*3.1. Factors Influencing Maximum Shear Modulus*

Figure 4 present the comprehensive view of the measured *G*max values of silty sand with different *FC*, *e*, and σ 0 c0. A remarkable finding from the figure is that *FC*, *e*, or σ 0 c0 all has a significant impact on *G*max, the increase of *e* will significantly reduce *G*max for silty sand at different *FC* and σ 0 c0. Furthermore, in each plot, the five trend lines describe the effect of *e* on *G*max, and the range of trend lines revealed the influence of varying σ 0 c0. Under otherwise similar conditions, *G*max decreases with increasing *e* or *FC*, but increases with increasing *FC*. The existing explanation that: as *e* increases, the dense state changes from compact to loose, which reduces the amount of force chain between particles, contribute to a attenuation in the stiffness of silty sand; while at a fixed *e*, the amount of sand grains composed of soil skeleton is constant as *FC* increases, a certain amount of grains participate in the composition of soil skeleton, and the grain contact area increases, eventually leading to an increase in *G*max. In addition, the relationship between *G*max and *e* is insensitive to σ 0 c0, but obviously sensitive to *FC*. According to Yang and Liu (2016) [14], there is a linear function relationship between *G*max with *e* for Toyoura silty sand, and the void ratio dependence appears to be similar to silty sand with different *FC*. Incorporating the test results in the study, an obvious soil-specific relationship between *G*max and *e* can be found, and a more comprehensive study needs to be conducted for addressing this concern. *3.1. Factors Influencing Maximum Shear Modulus* Figure 4 present the comprehensive view of the measured *G*max values of silty sand with different *FC*, *e*, and c0 . A remarkable finding from the figure is that *FC*, *e*, or c0 all has a significant impact on *G*max, the increase of *e* will significantly reduce *G*max for silty sand at different *FC* and c0 . Furthermore, in each plot, the five trend lines describe the effect of *e* on *G*max, and the range of trend lines revealed the influence of varying c0 . Under otherwise similar conditions, *G*max decreases with increasing *e* or *FC*, but increases with increasing *FC*. The existing explanation that: as *e* increases, the dense state changes from compact to loose, which reduces the amount of force chain between particles, contribute to a attenuation in the stiffness of silty sand; while at a fixed *e*, the amount of sand grains composed of soil skeleton is constant as *FC* increases, a certain amount of grains participate in the composition of soil skeleton, and the grain contact area increases, eventually leading to an increase in *G*max. In addition, the relationship between *G*max and *e* is insensitive to c0 , but obviously sensitive to *FC*. According to Yang and Liu (2016) [14], there is a linear function relationship between *G*max with *e* for Toyoura silty sand, and the void ratio dependence appears to be similar to silty sand with different *FC*. Incorporating the test results in the study, an obvious soil-specific relationship between *G*max and *e* can be found, and a more comprehensive study needs to be conducted for addressing this concern.

*J. Mar. Sci. Eng.* **2020**, *8*, x FOR PEER REVIEW 5 of 13

**Figure 4.** The relationship between *G*max and *e* for Nantong marine silty sand with different *FC*: (**a**) *FC* = 0%; (**b**) *FC* = 10%; (**c**) *FC* = 20%; (**d**) *FC* = 20%. **Figure 4.** The relationship between *G*max and *e* for Nantong marine silty sand with different *FC*: (**a**) *FC* = 0%; (**b**) *FC* = 10%; (**c**) *FC* = 20%; (**d**) *FC* = 20%.

For silty sand at a specific *FC*, given that *G*max is dependent on both *e* and c0 , *e* must be taken into account when quantifying the impact of c0 . Therefore, a void ratio function *F*(*e*) was introduced to characterize the influence of *e* on *G*max: For silty sand at a specific *FC*, given that *G*max is dependent on both *e* and σ 0 c0, *e* must be taken into account when quantifying the impact of σ 0 c0. Therefore, a void ratio function *F*(*e*) was introduced to characterize the influence of *e* on *G*max:

$$F(e) = \frac{(c - e)^2}{1 + e} \tag{2}$$

where *c* is a soil-specific fitting parameter dependent on the particle shape—2.97 for angular particles and 2.17 for rounded particles [5,15]. Considering that the particles of marine silty sand are where *c* is a soil-specific fitting parameter dependent on the particle shape—2.97 for angular particles and 2.17 for rounded particles [5,15]. Considering that the particles of marine silty sand are angular (Figure 2b), *c* = 2.97 was used. An empirical relation for *G*max prediction, incorporating material, particle shape, *e* and σ 0 c0, was proposed originally by Hardin and Black (1966) [5], then a more general form was developed based on the research of Iwasaki and Tatsuoka (1977) [11], Seed et al. (1986) [8], Youn et al. (2008) [9], Yang and Gu (2013) [10], Wichtmann et al. (2015) [12], and Payan et al. (2016) [1]:

$$G\_{\text{max}} = A \frac{(c - e)^2}{1 + e} \left(\frac{\sigma\_{\text{c0}}'}{P\_{\text{a}}}\right)^n \tag{3}$$

where *A* = material constant depends on soil type; *P*<sup>a</sup> = atmospheric pressure (≈100 kPa); *n* = stress exponent, the values of *n* typically distribute between 0·35 and 0·6 for silty sand. Iwasaki and Tatsuoka (1977) [11] and Yang and Liu (2016) [14] present a common phenomenon that the stress exponent *n* is a soil-specific constant.

