consolidation Conditions

Abstract

The shear modulus is an essential parameter that reflects the mechanical properties of the soil. However, little is known about the shear modulus of coral sand, especially under complex consolidation conditions. In this paper, we present the results of a multi-stage strain-controlled undrained cyclic shear test on saturated coral sand. The influences of several consolidation state parameters: effective mean principal stress ($p_0^{\prime }$), consolidation ratio (k_c), consolidation direction angle (α₀), and coefficient of intermediate principal stress (b) on the maximum shear modulus (G₀), the reference shear strain (γ_r) and the reduction of shear modulus (G) have been investigated. For a specified shear strain level, G will increase with increasing $p_0^{\prime }$ and k_c, but decrease with increasing α₀ and b. However, the difference between G for various α₀ and b can be reduced by the increase of shear strain amplitude (γ_a). G₀ shows an increasing trend with the increase of $p_0^{\prime }$ and k_c; on the contrary, with the increase of α₀ and b, G₀ shows a decreasing trend. To quantify the effect of consolidation state parameters on G₀, a new index (μ_G0) with four parameters (λ₁, λ₂, λ₃, λ₄) which is related to $p_0^{\prime }$, k_c, α₀, b is proposed to modify the prediction model of G₀ in literature. Similarly, the values of γ_r under different consolidation conditions are also evaluated comprehensively by the four parameters, and the related index (μ_γr) is used to predict γ_r for various consolidation state parameters. A new finding is that there is an identical relationship between normalized shear modulus G/G₀ and normalized shear strain γ_a/γ_r for various consolidation state parameters and the Davidenkov model can describe the G/G₀–γ_a/γ_r curves. By using the prediction model proposed in this paper, an excellent prediction of G can be obtained and the deviation between measured and predicted G is all within ±10%.

1. Introduction

Carbonate soils, most commonly calcium carbonate, are usually divided into three categories based on the calcium carbonate (CaCO₃) content: calcareous sand (<50%), siliceous carbonate sand (50–90%), and carbonate sand (>90%) [1]. Carbonate sand, which originates mainly from coral reefs, is referred to as coral sand. Coral sand is mostly distributed in the tropical ocean [2,3] and is the common reclamation material in ocean engineering [4,5,6]. Due to the special formation progress of coral sand, it has the particles features of intraparticle pores, high friction angle, high compressibility, low hardness, and fragileness at high-stress levels [3,7,8,9,10,11], which makes the mechanical properties of coral sand quite different from those of terrigenous sand.

The marine environment is extremely complex including the complexity of coral sand deposition and the dynamic loadings on a coral sand foundation. Waves, tsunamis, and earthquakes can threaten the stability of the coral sand foundation, and then affect the safety of ocean engineering. So it is vital to explore the undrained dynamic response of saturated coral sand. The maximum shear modulus G₀ and the reduction of shear modulus (G/G₀) from small to large strain range are the essential parameters in seismic response analysis studies. Many researchers have studied the dynamic shear modulus of silica sand. The maximum shear modulus G₀ is referred to as the shear modulus at small strain (γ_a < 10⁻⁶), which is a key parameter in the prediction of G. The previous researches reveal that the value of G₀ is related to the void ratio e and the effective confining pressure ${\sigma }_0^{\prime }$ [12,13,14,15,16]. The void ratio reflects the influence of soil property on G₀, and the effective confining pressure characterizes the effect of consolidation condition on G₀. Hara et al. [17] obtained the G₀ of cohesive soils by using shear wave velocity V_s in laboratory and in situ tests (G₀ = ρV_s²), and a unique correlation is established between G₀ and shear strength S_u. Hardin and Kalinski [18] investigate the shear modulus of gravelly soils including uniform and graded crushed limestone gravel, graded river gravel, standard Ottawa and crushed limestone sands, and gravel–sand–silt mixtures. The modified three-dimensional constitutive equations for the elasticity of particulate materials are used to evaluate G₀ for various types of soils. Goudarzy et al. [19,20] explored the effect of the non-plastic fines content of granular on the maximum shear modulus. The G₀ of clean Hostun sand and Hostun sand mixed with 5%, 10%, 20%, 30%, and 40% fines are analyzed. To evaluate the influence of fine content on G₀, the equivalent void ratio (e*) is introduced to replace e in the Hardin model. The modified model has a good prediction for the Hostun sand. Yan et al. [21] investigated the small-strain shear modulus of unsaturated silty-fine sand. They find that there exists an optimum saturation (S_r)_opt in silty-fine sand. G₀ reaches its maximum value when the silty-fine sand is in optimum saturation and an improved prediction model for G₀ is proposed for sand under different saturation degrees.

