**1. Introduction**

The small-strain stiffness *G*max of marine deposits plays a fundamental role in liquefaction potential assessment, site seismic response analyses, and the design of marine structures (e.g., pipeline, immersed tunnel, caisson foundation) subjected to storm or earthquake loading [1–4]. Generally, *G*max is defined as the stiffness of soil at small-strain level of 10−<sup>6</sup> , where the soil properties are considered to exhibit pure elasticity. Hardin and his co-authors [5–7] conducted comprehensive studies on *G*max of clean, uniform, quartz sands through well-controlled resonant column tests, and these investigations indicated that the global void ratio *e* and initial effective confining pressure σ 0 c0 are considered to be the most important ones among the various factors that may influence *G*max. Similar results were also presented by Seed et al. (1986) [8], Youn et al. (2008) [9], Yang and Gu (2013) [10], and Payan et al. (2016) [1].

While a large number of attempts have been carried out to characterize *G*max for clean sands, systematic studies on silty sand with different fines content (*FC*) are relatively few, despite the fact that naturally deposited sands are not clean, but contain a certain amount of fine particles [11–14]. A systematic study was first implemented by Iwasaki and Tatsuoka (1977) [11] to study the *G*max influence factors of Iruma silty sand. Their results showed that *G*max decreased with increasing *FC*, and at given *e* and σ 0 c0, *G*max exhibited a decreasing trend as uniformity coefficient *C*<sup>u</sup> increasing. The state

parameter of skeleton void ratio *e*sk was introduced by Wichtmann et al. (2015) [12] to uniquely characterize *G*max of silty sand. However, as discussed by Rahman et al. (2008) [13] and Yang and Liu (2016) [14], the application of *e*sk might contribute to underestimation of *G*max at high *FC*. Goudarzy et al. (2017) [15] developed a new *G*max prediction method based on the binary packing state. A series of bender element tests has been conducted on Ottawa sand with *FC* = 5%–20% by Salgado et al. (2000) [16], the test results revealed that *G*max decreases dramatically with the increasing of *FC* at a constant relative density and σ 0 c0. Salgado et al. (2000) [16] introduced a state parameter ψ to estimate *G*max in the framework of critical state soil mechanics by taking account of the stress dependence. However, compared with the Goudarzy et al. (2017) method [15], the introduction of state parameter ψ requires determination of the critical state line, thus complicating the application of this method [16]. state parameter of skeleton void ratio *e*sk was introduced by Wichtmann et al. (2015) [12] to uniquely characterize *G*max of silty sand. However, as discussed by Rahman et al. (2008) [13] and Yang and Liu (2016) [14], the application of *e*sk might contribute to underestimation of *G*max at high *FC*. Goudarzy et al. (2017) [15] developed a new *G*max prediction method based on the binary packing state. A series of bender element tests has been conducted on Ottawa sand with *FC =* 5%–20% by Salgado et al. (2000) [16], the test results revealed that *G*max decreases dramatically with the increasing of *FC* at a constant relative density and c0 . Salgado et al. (2000) [16] introduced a state parameter *ψ* to estimate *G*max in the framework of critical state soil mechanics by taking account of the stress dependence. However, compared with the Goudarzy et al. (2017) method [15], the introduction of state parameter *ψ* requires determination of the critical state line, thus complicating the application of this method [16].

Many natural silty sands contain a significant amount of fines. This is particularly true for marine deposits, which in most cases behave as composite soils. Therefore, study is needed on whether the *G*max prediction method established for clean sand is also applicable to that of marine silty sand. The main purpose of this study is to explore how *FC*, initial effective confining pressure (σ 0 c0), and global void ratio (*e*) affect the *G*max of marine silty sand and whether the *G*max of silty sand can be predicted within the established framework based on clean sand. In addition, the influence of parameters in the Hardin model for *G*max prediction was discussed in a traditional way. In particular, the binary packing state concept [17–19] is implemented to establish the modified Hardin model for evaluation of *G*max of marine silty sand. For this purpose, a series of bender element tests were conducted on marine silty sand with *FC* = 0%~30%. Many natural silty sands contain a significant amount of fines. This is particularly true for marine deposits, which in most cases behave as composite soils. Therefore, study is needed on whether the *G*max prediction method established for clean sand is also applicable to that of marine silty sand. The main purpose of this study is to explore how *FC*, initial effective confining pressure ( c0 ), and global void ratio (*e*) affect the *G*max of marine silty sand and whether the *G*max of silty sand can be predicted within the established framework based on clean sand. In addition, the influence of parameters in the Hardin model for *G*max prediction was discussed in a traditional way. In particular, the binary packing state concept [17–19] is implemented to establish the modified Hardin model for evaluation of *G*max of marine silty sand. For this purpose, a series of bender element tests were conducted on marine silty sand with *FC =* 0%~30%.

