A Simplified Silty Sand Model

Abstract

A unified critical state model has been developed for both clean sand and silty sand using the modified Cam-clay model (MCC). The main feature of the proposed model is a new critical state line equation in the e-ln(p) plane that is capable of handling both straight and curved test results. With this feature, the error in calculating plastic volumetric strain is, in theory, eliminated. Another crucial feature of the model is the transformed stress tensor based on the SMP (spatially mobilized plane) criterion, which takes into account the proper shear yield and failure of soil under three-dimensional stresses. Additionally, the proposed model applies the intergranular void ratio with the fines influence factor for silty sand. Only eight soil parameters are required for clean sand, and a total number of twelve soil parameters are needed for silty sand. This model not only enhances the predictive accuracy for granular soils but also broadens the applicability of the model to encompass silty sand in both drained and undrained analyses.

1. Introduction

The geotechnical engineering problems can be solved by many types of solutions, e.g., analytical solution [1,2,3], semi-analytical solution [4,5] and numerical approachs [6,7,8]. Recent technological advancements lead to sophisticated construction including phrased construction. The finite element method (FEM) is one of the powerful numerical methods applicable for the complex in both the geometry and loading of the problem. Nevertheless, the rigorousness of the solutions from FEM depends largely on the constitutive models incorporating the FEM-based analysis. Constitutive models are generally developed for specific types of geomaterial, such as clay and sand. The majority of the models are developed solely for each type of soil. However, in nature, the soil deposits are composed of more than one type, and silty sand is a good example. In order to accurately predict the behavior of silty sand, the critical state model’s complexity rises to require twenty or more soil characteristics [9], thus making it unapproachable to general users. Therefore, it is crucial to develop a simple yet practical model for silty sand.

The modified Cam-clay (MCC) model was originally developed to describe the stress–strain strength characteristics of normally consolidated clay based on Roscoe and Burland’s critical state theory [10]. A number of modifications have been made to the MCC to improve its ability to model geomaterials, such as [11,12,13,14,15,16,17,18,19,20,21]. Within the MCC and its family, the critical state void ratio ($e_{cs}$) defined by the critical state line (CSL) is assumed to be a straight line lower than the normally consolidated line (NCL) with the same slope λ, as shown in Figure 1a. The distance between the NCL and CSL is assumed to be $(\lambda -\kappa )\ln \left(2\right)$, where κ is the slope of the unloading-reloading portion in the isotropic compression test result. Equation (1) contains two fitting parameters: Γ, the limiting critical state void ratio, usually defined as $e_{cs}$ at $p$ = 1.0, and $\lambda$, which represents the constant slope of the CSL. However, unlike clay, the literature shows that the CSL of granular soil is not a straight line, e.g., [22,23,24,25,26,27,28]. A power function curve shown in Equation (2) provides a better representation for granular soil. In Equation (2), Γ acts as the upper bound of $e_{cs},$ while $\lambda , p_a and \xi$ are the fitting parameters. Although $p_a$ is recommended, it is not necessary to have the atmospheric pressure (approximately 101 kPa). Thus, Equation (2) requires three fitting parameters. It should be noted that the slope of CSL in Equation (2) is not a constant but a varied value $\overline{\lambda }$. A comparison between Equations (1) and (2) is demonstrated in Figure 1b. Using Equation (1) for granular soil would result in a significant error in the volumetric plastic strain ($∆{\epsilon }_v^p=∆e^p/(1+e_0))$ between two points on the CSL such as points A and B in Figure 1b. Therefore, a power function curve is popularly used for sand or silty sand. However, this form of the power function can be problematic in sand models when dealing with high pressure because it cannot provide a straight portion after the curved portion. Figure 2a shows the concavity on the extension of Equation (2) after fitting the laboratory data [11], where the dashed line deviated from the trend of a straight line. To achieve a proper function of $e_{cs}$ in the e-ln(p) for granular soil, including clean sand and silty sand, without any error in the volumetric plastic strain calculation, a curve illustrated in Figure 2b is required.