Iwasaki et al. [22] evaluated the degree of shear modulus reduction by using two types of equipment. The test results show that the shear modulus at 10⁻⁴ shear strain amplitude measured by different equipment is identical, and a simplified procedure for prediction G is proposed. Lanzo et al. [23] reported the test results of two reconstituted grains of sand and three laboratory-made clays. The results indicated that with the increase of ${\sigma }_0^{\prime }$ and the over-consolidation ratio (OCR), the normalized shear modulus reduction curve (G/G₀–γ_a) generally increases. Furthermore, with the increase of plasticity index (PI), the effect of ${\sigma }_0^{\prime }$, and OCR on shear modulus reduction become small. Senerakis et al. [24] found that the G/G₀–γ_a of volcanic sands were more linear than those of quartz sands. This trend was more pronounced at lower ${\sigma }_0^{\prime }$ and at higher values of the coefficient of uniformity (C_u). The parameters of ${\sigma }_0^{\prime }$ and C_u may not be used in the modified hyperbolic model to quantify the rate of modulus reduction, and it is probably due to the unique fabric of volcanic sands. Chen et al. [25] investigated the characteristics of shear modulus reduction under isotropic and anisotropic consolidations for silt. The correlations between G/G₀ and γ_a for various complex stress conditions are distinct. However, the correlation between G/G₀ and γ_a/γ_r is identical regardless of the consolidation state parameters.

In recent years, with the rise of ocean engineering, the mechanical properties of coral sand become the research focus in marine geotechnical engineering. Controlling the quality of hydraulic filling materials is the key to ensuring the construction safety of marine structures [6]. For soil mass, the shear strength is always the most critical mechanical parameter. Many studies have confirmed that the particle size and confining pressure can affect the shear strength of coral sand [26,27]. However, these researchers focused on the shear characteristics of soils under monotonic shear. Since large-scale deformation of saturated coral sand has been observed in the 1993 Mw 7.7 Guam earthquake [28], the 2006 Mw 6.7 and Mw 6.0 Hawaii earthquakes [29], and the 2010 Mw 7.0 Haiti earthquake [30], the dynamic characteristics of coral sand foundation also need further discuss, and the dynamic shear modulus of coral sand is the most essential, which can establish the constitutive relation and guide the construction of offshore engineering. In fact, engineers have long been concerned about the maximum shear modulus together with the dynamic shear modulus reduction of coral sand foundations. Giang et al. [31] found that particle shapes could affect G₀. Due to the unique particle characteristics of coral sand, the G₀ of coral sand is much higher than that of silica sand. The previous prediction model will underestimate or overestimate the G₀ of coral sand. Chen et al. [32] carried out different types of strain-controlled undrained cyclic triaxial tests, and a correlation-function-based method is proposed to calculate the secant shear modulus, which shows a good prediction of shear modulus attenuation. Wu et al. [5] studied the dynamic shear modulus of saturated marine coral sand with different fines contents (FC) and relative densities (D_r). To quantify the influence of both parameters mentioned above, the equivalent skeleton void ratio e*_sk is introduced, and a unique correlation can be observed between G₀, ${\sigma }_0^{\prime }$, and e*_sk. However, most of these studies focused on the effect of soil properties such as confining pressure, relative density, void ratio, and coefficient of uniformity on the shear modulus of coral sand. The mechanism of the effect of consolidation state parameters on the shear modulus reduction is still unclear.

In this paper, we present a comprehensive experimental study of the maximum shear modulus and the reduction of the shear modulus with increasing shear strain. The influence of effective mean principal stress ($p_0^{\prime }$), consolidation ratio (k_c), consolidation direction angle (α₀), and coefficient of intermediate principal stress (b) on the shear modulus of coral sand is systematically investigated. The cyclic deformation characteristics of the specimens are also analyzed. The cyclic deformation properties of the specimens were also analyzed. The conclusions drawn in this study can lead to a better understanding of the undrained cyclic behavior of saturated coral sand and provide some key parameters for the analysis of the seismic response of coral sand foundations.

