*3.2. Modified Hardin Model Based on Binary Packing Model 3.2. Modified Hardin Model Based on Binary Packing Model*

parameter *b* [13,15,35]:

where *r* = 1/*χ*, and *k* = 1 − *r*

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

The binary packing state concept [17,31] is adopted herein to interpret the behavior of granular soil. For the binary packing system, the *FC*th has been introduced to distinguish the difference of "coarse-dominated behavior" from "fines-dominated behavior" for silty sand with various *FC* [32,33]. The *FC*th can be determined empirically by semi-experience formula [13]: The binary packing state concept [17,31] is adopted herein to interpret the behavior of granular soil. For the binary packing system, the *FC*th has been introduced to distinguish the difference of "coarse-dominated behavior" from "fines-dominated behavior" for silty sand with various *FC* [32,33]. The *FC*th can be determined empirically by semi-experience formula [13]:

$$\text{FC}\_{\text{th}} = 0.40 \times \left( \frac{1}{1 + \exp(0.5 - 0.13 \cdot \chi)} + \frac{1}{\chi} \right) \tag{6}$$

**(7)**

**(8)**

where s f *d d* 10 50 is the particle size disparity ratio, s *d*10 is the grain size at 10% finer for clean sand, f *d*50 is the grain size at 50% finer for pure fines. where χ = *d* s <sup>10</sup>/*d* f <sup>50</sup> is the particle size disparity ratio, *d* s <sup>10</sup> is the grain size at 10% finer for clean sand, *d* f <sup>50</sup> is the grain size at 50% finer for pure fines.

 1

*e b FC*

The physical meaning of *b* is the fraction of fines that participate in the force chain between soil grains and 0 ≤ *b* ≤ 1. Equation (7) is based on coarse-dominated behavior soil fabric, this meaning *b*  requires *FC* < *FC*th. Rahman and his co-authors developed a semi-empirical relation to predict the

type. The experimental results, presented by Lashkari (2014), suggested that a *μ* of 0.30 and *n*<sup>b</sup> of 1.0 satisfy a large dataset and were later verified with new datasets. Goudarzy et al. (2016) acknowledged that these parameters might vary for different types of soil, the *μ* and *n*<sup>b</sup> value were

instead of *e*, can well capture various aspects of the mechanical behavior of silty sand [36]. Notably, the binary packing state parameter has been introduced to uniquely quantify the critical state line, steady state line, and liquefaction resistance, etc. of the silty sand with different *FC*. Hence, an effort has been made to investigate whether *e*\* determined by Rahman's approach can better characterize

*FC FC FC b r*

*b FC*

b th

 

*k FC*

th

. It has been well recognized that *e*\*,

, *μ* and *n*<sup>b</sup> are the fitting parameters which depend on the specific soil

*n r*

1 1

\*

*e*

1 exp

0.25

*G*max by replacing *e* with *e*\* in Equation (3):

optimized in Equation (8) to obtain the maximum value of *R*<sup>2</sup>

As *FC* increases, fines may come in between the contact of sand grains and participate in the

As *FC* increases, fines may come in between the contact of sand grains and participate in the force chain. Thus, the effect of fines on the force transfer mechanism is considered by introducing an alternative equivalent skeleton void ratio *e*\* [31,34], as defined by Equation (7).

$$e\* = \frac{e + (1 - b) \cdot FC}{1 - (1 - b) \cdot FC} \tag{7}$$

The physical meaning of *b* is the fraction of fines that participate in the force chain between soil grains and 0 ≤ *b* ≤ 1. Equation (7) is based on coarse-dominated behavior soil fabric, this meaning *b* requires *FC* < *FC*th. Rahman and his co-authors developed a semi-empirical relation to predict the parameter *b* [13,15,35]:

$$b = \left\{ 1 - \exp\left( -\mu \frac{\left(FC/FC\_{\text{th}}\right)^{\eta\_{\text{b}}}}{k} \right) \right\} \left( r \times \frac{FC}{FC\_{\text{th}}}\right)^{r} \tag{8}$$