#### **2. Materials and Methods 2. Materials and Methods**

#### *2.1. Testing Apparatus 2.1. Testing Apparatus*

*2.2. Tested Materials*

The measurement of shear wave velocity (*V*s) or the associated *G*max was performed using a pair of piezoceramic bender elements (BE) installed in the cell chamber of a dynamic hollow/solid cylinder apparatus (HCA) [20], as shown in Figure 1. For each of the BE tests, a set of sinusoid signals from 1 to 40 kHz, rather than a single signal, was used as the excitation, and the received signals corresponding to these excitation frequencies were examined in whole to better identify the travel time of the shear wave, then, *G*max can be calculated as following [16]. The measurement of shear wave velocity (*V*s) or the associated *G*max was performed using a pair of piezoceramic bender elements (BE) installed in the cell chamber of a dynamic hollow/solid cylinder apparatus (HCA) [20], as shown in Figure 1. For each of the BE tests, a set of sinusoid signals from 1 to 40 kHz, rather than a single signal, was used as the excitation, and the received signals corresponding to these excitation frequencies were examined in whole to better identify the travel time of the shear wave, then, *G*max can be calculated as following [16].

$$G\_{\text{max}} = \rho V\_{\text{s}}^2 \tag{1}$$

**Figure 1.** GCTS HCA-300 dynamic hollow cylinder-TSH testing system and bender element system. **Figure 1.** GCTS HCA-300 dynamic hollow cylinder-TSH testing system and bender element system.

#### *2.2. Tested Materials J. Mar. Sci. Eng.* **2020**, *8*, x FOR PEER REVIEW 3 of 13

Nantong marine sand was used as clean sand and Nantong marine silt with sub-angular particles was used as pure fines to investigate the effects of *FC* on the *G*max of silty sand. Figure 2 shows the grain size distributions and scanning electron microscopy image of clean sand and pure fines, and the material properties are given in Table 1. Although the ASTM D4253 [21] and D4254 [22] test methods for the determination of minimum and maximum void ratios (*e*min and *e*max) are applicable to silty sand with *FC* < 15%, these methods were also used for silty sands with *FC* ≥ 15% in order to provide consistent measurements [23]. The clean sand was mixed with non-plastic Nantong silt (pure fines) corresponding to various *FC* from 0% to 30% by mass. The *e*min and *e*max of the silty sand are shown in Table 2. Nantong marine sand was used as clean sand and Nantong marine silt with sub-angular particles was used as pure fines to investigate the effects of *FC* on the *G*max of silty sand. Figure 2 shows the grain size distributions and scanning electron microscopy image of clean sand and pure fines, and the material properties are given in Table 1. Although the ASTM D4253 [21] and D4254 [22] test methods for the determination of minimum and maximum void ratios (*e*min and *e*max) are applicable to silty sand with *FC* < 15%, these methods were also used for silty sands with *FC* ≥ 15% in order to provide consistent measurements [23]. The clean sand was mixed with non-plastic Nantong silt (pure fines) corresponding to various *FC* from 0% to 30% by mass. The *e*min and *e*max of the silty sand are shown in Table 2.

**Figure 2.** Scanning electron microscopy image and grain size distributions of clean sand, pure fines, and marine silty sand with different fines content: (**a**) grain size distribution; (**b**) scanning electron microscopy image. **Figure 2.** Scanning electron microscopy image and grain size distributions of clean sand, pure fines, and marine silty sand with different fines content: (**a**) grain size distribution; (**b**) scanning electron microscopy image.


**Table 1.** Index properties of clean sand and pure fines. **Table 1.** Index properties of clean sand and pure fines.

min *e* 0.662 0.764 **Table 2.** Physical index of Nantong marine silty sand with different *FC*.