2. Test Materials and Apparatus

2.1. Test Material

The tested coral sand material is sourced from the Nansha Islands, South China Sea. Figure 1 shows the particle distribution curve, photograph, and scanning electron microscope (SEM) picture of tested coral sand. The particle shape of the tested coral sand varies from long strips to round, and some coral and marine fragments can also be observed. From a micro perspective (the SEM picture), the tested coral sand particles have rough surfaces and are attached to fine particles. The specific gravity (G_s) of coral sand is 2.80, maximum and minimum void ratio (e_max, e_min) is 1.72 and 0.99 respectively, according to the ASTM standards of D4253-14 and D4254-14 [33,34]. Since the maximum particle size used in the specimen cannot be greater than 1/6 to 1/7 of the specimen diameter, the maximum particle size of coral sand used in this test is 2 mm. For the particle distribution curve used in this test, the mean particle size (d₅₀) is 0.31 mm, coefficient of uniformity (C_u) and curvature coefficient (C_c) is 4.67 and 0.86 respectively. Following the Unified Soil Classification System [35], the tested coral sand is classified as poorly graded sand (SP).

2.2. Test Apparatus and Stress Distribution

An advanced hollow cylinder torsional shear apparatus manufactured by GDS Instruments (Hook, Hampshire, United Kingdom) is used in this study [Figure 2a]. The instrument is composed of four parts: pressure chamber, host, pressure controller, and data acquisition system. This apparatus can dynamically and independently control the axial load (W), torque (M_T), outer cell pressure (p_o), and inner cell pressure (p_i). The distributions of these four cyclic loadings on the specimen are shown in Figure 2b.

Table 1. Performance indexes of the apparatus used in this study.

Controller	Capacity	Deviation	Precision
Axial load	3 kN	0.1% F.S	3 N
Axial displacement	±40 mm	0.1% F.S	1 μm
Torque	30 Nm	0.1% F.S	0.03 Nm
Rotation displacement	360°	0.057% F.S	0.04°
Axial/Rotation frequency	≤5 Hz	-	-
Outer/inner cell pressure	1 MPa	0.1% F.S	1 kPa
Outer/inner cell volume	200 mL	0.25% F.S	0.001 mL
Back pressure	1 MPa	0.1% F.S	1 kPa
Back volume	200 mL	0.1% F.S	0.001 mL
Pore pressure	1 MPa	0.1% F.S	1 kPa

F.S = full scale.

gives the performance parameters of the GDS apparatus including the capacity, deviation, and precision of the sensors. The four corresponding stress components (vertical stress σ_z, radial stress σ_r, circumferential stress σ_θ, and torsional shear stress τ_zθ) and three principal stresses (σ₁, σ₂, σ₃) are illustrated in Figure 2c,d.

Table 2. Equations for data interpretation.

Stress Component	Principal Stress	Stress Characteristic Parameter
Vertical ${\sigma }_{\mathrm{z}}=\frac{W}{\pi (r_{\mathrm{o}}{}^2-r_{\mathrm{i}}{}^2)}+\frac{p_{\mathrm{o}}{r}_{\mathrm{o}}{}^2-p_{\mathrm{i}}{r}_{\mathrm{i}}{}^2}{(r_{\mathrm{o}}{}^2-r_{\mathrm{i}}{}^2)}$	Major ${\sigma }_1=\frac{{\sigma }_{\mathrm{z}}+{\sigma }_{\mathsf{\theta }}}{2}+\sqrt{{\left(\frac{{\sigma }_{\mathrm{z}}-{\sigma }_{\mathsf{\theta }}}{2}\right)}^2+{\tau }_{{}_{\mathrm{z}\mathsf{\theta }}}^2}$	Mean principal stress $p=\frac{{\sigma }_1+{\sigma }_2+{\sigma }_3}{3}$
Radial ${\sigma }_{\mathrm{r}}=\frac{p_{\mathrm{o}}{r}_{\mathrm{o}}+p_{\mathrm{i}}{r}_{\mathrm{i}}}{r_{\mathrm{o}}+r_{\mathrm{i}}}$	Intermediate principal ${\sigma }_2={\sigma }_{\mathrm{r}}$	Ratio of major and minor principal stress $k_{\mathrm{c}}={\sigma }_1/{\sigma }_3$
Circumferential ${\sigma }_{\mathsf{\theta }}=\frac{p_{\mathrm{o}}{r}_{\mathrm{o}}-p_{\mathrm{i}}{r}_{\mathrm{i}}}{r_{\mathrm{o}}-r_{\mathrm{i}}}$	Minor principal ${\sigma }_3=\frac{{\sigma }_{\mathrm{z}}+{\sigma }_{\mathsf{\theta }}}{2}-\sqrt{{\left(\frac{{\sigma }_{\mathrm{z}}-{\sigma }_{\mathsf{\theta }}}{2}\right)}^2+{\tau }_{{}_{\mathrm{z}\mathsf{\theta }}}^2}$	Coefficient of intermediate principal stress $b=\left({\sigma }_2-{\sigma }_3\right)/\left({\sigma }_1-{\sigma }_3\right)$
Shear ${\tau }_{\mathrm{z}\mathsf{\theta }}=\frac{3M_{\mathrm{T}}}{2\pi (r_{\mathrm{o}}{}^3-r_{\mathrm{i}}{}^3)}$		Direction angle of principal stress $\alpha =\frac{1}{2}\arctan \left(\frac{2{\tau }_{\mathrm{z}\mathsf{\theta }}}{{\sigma }_z-{\sigma }_{\mathsf{\theta }}}\right)$