$$e_{cs}=\Gamma -\lambda \ln \left(p\right)$$

(1)

$$e_{cs}=\Gamma -\lambda {\left(p/p_{ref}\right)}^{\xi }$$

(2)

Matsuoka et al. [13] were the first to revise the shear yield and shear failure in the MCC model from the extended Mises criterion to the SMP criterion by introducing the transformed stress. Yao et al. [14] successfully applied the transformed stress of [13] to model loose and dense sand, depending on two key factors, i.e., the initial density ($e_0$) and mean effective stress ($p_0$). Furthermore, Yao et al. [15] proposed another transformed stress, in conjunction with a new hardening parameter, to the MCC model to model both clay and sand. Finally, in this study, a new curve of the CSL in the e-ln(p) is derived, which has the ability to fit a straight line, curved line, and a curved line connected with a straight line. Therefore, the sand model developed by Yao et al. [14] can be extended for both clean sand and silty sand. Additionally, we have made some revisions to the work of Yao et al. [14] to take into account the effect of the participation of fines in sand by applying the concept of the equivalent intergranular void ratio ($e^{*}$) proposed by Thevanayagam and Martin [29] and the fines influence factor (β) proposed by Lashkali [9]. The roundness of both sand and fines can settle the arguments in the load bearing mechanism of many silty sands in the literature, according to the work of Lashkali [9]. With these ideas, it is possible to oversimplify the soil characteristics in the critical state model of silty sand.

2. The Critical State Line in e-ln(p) Plane

As mentioned earlier, the MCC model adopted a straight line of the CSL, but a curved line is valid for sand and silty sand. To address this, the proposed CSL should have the following features:

(1)

The equation should provide both the curved portion and the connecting straight line.

(2)

One of the fitting parameters should be the desired slope of the straight portion.

(3)

The equation should consist of parameters that can adjust to fit various curvatures and locations.

(4)

All parameters should be determined from a conventional test.

By incorporating these features, the proposed CSL can better capture the behavior of sand and silty sand and provide more accurate predictions for engineering analyses.

Equation (3) presents the new CSL with four fitting parameters: Γ, the apparent maximum void ratio which controls the vertical translation of the curve;$p_{ref}$, the reference pressure which is a positive value used to normalize the mean effective stress controlling the horizontal translation; θ, the curvature index which controls the curvature of the upper portion; and α, the half-slope of the straight portion which controls the slope of the lower portion. The term “apparent maximum” is used for the Γ because no data point from the test result can be greater than this value. It is recommended that the value of λ/2 or a value close to λ/2, where λ is the slope of CSL in Equation (1). To investigate Equation (3), variations of all fitting parameters were demonstrated in Figure 3, Figure 4, Figure 5 and Figure 6. Notably, when θ = 0, the curvature of the CSL vanishes and Equation (3) becomes a straight line with a slope of 2α, as shown in Equation (4). Furthermore, when θ = 0 and $p_{ref}=1$, Equation (3) reduces to Equation (1) with α = λ/2. In Figure 7, the test results of Toyoura sand [11] were plotted against Equations (1)–(3) to evaluate their performance. Equations (5) and (6) present the slope $\overline{\lambda }$ of Equation (3) in terms of tangential and secant, respectively.

$$e=\Gamma -\alpha \ln \left(\frac{p}{p_{ref}}\right)-\sqrt{\theta +{\left(\alpha \ln \left(\frac{p}{p_{ref}}\right)\right)}^2}$$

(3)

$$e=\Gamma -2\alpha \ln \left(\frac{p}{p_{ref}}\right)$$

(4)

All four fitting parameters can be determined by the conventional isotropic compression test. While Equation (3) requires more parameters than Equations (1) and (2), all four parameters can be determined by fitting the same test data used for Equations (1) and (2), without requiring additional testing. Therefore, Equation (3) satisfies all the aforementioned features.

$${\overline{\lambda }}_t=\frac{\partial e}{\partial \left(lnp\right)}=\alpha \left[1+\frac{\alpha \ln \left(\frac{p}{p_{ref}}\right)}{\sqrt{\theta +{\left(\alpha \ln \left(\frac{p}{p_{ref}}\right)\right)}^2}}\right]$$