where *r* = 1/χ, and *k* = 1 − *r* 0.25 , µ and *n*<sup>b</sup> are the fitting parameters which depend on the specific soil type. The experimental results, presented by Lashkari (2014), suggested that a µ of 0.30 and *n*<sup>b</sup> of 1.0 satisfy a large dataset and were later verified with new datasets. Goudarzy et al. (2016) acknowledged that these parameters might vary for different types of soil, the µ and *n*<sup>b</sup> value were optimized in Equation (8) to obtain the maximum value of *R* 2 . It has been well recognized that *e*\*, instead of *e*, can well capture various aspects of the mechanical behavior of silty sand [36]. Notably, the binary packing state parameter has been introduced to uniquely quantify the critical state line, steady state line, and liquefaction resistance, etc. of the silty sand with different *FC*. Hence, an effort has been made to investigate whether *e*\* determined by Rahman's approach can better characterize *G*max by replacing *e* with *e*\* in Equation (3):

$$F(e\*) = (c - e\*)^2 / (1 + e\*)\tag{9}$$

Figure 8 show the relationship between *G*max, *F*(*e*\*), and normalized effective confining stress (σ 0 c0/*P*a) *<sup>n</sup>* of silty sands. Despite the variation in *FC*, *e*, or σ 0 c0 of the specimens, all of the test data points are located in a narrow surface, which means that *e*\* appears to adequately capture the effects of *FC*, *e*, and particle gradations when *FC* < *FC*th. Therefore, the modified Hardin model based on the binary packing state parameter can be established: 2 \* \* \* *F e c e e* 1 (9) Figure 8 show the relationship between *G*max, *F*(*e* \*), and normalized effective confining stress ( c0 /*P*a) *<sup>n</sup>* of silty sands. Despite the variation in *FC*, *e*, or c0 of the specimens, all of the test data points are located in a narrow surface, which means that *e* \* appears to adequately capture the effects

of *FC*, *e*, and particle gradations when *FC* < *FC*th. Therefore, the modified Hardin model based on the

*G*max = *A* ∗ (*c* − *e*∗) 2 (1 + *e*∗) σ 0 c0 *P*a !*n* (10) binary packing state parameter can be established: 2 \* c0 \* *n c e G A* 

(10)

*A*\* = 59.3 MPa and R-square = 0.938 for Nantong silty sand, *n* was determined using Equation (4). a 1 \* *e P A\** = 59.3 MPa and R-square =0.938 for Nantong silty sand, *n* was determined using Equation (4).

max

**Figure 8.** The modified Hardin model for Nantong marine silty sand with different *FC* in *G*max-*F*(*e*\*)- c0 /*P*<sup>a</sup> space. **Figure 8.** The modified Hardin model for Nantong marine silty sand with different *FC* in *G*max-*F*(*e*\*)-σ 0 c0/*P*<sup>a</sup> space.

*G*max values are basically consistent. Considering the complexity of the effect of fines and man-made errors, such an error is acceptable. Therefore, the modified Hardin model can be used to predict the

*G*max of silty sand when *FC* < *FC*th in a simple yet reliable way.

50

100

150

200

Predicted small-strain stiffness by

modified Hardin model, *G*max(MPa)

250

300

350

To validate the accuracy of the modified Hardin model, the comparison between the predicted

+10%


*FC* = 0% *FC* = 10% *FC* = 20% *FC* = 30%

1 1

50 100 150 200 250 300 350 Measured small-strain stiffness,*G*max(MPa)

c0 /*P*<sup>a</sup> space.