gives the equations of data interpretation for all consolidation loadings, the stress components, and the consolidation state parameters. More details about this apparatus can be found in the literature [25,36,37].

3. Specimen Preparation, Saturation, and Consolidation

The hollow cylinder specimen in this study has a height of 200 mm and an inner and outer diameter of 60 mm and 100 mm. The specimen preparation method can significantly affect the mechanical properties of soil [38,39]. To ensure the uniformity of each specimen, the dry deposition method was adopted for preparing the coral sand specimens. The specimens were prepared by pouring the dried sand in seven layers into the hollow space between two molds via a funnel with a near-zero falling head. To investigate the influence of consolidation characteristic parameters on the dynamic shear modulus, the target initial relative density (D_r) of each specimen is 45%, and the corresponding void ratio is 1.355. The actual relative density after consolidation (D_rc) for all specimens ranged from 49.18% to 52.88% (see

Table 3. Scheme of multi-stage strain-controlled undrained cyclic shear tests.

Test No.	End of Consolidation				Soil Property		Test Result		Fitting Parameter
	${\mathit{p}}_0^{\prime }$ (kPa)	kc	αc/(°)	b	Drc (%)	e	G0 (MPa)	γr (×10−4)	C1	C2
01	50	1.0	-	-	49.18	1.361	51.6	5.464	0.46	0.92
02	100				49.45	1.359	68.1	6.056	0.48	0.98
03	200				51.64	1.343	101.1	11.844	0.48	1.02
04	300				52.47	1.337	121.7	15.264	0.48	1.01
05	100	1.5	0	0	50.96	1.348	88.1	8.758	0.52	0.98
06		2.0			51.51	1.344	91.2	12.671	0.52	1.00
07		2.5			52.88	1.334	93.7	14.613	0.52	1.01
08	100	1.5	30		51.37	1.345	82.3	9.497	0.49	1.01
09			45		51.78	1.342	75.6	9.731	0.52	1.04
10			60		51.64	1.343	74.3	10.396	0.50	1.04
11			90		51.51	1.344	66.3	11.650	0.50	1.00
12	100	1.5	0	0.25	51.78	1.342	84.2	9.315	0.47	1.02
13				0.50	51.37	1.345	81.8	9.913	0.52	0.98
14				0.75	51.64	1.343	77.5	10.582	0.49	1.05
15				1.00	51.37	1.345	75.2	11.874	0.50	0.96

), and the corresponding void ratio ranged from 1.334 to 1.361. Note that the difference in the relative density before and after the consolidation is about 5%. It is probably due to the high compressibility of coral sand.

After the specimen preparation was complete, the specimen was carefully set in the pressure chamber. The combination method of flushing CO₂ and de-aired water was used to make the specimen attain preliminary saturation, then applying back pressure (p_b) to 400 kPa in three steps (end of step I: p_b = 100 kPa, p_o = p_i = 110 kPa; end of step II: p_b = 200 kPa, p_o = p_i = 210 kPa; end of step III: p_b = 400 kPa, p_o = p_i = 410 kPa); until the Skempton’s B-value ≥ 0.97, which is a criterion indicating the specimen is completely saturated. The effective stress of the specimen always remains at 10 kPa during the saturation. After saturation, the specimens were firstly isotropically consolidated under different effective mean principal stress $p_0^{\prime }$ (50, 100, 200, 300 kPa), and then anisotropically consolidated according to the scheme under various consolidation ratios of major and minor principal stress k_c (1.0, 1.5, 2.0, 2.5), direction angle of principal stress α₀ (0°, 30°, 45°, 60°, 90°) and coefficient of intermediate principal stress b (0, 0.25, 0.5, 0.75, 1.0). The whole consolidation process lasts for several hours. When the strain components and back pressure volume are stable, the coral sand specimen can be considered wholly consolidated. The consolidation stress paths are shown in Figure 3. The values of W, M_T, p_o, and p_i for each consolidation condition are illustrated in Table 3. The saturation and consolidation for all the tested specimens were performed following the procedures in the ASTM standard [40].