In order to explore the distribution of *A* and *n* values, the *G*max values of silty sand are plotted as function of σ 0 c0/*P*<sup>a</sup> and *F*(*e*) in Figure 5. Under otherwise identical conditions, *G*max increases with increasing in normalized effective confining stress σ 0 c0/*P*<sup>a</sup> and void ratio function *F*(*e*). In addition, *R*-square of the Hardin model are all greater than 0.9, which means that the Hardin model can characterize the influence of *e* and σ 0 c0 on *G*max of silty sand at a specific *FC* well. However, for a specific silty sand, the exponent *n* is insensitive to *FC* and *e*, which is consistent with the results demonstrated by Iwasaki and Tatsuoka (1977) [11] and Yang and Liu (2016) [14]. The exponent *n*, reflecting the incremental rate of *G*max due to the enhancement of σ 0 c0, is highly dependent on the types of silty sand and present as a soil-specific constant. Using the generalized nonlinear regression model for the test data of marine silty sand tested in this study and six silty sands compiled from the literature, the soil-specific constant *n* is closely related to the synthesizing material parameter *C* s u ·*C* f <sup>u</sup> of sandy soils (as shown in Figure 6). It is seen that *n* increases with the increase of *C* s u · *C* f u , indicating a logarithmic function relation. The soil-specific constant *n* can be determined empirically by the following equation:

*J. Mar. Sci. Eng.* **2020**, *8*, x FOR PEER REVIEW 7 of 13

$$m = 0.086 \ln \left( \mathbf{C\_u^s} \cdot \mathbf{C\_u^f} \right) + 0.302, R^2 = 0.98 \tag{4}$$

**Figure 5.** The Hardin model for Nantong marine silty sand in *G*max-*F*(*e*\*)- c0 /*P*<sup>a</sup> space: (**a**) *FC* = 0%; (**b**) *FC* = 10%; (**c**) *FC* = 20%; (**d**) *FC* = 20%. s f 2 =0.086ln 0.302, 0.98 u u *n C C R* **Figure 5.** The Hardin model for Nantong marine silty sand in *G*max-*F*(*e*\*)-σ 0 c0/*P*<sup>a</sup> space: (**a**) *FC* = 0%; (**b**) *FC* = 10%; (**c**) *FC* = 20%; (**d**) *FC* = 20%.

0 10 20 30 40

It is worth noting that the addition of *FC* will obviously alter the material-specific fitting

*A FC A m FC* <sup>0</sup> exp

where, the value of *A*<sup>0</sup> represents the parameter *A* for clean sand (*FC* = 0%) in the Hardin model, *m* is the fitting parameter and the value of *m* is −1.52 for Nantong silty sand. It is worth noting that care should be exercised when the Hardin model is directly used for predicting the *G*max of silty sand, considering the sensitivity of the *A* to *FC*. Therefore, a modified Hardin model needs to be explored

**Figure 6.** Variation of stress exponent *n* versus the synthesizing material property parameter

parameter *A*, which describe the increment ratio of *G*max/*F*(*e*) caused by the increasing of (

(Figure 7), and a fairly good exponential relationship can be given as following:

0.3

0.4

0.5

Stress exponen, *n*

for unified charactering *G*max of silty sand with different *FC*.

0.6

0.7

W, Payan et al. (2017) B2, Payan et al. (2017) Goudarzy et al. (2017) Wichtmann et al. (2015) Yang and Liu. (2016)

> s f *C C* u u **.**

> > c0 /*P*a) n

> > > (5)

Nantong marine silty sand in this study B1, Payan et al. (2017)

(**b**) *FC* = 10%; (**c**) *FC* = 20%; (**d**) *FC* = 20%.

**Figure 6.** Variation of stress exponent *n* versus the synthesizing material property parameter s f *C C* u u **. Figure 6.** Variation of stress exponent *n* versus the synthesizing material property parameter *C* s u · *C* f u .

It is worth noting that the addition of *FC* will obviously alter the material-specific fitting parameter *A*, which describe the increment ratio of *G*max/*F*(*e*) caused by the increasing of ( c0 /*P*a) n (Figure 7), and a fairly good exponential relationship can be given as following: It is worth noting that the addition of *FC* will obviously alter the material-specific fitting parameter *A*, which describe the increment ratio of *G*max/*F*(*e*) caused by the increasing of (σ 0 c0/*P*a) *n* (Figure 7), and a fairly good exponential relationship can be given as following:

$$A(FC) = A\_0 \times \exp(m \cdot FC) \tag{5}$$

 c0 

/*P*<sup>a</sup> space: (**a**) *FC* = 0%;

where, the value of *A*<sup>0</sup> represents the parameter *A* for clean sand (*FC* = 0%) in the Hardin model, *m* is the fitting parameter and the value of *m* is −1.52 for Nantong silty sand. It is worth noting that care should be exercised when the Hardin model is directly used for predicting the *G*max of silty sand, considering the sensitivity of the *A* to *FC*. Therefore, a modified Hardin model needs to be explored where, the value of *A*<sup>0</sup> represents the parameter *A* for clean sand (*FC* = 0%) in the Hardin model, *m* is the fitting parameter and the value of *m* is −1.52 for Nantong silty sand. It is worth noting that care should be exercised when the Hardin model is directly used for predicting the *G*max of silty sand, considering the sensitivity of the *A* to *FC*. Therefore, a modified Hardin model needs to be explored for unified charactering *G*max of silty sand with different *FC*. *J. Mar. Sci. Eng.* **2020**, *8*, x FOR PEER REVIEW 8 of 13

**Figure 7.** The relationship between *A* of the Hardin model and *FC* for Nantong marine silty sand. **Figure 7.** The relationship between *A* of the Hardin model and *FC* for Nantong marine silty sand.