( c0 /*P*a)

To validate the accuracy of the modified Hardin model, the comparison between the predicted *G*max in Equation (10) and the measured *G*max are presented in Figure 9. Almost all of the data pairs are close to the bisecting line, with the errors within 10%, indicating that the measured and predicted *G*max values are basically consistent. Considering the complexity of the effect of fines and man-made errors, such an error is acceptable. Therefore, the modified Hardin model can be used to predict the *G*max of silty sand when *FC* < *FC*th in a simple yet reliable way. To validate the accuracy of the modified Hardin model, the comparison between the predicted *G*max in Equation (10) and the measured *G*max are presented in Figure 9. Almost all of the data pairs are close to the bisecting line, with the errors within 10%, indicating that the measured and predicted *G*max values are basically consistent. Considering the complexity of the effect of fines and man-made errors, such an error is acceptable. Therefore, the modified Hardin model can be used to predict the *G*max of silty sand when *FC* < *FC*th in a simple yet reliable way.

*J. Mar. Sci. Eng.* **2020**, *8*, x FOR PEER REVIEW 9 of 13

 2 \* \* \* *F e c e e* 1

of *FC*, *e*, and particle gradations when *FC* < *FC*th. Therefore, the modified Hardin model based on the

 

*c e*

*A\** = 59.3 MPa and R-square =0.938 for Nantong silty sand, *n* was determined using Equation (4).

\*

2 \* c0

*e P* 

\* 1 \*

 

 c0 

a

*n*

Figure 8 show the relationship between *G*max, *F*(*e*

points are located in a narrow surface, which means that *e*

binary packing state parameter can be established:

*<sup>n</sup>* of silty sands. Despite the variation in *FC*, *e*, or

max

*G A*

(9)

(10)

), and normalized effective confining stress

\* appears to adequately capture the effects

of the specimens, all of the test data

**Figure 9.** Comparison of the measured *G*max and the predicted *G*max using the modified Hardin model.

Given the complexity of material properties, further work to validate the applicability of the modified Hardin model evaluation *G*max by using experimental data is worthwhile. The similar *G*max testing series were carried out on four types of silty sand by Goudarzy et al. (2016) [15], Salgado et al. (2000) [16], Chien and Oh (1998) [37], and Thevanayagam and Liang (2001) [38]. Table 4 presents the physical index properties and fitting parameters of Nantong marine silty sand tested in this study and four silty sands using compiled data from the literature. Best fitting values of µ and *n*<sup>b</sup> in Equation (8) are 0.27~0.34 and 0.89~1.08, and the R-square value of the modified Hardin model for experimental data compiled from the literature are all over 0.9, which means the modified Hardin model can characterize *G*max for different types of silty sands well. It should be noted that *A*\* for different types of silty sand presents an obvious soil-specific diversification. In addition, as shown in Figure 10, a power function relationship between *A*\* and the synthesizing material property parameters ln(*e*range(s)·*C*u(s)·χ) was established:

$$A\* = 54.6 \times \left[ \ln \left( e\_{\text{range}(s)} \cdot \mathbb{C}\_{\mathbf{u}(s)} \cdot \chi \right) \right]^{-0.43} \tag{11}$$


**Table 4.** Physical index properties and fitting parameters of silty sands considered in this study.

Note: *e*range(s)—void ratio range of clean sand (=*e*max − *e*min); *C*u(s)—uniformity coefficient of clean sand; µ and *n*b—fitting parameters in Equation (8); *A*\*—fitting parameter in Equation (10); *R* <sup>2</sup>—coefficient of determination for Equation (10).

Thus, the modified *G*max prediction method based on the binary packing model can be established by combining Equations (4), (10), and (11), only considering basic indices of the clean sand and pure property parameters ln(*e*range(s)*∙C*u(s)*∙χ*) was established:

**Data from Material**

This study Nantong sand + Nantong silt

powder

fines. It is worth noting that the application of the binary packing model should not be limited to the evaluation of *G*max. Existing test results show that *e*\* presents a unified correlation with static liquefaction characteristics [39], drained and undrained triaxial compression behaviors [40], critical strength [41], liquefaction strength [42], etc., of silty sand, and the proposed procedure in this paper provides a significant improvement in the evaluation of the above mechanical properties in geotechnical engineering practice. Salgado et al. (2000) Ottawa sand + Sil-co-Sil 0.30 1.48 11.8 0.34 0.92 44.7 0.895 Chien and Oh (1998) Yunling sand+Yunling silt 0.55 1.69 2.17 0.27 1.08 64.9 0.883 Thevanayagam and Liang(2001) Foundary sand + Sil-co-Sil 0.19 1.69 17.1 0.29 0.89 43.2 0.902 Note: *e*range(s)—void ratio range of clean sand (=*e*max − *e*min); *C*u(s)—uniformity coefficient of clean sand; *μ* and *n*b—fitting parameters in Equation (8); *A*\*—fitting parameter in Equation (10); R<sup>2</sup>—coefficient of determination for Equation (10).

*J. Mar. Sci. Eng.* **2020**, *8*, x FOR PEER REVIEW 10 of 13

**Figure 9.** Comparison of the measured *G*max and the predicted *G*max using the modified Hardin model.

Given the complexity of material properties, further work to validate the applicability of the modified Hardin model evaluation *G*max by using experimental data is worthwhile. The similar *G*max testing series were carried out on four types of silty sand by Goudarzy et al. (2016) [15], Salgado et al. (2000) [16], Chien and Oh (1998) [37], and Thevanayagam and Liang (2001) [38]. Table 4 presents the physical index properties and fitting parameters of Nantong marine silty sand tested in this study and four silty sands using compiled data from the literature. Best fitting values of *μ* and *n*b in Equation (8) are 0.27~0.34 and 0.89~1.08, and the R-square value of the modified Hardin model for experimental data compiled from the literature are all over 0.9, which means the modified Hardin model can characterize *G*max for different types of silty sands well. It should be noted that *A\** for different types of silty sand presents an obvious soil-specific diversification. In addition, as shown in Figure 10, a power function relationship between *A***\*** and the synthesizing material

> -0.43 \* range s <sup>u</sup> <sup>s</sup>

**Table 4.** Physical index properties and fitting parameters of silty sands considered in this study.

Index Properties In

Equation (8)

*e*range(s) *C*u(s) *χ μ n*<sup>b</sup> *A*\* *R*<sup>2</sup>

0.60 1.67 2.0 0.32 0.94 62.1 0.932

0.35 2.01 63.3 0.33 1.05 30.3 0.943

(11)

In Equation (10)

*A e C* 54.6 ln

**Figure 10.** The relationship between *A\** of the modified Hardin model and ln(*e*range(s)*∙C*u(s)*∙χ*). **Figure 10.** The relationship between *A*\* of the modified Hardin model and ln(*e*range(s)·*C*u(s)·χ).

#### Thus, the modified *G*max prediction method based on the binary packing model can be **4. Conclusions**

established by combining Equations (4), (10), and (11), only considering basic indices of the clean sand and pure fines. It is worth noting that the application of the binary packing model should not be limited to the evaluation of *G*max. Existing test results show that *e*\* presents a unified correlation with static liquefaction characteristics [39], drained and undrained triaxial compression behaviors In order to investigate how *e*, *FC* and σ 0 c0 alter the *G*max of marine silty sand, comprehensive bender element tests were performed under isotropic consolidation, and a modified procedure based on the Hardin model was established to predict the *G*max. The main obtained results are summarized as follows.


**Author Contributions:** Conceptualization, Q.W. and K.Z.; validation, Q.L.; formal analysis, Q.L. and Q.G.; writing—original draft preparation, Q.W.; writing—review and editing, K.Z.; supervision, P.C. and G.C.; funding acquisition, K.Z. and G.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Projects of the National Natural Science Foundation of China (NSFC), grant number 51978335, and by the Open Research Fund of State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, grant number Z019010.

**Acknowledgments:** The study in this paper was partly supported by the National Key Basic Research Program of China (Grant No. 2011CB013605). This financial support is highly appreciated.

**Conflicts of Interest:** The authors declare no conflict of interest.