4. Multi-Stage Strain-Controlled Undrained Cyclic Shear Tests

It should be noted that under high-frequency seismic load, the saturated sand will generate excess pore water pressure, which will lead to the reduction of effective stress and the destruction of the foundation. In order to truly simulate the dynamic response of saturated coral sand under seismic load in the marine environment, it is more reasonable to choose the undrained test in this study. For each consolidation condition, multi-stage strain-controlled undrained cyclic torsional shear tests were performed. Table 3 lists the detailed procedure for the strain-controlled undrained cyclic shear test. 17 to 20 stages of sinusoidal cyclic strain were applied to each specimen, with ten cycles per stage. ASTM D5311/D5311M standard recommended that a range of cyclic loading frequency f = 0.1–2.0 Hz, and the frequency of 1.0 Hz is preferred [40]. Moreover, many previous studies also used a 1.0 Hz sinusoidal wave as the cyclic loading condition to observe the dynamic response of soil under earthquake events [41,42,43]. So, the cyclic loading frequency of 1.0 Hz was adopted for all tests in this study to simulate the seismic load. The torsional strain amplitudes change from small strain (0.001%) to large strain (0.5%). After each cyclic loading phase, the specimen was reconsolidated to the initial stress state for 10 min. This stage is intended to dissipate the excess pore water pressure (EPWP) at a large shear strain, which could affect the test result of the next stage [44]. During the drained re-consolidation stage, the volume of the specimen decreased, leading to a slight increase in D_r. However, the value of G can be less affected by density for multi-stage cyclic loading, which has been confirmed in literature [41,45]. Efforts have been made during the whole experiment process to ensure the consistency of the specimen to the greatest extent.

5. Test Results and Analysis

5.1. Typical Results of The Strain-Controlled Test

Figure 4 shows the typical test result of strain-controlled undrained cyclic torsional shear (No. 01). As the number of stages increases, the shear strain amplitude (γ_a) and shear stress amplitude (τ_a) also increase slowly, but not significantly. When γ_a reaches 1 × 10⁻⁴, the shear stress has an observable development; when the strain reaches 2.89 × 10⁻⁴, the EPWP (u_e) starts to develop significantly (more than 2 kPa). Note that the test is strain-controlled, τ_a for each cycle stage is constant and can eventually return to the origin. However, the cyclic shear stress may decrease during the cycle stage and produce deviation in the end. Figure 4d(I) depicts the development of the shear stress-strain relationship in the strain-controlled undrained cyclic shear test. With the increase of shear strain magnitude γ_a, the area of the hysteresis loop becomes large, indicating the gradual decrease in shear strength of the coral sand specimen. Figure 4d(II–IV) is the stress-strain relationship corresponding to the shear strain magnitude of 5 × 10⁻⁵, 5 × 10⁻⁴, and 5 × 10⁻³, respectively. The nonlinear of the stress-strain relationship becomes obvious with the increase of γ_a, and the shear stress amplitude gradually decreases at large γ_a. This is because a large strain amplitude causes the EPWP of the coral sand specimen to rise and then increase the flow properties of the soil.

Figure 5 demonstrates the idealized shear stress-strain response of soil under cyclic torsional shear loading. The strain-dependent secant shear modulus is used to study the modulus attenuation of coral sand in this test, which is commonly used in previous studies.