*C*<sup>c</sup> 0.796 0.829 1.453 1.752 *C*<sup>u</sup> 1.646 1.681 2.826 3.201

#### *2.3. Specimen Preparation, Saturation and Consolidation* segregation and enhancing uniformity [24], all specimens of the tested silty sands were prepared by

The bender element tests were conducted on specimens with 100 × 200 mm (diameter × height), and all specimens of the tested silty sands were prepared by the moist tamping method; considering this method can ensure a very wide range of *e* for the specimens and contribute to preventing segregation and enhancing uniformity [24], all specimens of the tested silty sands were prepared by the moist tamping method using an under-compaction procedure. All samples were tested under saturated rather than other conditions, as the former is more practical [14]; in order to saturate the specimen fully, carbon dioxide flushing from bottom to top of the specimen was applied firstly; then, de-aired water flushing followed immediately [19]; finally, back pressure saturation at the back pressure of 400 kPa was used to guarantee Skempton's *B*-value greater than 0.95 [25]. After saturation, all the specimens were isotropically consolidated. the moist tamping method using an under-compaction procedure. All samples were tested under saturated rather than other conditions, as the former is more practical [14]; in order to saturate the specimen fully, carbon dioxide flushing from bottom to top of the specimen was applied firstly; then, de-aired water flushing followed immediately [19]; finally, back pressure saturation at the back pressure of 400 kPa was used to guarantee Skempton's *B*-value greater than 0.95 [25]. After saturation, all the specimens were isotropically consolidated. *2.4. Testing Program and Process* For the bender element tests, the 10 kHz excitation signal was found to consistently yield a clear arrival of the shear wave for both clean sand and silty sand with various *FC*, which is consistent with

*J. Mar. Sci. Eng.* **2020**, *8*, x FOR PEER REVIEW 4 of 13

this method can ensure a very wide range of *e* for the specimens and contribute to preventing

#### *2.4. Testing Program and Process* the test results of Yang and Liu (2016) [14]. Figure 3 presents a set of typical received signals captured from the bender element in different silty sand specimens. The first arrival time method

For the bender element tests, the 10 kHz excitation signal was found to consistently yield a clear arrival of the shear wave for both clean sand and silty sand with various *FC*, which is consistent with the test results of Yang and Liu (2016) [14]. Figure 3 presents a set of typical received signals captured from the bender element in different silty sand specimens. The first arrival time method was introduced to determine the shear wave travel time in this study [26–28], and the zero after first bump point corresponds to Point C marked in Figure 3, suggested by Yoo et al. (2018) [29] and Lee and Santamarina (2005) [30], was selected as the shear wave arrival time. was introduced to determine the shear wave travel time in this study [26–28], and the zero after first bump point corresponds to Point C marked in Figure 3, suggested by Yoo et al. (2018) [29] and Lee and Santamarina (2005) [30], was selected as the shear wave arrival time. In order to investigate the influences of *FC*, *e*, and c0 on *G*max of silty sand, *FC* = 0, 10, 20, and 30% were considered, and three samples were prepared at different *e* for silty sand at a fixed *FC*. The *G*max were measured subjected to c0 at 100, 200, 250, 300, and 400 kPa in five stages, Table 3 details the test conditions.

**Figure 3.** Shear wave signals in specimen for case ID: S11. **Figure 3.** Shear wave signals in specimen for case ID: S11.

**Table 3.** Schemes of bender element tests for Nantong marine silty sand. **ID** *FC***/%** *D***r/%** *e ρ(***g/cm<sup>3</sup> )** *b* **Value** *e* **\*** c0 **/kPa** S1 0 35 1.07 1.286 0 1.286 In order to investigate the influences of *FC*, *e*, and σ 0 c0 on *G*max of silty sand, *FC* = 0, 10, 20, and 30% were considered, and three samples were prepared at different *e* for silty sand at a fixed *FC*. The *G*max were measured subjected to σ 0 c0 at 100, 200, 250, 300, and 400 kPa in five stages, Table 3 details the test conditions.


S2 0 50 0.97 3 1.352 0 1.352 S3 0 60 0.89 1.412 0 1.412 **Table 3.** Schemes of bender element tests for Nantong marine silty sand.

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