5.2. The Characteristics of Dynamic Shear Modulus under Various Consolidation Conditions

Figure 6 presents the relationship between G and γ_a of saturated isotropically and anisotropically consolidated coral sand for the undrained cyclic torsional shear tests. Clearly, G decreases with the increase of γ_a for all tests, and various consolidation state parameters ($p_0^{\prime }$, k_c, α₀, b) have different effects on shear modulus reduction. Under isotropic consolidation, G increases with increasing $p_0^{\prime }$ for a specified shear strain level, and the rates of shear modulus reduction for various $p_0^{\prime }$ are almost the same. This is consistent with the test result of Kokusho [13] for Toyoura sand and Chen et al. [25] for Nanjing fine sand. However, under anisotropic consolidation, the effect of k_c, α₀, and b on the shear modulus reduction shows different regular. When other parameters ($p_0^{\prime }$, α₀, b) are constant, G increases with the increase of k_c. Note that with the increase of k_c, the difference between G becomes small. This can be assumed that when k_c > 1.0, the void between coral sand particles begin to decrease gradually, which leads to the rearrangement of particles and the increase of friction between particles, making the shear modulus of coral sand increase significantly. However, when k_c continues to increase, the rearrangement between particles becomes more and more difficult, and the static shear stress only increases the friction between particles. As a result, the differences between the shear modulus for the various k_c become small as the k_c continues to increase. Yuan et al. (2005) and Sun et al. (2013) obtained similar test results on silt, silty clay, and sludgy soil. They also proposed a prediction model to describe the trend of G₀ with the increase of k_c. [46,47] On the contrary, when α₀ and b increase, the shear modulus for a specified shear strain level shows a decreasing trend. However, this gap becomes smaller as the shear strain increases. When shear strain is over 6 × 10⁻⁴, there is no obvious difference of G value under various α₀ or b. Figure 6 also illustrates the relationship between G and γ_a of saturated Nanjing fine sand under similar consolidation conditions, which is shown in gray symbols. The physical properties of this sandy soil can be found in Chen (2016). It is clear that the regular shear modulus reduction is quite similar for both sandy soils. However, the development trend of the shear modulus for Nanjing fine sand is significantly lower than that for coral sand under the same k_c, α₀, b. However, for the same $p_0^{\prime }$, the modulus attenuation trend of the two sandy soils is quite close.

5.3. The Maximum Dynamic Shear Modulus under Various Consolidation Conditions

The maximum dynamic shear modulus G₀ is an essential parameter in the undrained dynamic shear response of saturated sandy soils. G₀ is generally considered as the shear modulus at a small strain (less than 10⁻⁶). To investigate the correlation between G₀ and various consolidation state parameters, a total of 15 strain-controlled undrained cyclic torsional shear tests were performed using the GDS hollow cylinder apparatus. Due to the limited testing accuracy of this apparatus, the dynamic shear modulus less than 1 × 10⁻⁵ is hardly measured, and G₀ cannot be obtained directly. However, according to the hyperbolic model proposed by Hardin and Drnevich [48], G₀ can be calculated by

$$G_0=\underset{\gamma \to 0}{\lim }\frac{1}{a_1+b_1\gamma },$$

(1)

where a₁ and b₁ are the linear fitting parameter. Hardin also gives the model to predict the G₀ of sandy soils:

$$G_0=AF\left(e\right){\left(\frac{{\sigma }_0^{\prime }}{P_{\mathrm{a}}}\right)}^n{P}_{\mathrm{a}},$$

(2)

where A and n are the material parameters, ${\sigma }_0^{\prime }$ is effective confining pressure in the triaxial test, P_a is the atmospheric pressure (P_a = 100 kPa in this study), and F(e) is a function related to void ratio (e). As mentioned above, the consolidation state parameters ($p_0^{\prime }$, k_c, α₀, b) have different effects on the dynamic shear modulus, and Equation (2) cannot completely predict G₀ under complex consolidation conditions.

Figure 7 shows the correlations between G₀ and various consolidation parameters. Clearly, with the increase of mean effective principal stress and consolidation ratio, G₀ shows a gradually increasing trend. On contrary, with the increase of consolidation direction angle and coefficient of intermediate principal stress, G₀ shows a decreasing tendency.

To quantify the influence of the above parameters on G₀, a new index denoted as μ_G0 is introduced, which can be expressed as follows:

$${\mu }_{\mathrm{G}0}={\lambda }_1{}^{n_1}{\lambda }_2{}^{n_2}{\lambda }_3{}^{n_3}{\lambda }_4{}^{n_4},$$

(3)

where

$${\lambda }_1=\frac{p_0^{\prime }}{P_{\mathrm{a}}},{\lambda }_2=k_{\mathrm{c}},{\lambda }_3=\cos (\frac{{\alpha }_0}{2}),{\lambda }_4=\frac{1}{2+b}$$

and n₁, n₂, n₃, and n₄ are the weights for various parameters. Figure 8 shows the correlation between the measured G₀ and μ_G0. For the coral sand in this study, when n₁ = 0.527, n₂ = 0.419, n₃ = 0.760, n₄ = 0.220, there is a strong linear relationship between G₀ and μ_G0 under complex consolidation conditions.

Furthermore, F(e) in Equation (2) can be equal to e^a for simplicity according to the literature [14,16]. Thus, the prediction model of G₀ can be rewritten as:

$$G_0=A{e}^{\mathrm{a}}{\mu }_{G0}{P}_{\mathrm{a}},$$

(4)

For the tested coral sand, the appropriate values of A and a are 0.57 and 1.25, respectively. For the isotropic condition, it is difficult to define the value of α₀ and b in hollow cylinder specimens through the calculation equations. Since the σ₂ is always equal to σ₃ in cylinder specimen for the triaxial test, it is reasonable to make α₀ = 0°, and b = 0 under isotropic conditions. When n₂ = n₃ = n₄ = 0 (triaxial test), Equation (4) degenerates to Equation (2). The predicted G₀ for various consolidation conditions by using Equations (3) and (4) are illustrated in Table 3. Figure 9a shows the comparison between the measured and predicted G₀ of saturated coral sand. It is striking to find that all the test data are close to the line of y = x, within ±10% deviation, indicating a good prediction of the proposed model on G₀. The prediction results of Nanjing fine sand are also shown in Figure 9b. When n₁ = 1.0, n₂ = 0.7, n₃ = 1.5, and n₄ = 0.2, an acceptable prediction of G₀ within ±20% deviation can be obtained by using Equation (4).

5.4. The Prediction Model of Dynamic Shear Modulus Reduction

Figure 10 shows the relationships between normalized shear modulus G/G₀ and γ_a of saturated isotropically and anisotropically consolidated coral sand. It seems that the G/G₀ versus γ_a curves shift to the right with the increase of $p_0^{\prime }$ and k_c. However, with the increase of α₀ and b, the G/G₀–γ_a curves have little change. This implies that the feature of dynamic shear modulus reduction can be significantly affected by $p_0^{\prime }$ and k_c, rather than α₀ and b, and the consolidation state parameters of α₀ and b mainly affect the maximum shear modulus of coral sand.

Figure 11 presents all the test data under various consolidation conditions. The boundaries of dynamic shear modulus for various silica grains of sand in literature are also shown in this figure [12,13,49,50]. The distribution of test data is slightly beyond the upper boundary given by Yuan. This indicates the differences in dynamic mechanical properties between coral sand and silica sand.

The reference shear strain γ_r refers to the value of γ_a corresponding to G/G₀ = 0.5, which is a common index to normalize the trend of dynamic shear modulus. Figure 12 plots the normalized shear modulus G/G₀, and the normalized shear strain γ_a/γ_r curves under various complex consolidation conditions. It is clear to find that the G/G₀–γ_a/γ_r curves can fall in a very narrow band, indicating the validity of γ_r in characterizing the reduction of dynamic shear modulus. In traditional triaxial tests, the strain-dependent shear modulus reduction can be described by the Davidenkov model [51], which is expressed as

$$\frac{G}{G_0}=1-{\left[\frac{{\left({\gamma }_{\mathrm{a}}/{\gamma }_{\mathrm{r}}\right)}^{2C_1}}{1+{\left({\gamma }_{\mathrm{a}}/{\gamma }_{\mathrm{r}}\right)}^{2C_1}}\right]}^{C_2},$$

(5)

where C₁ and C₂ are the fitting parameters. The values of C₁ and C₂ for each test are illustrated in Table 3. Since the values of C₁ and C₂ for various consolidation conditions are very close, it is recommended to take the average value of C₁ (=0.5) and C₂ (=1.00) as the fitting value for all cases. Under these circumstances, Equation (5) degenerates into a hyperbolic model.

It can be seen from the above results that the reference shear strain γ_r is an important parameter that evaluates the trend of shear modulus reduction. Actually, many studies have confirmed that γ_r is related to effective confining pressure ${\sigma }_0^{\prime }$ and fine content FC. However, the influences of different consolidation state parameters under complex consolidation conditions on γ_r are rarely investigated. Figure 13 shows the correlation between γ_r and four consolidation state parameters ($p_0^{\prime }$, k_c, α₀, b). Obviously, when the other three parameters remain constant, the values of γ_r become large with the increase of any one of those parameters. The value of γ_r ranges from 5.464 × 10⁻⁴ to 15.264 × 10⁻⁴, and it means the consolidation state parameters affect γ_r to a certain degree. Moreover, compared with α₀ and b, $p_0^{\prime }$, and k_c have a greater impact on γ_r.

Similarly, to quantify the influence of different consolidation state parameters on γ_r, another consolidation index μ_γr is introduced, which can be expressed as follows:

$${\mu }_{\gamma \mathrm{r}}={\left({\lambda }_1{\lambda }_2\right)/\left({\lambda }_3{\lambda }_4\right)}^m,$$

(6)

where λ₁, λ₂, λ₃, and λ₄ have been defined in Equation (3), and m is a calibration parameter that is determined by the correlation between consolidation state parameters and γ_r. Figure 14 shows the correlation between γ_r and μ_γr. When m = 1.0, with the increase of μ₂, γ_r also presents the increasing tendency, and a strong linear relationship can be found in this figure:

$${\gamma }_{\mathrm{r}}=D_1\times {\mu }_{\gamma \mathrm{r}}+D_2,$$

(7)

where D₁ and D₂ are the linear fitting parameters, and D₁ = 2.1807 × 10⁻⁴, D₂ = 2.5403 × 10⁻⁴ for the tested coral sand.

Thus, based on the analysis of the consolidation state parameters, a prediction model of G for an undrained strain-controlled torsional shear test is established by Equations (3)–(7). This new model takes the effective mean principal stress ($p_0^{\prime }$), consolidation ratio (k_c), consolidation direction angle (α₀), and coefficient of intermediate principal stress (b) into consideration. To verify the accuracy of the model, Figure 15a shows the correlation between the measured G in the multi-stage strain-controlled undrained cyclic shear tests and the predicted G calculated from the new model. All data are close to the line of y = x, and the deviations of measured and predicted are all within ±15%, indicating a good prediction of the proposed new model on G. Figure 15b also shows the prediction result of G for the saturated Nanjing Fine sand, when m = −1.0 in Equation (6), a good prediction result can be observed.

This study provides a comprehensive view of how the consolidation state parameters affect the dynamic shear modulus of saturated coral sand. However, the prediction model proposed in this study is still a semi-empirical formula, which is strongly related to the consolidation state parameters. Due to the lack of test data on the shear modulus under complex consolidation conditions, the correlation between fitting parameters in the formulas and physical properties also needs to be further discussed. More test data are welcomed to verify the applicability of the model and determine the physical meaning of the fitting parameters. According to the existing research results, the next step of the research will focus on determining the physical meaning of the fitting parameters and analyzing the stress state of soil elements to propose a more physical model to better predict the dynamic shear modulus of saturated sand under various consolidation conditions.

6. Conclusions

This study presented results for saturated coral sand from multi-stage strain-controlled undrained cyclic shear tests. The influences of the effective mean principal stress ($p_0^{\prime }$), consolidation ratio (k_c), consolidation direction angle (α₀), and coefficient of intermediate principal stress (b) on dynamic shear strain reduction in saturated coral sand are investigated, and the main conclusions are as follows.

• Shear strain modulus G decreases with the increase of γ_a for all tests, and the consolidation state parameters ($p_0^{\prime }$, k_c, α₀, b) have a significant effect on G. For a specified shear strain level, G generally increases with increasing $p_0^{\prime }$ and k_c, but decreases with increasing α₀ and b.
• The consolidation state parameters can affect the maximum shear modulus G₀ severely. Specifically, G₀ has a positive correlation with $p_0^{\prime }$ and k_c, and a negative correlation with α₀ and b. This regulation is consistent with that of G. To further analyze the influence of consolidation state parameters on G₀, a new index (μ₁) that describes the complex consolidation conditions is introduced, and the four parameters λ₁, λ₂, λ₃, λ₄ in the new index are used to quantify the effect of $p_0^{\prime }$, k_c, α₀, b on G₀, respectively. Based on this index, a new model of G₀ is established, and the test data in this study also proves the validity of this new prediction model.
• The reference shear strain γ_r under isotropic and anisotropic consolidations are also varied. Based on the parameters of λ₁, λ₂, λ₃, λ₄, another new index (μ₂) is also proposed, and a strong linear relationship can be observed between γ_r and μ₂. It is delighted to find that the relationships between normalized shear modulus G/G₀ and normalized shear strain γ_a/γ_r are almost identical. The Davidenkov model can be used to describe the G/G₀–γ_a/γ_r curves and for simplicity, the recommended parameters of C₁ and C₂ in this model are 0.50 and 1.00, respectively.
• The prediction model proposed in this paper can well describe the dynamic shear modulus reduction trend of the tested coral sand under isotropic and anisotropic consolidation conditions, and the deviation between measured and predicted G are all within ±10%, indicating the good prediction result of this new model. However, due to the lack of test data, more test data is needed to further confirm the effectiveness of the prediction model.
• The test data in this study can provide important parameters for island reef engineering. Correspondingly, the prediction of dynamic shear moduli with different strains can be a reference for the seismic design of the foundation. Due to the complex consolidation conditions of soils in natural environments, the prediction model of G₀ proposed in this study can be a guideline in engineering practice.