(5)

$${\overline{\lambda }}_s=-\frac{∆e}{∆\mathit{ln}\left(p\right)}=\alpha +\left\{\sqrt{\mathrm{\theta }+{\left(\alpha \mathit{ln}\left(\frac{p_1}{p_{ref}}\right)\right)}^2}-\sqrt{\mathrm{\theta }+{\left(\alpha \mathit{ln}\left(\frac{p_2}{p_{ref}}\right)\right)}^2}\right\}/ln\left(\frac{p_2}{p_1}\right)$$

(6)

In the MCC model and its family, the CSL and NCL are both assumed to be a straight line with a constant slope λ (see Figure 1) and the vertical distance of $(\lambda -\kappa )\ln \left(2\right)$ [14]. In the proposed model, once the curve of NCL is obtained using Equation (3), the CSL is assumed to locate below the NCL with a distance of $〈\overline{\lambda }-\kappa 〉\ln \left(2\right),$ where $<>$ represents the Macualey brackets (i.e., $\overline{\lambda }-\kappa$ = $\overline{\lambda }-\kappa$ when $\overline{\lambda }-\kappa \ge 0$ and zero otherwise). The shape and location of NCL compared to the CSL for this case are shown in Figure 8a. It is evident that both the CSL and NCL contain both curved and straight portions. Within the curve portion, the NCL and CSL have different slopes, and these lines are close together when the mean effective stress converges to a very small value ($p\to 0$). On the other hand, the CSL is parallel to the NCL with the same slope within the straight portion. This implies that the new CSL is not only applicable for granular soil but also for clay in the MCC model.

In addition, a curved of the reference control line (RCL) can also be derived from the CSL by assuming that the shape of the CSL and the RCL are identical. Therefore, both CSL and RCL have the same parameters, i.e., Γ, α, and θ. The remaining parameter of the RCL is $p_{ref}$ which can be determined by the horizontal translation of the CSL to the right until the vertical distance between the two straight portions of the CSL and RCL is equal to $〈\overline{\lambda }-\kappa 〉\ln \left(2\right)\approx 〈2\alpha -\kappa 〉\ln \left(2\right)$. Since the straight portions are parallel with the slope of $2\alpha$, then the horizontal distance between two straight lines is $\left(2\alpha -\kappa \right)\ln \left(2\right)/\left(2\alpha \right)$. Hence, the parameter $p_{ref\left(RCL\right)}=p_{ref\left(CSL\right)}\times 2^\left(\right(2\alpha -\kappa )/(2\alpha \left)\right)$.

3. Critical State Model for Clean Sand

In the work of Yao et al. [14], the NCL was used as a key tool to distinguish between loose and dense sand, using as a reference control line (RCL). Therefore, in this case, the terms loose and dense refer to sands that are looser or denser than the RCL. Due to changes in the RCL and CSL, some revisions were required for the model presented in [14]. Before describing the revisions in detail, a clear summary of sand behavior for both loose and dense conditions is stated as follows:

(1)

Sand on RCL exhibits only volume contraction during shearing, and the stress ratio $\eta =q/p$ reaches its maximum value at the critical state, where $\eta =M_c$ and $M_c$ is the slope of the critical state line on p-q plane.

(2)

Sand that is looser than the RCL (sheared from a point above the RCL) exhibits only volume contraction, but it has a larger amount than sand on the RCL, it finally reaches the same maximum stress ratio as sand on the RCL at $\eta =M_c$.

(3)

Sand that is denser than the RCL (sheared from a point below the RCL) exhibits volume contraction in early stage, followed by dilation (the stress ratio is greater than the maximum stress ratio of sand on the RCL, i.e., ($\eta =M_f)>M_c$, and finally converges to the stress ratio $\eta =M_c$.

It is evident that both sand on RCL and sand looser than RCL have no peak but reach the same maximum stress ratio at the critical state, while a peak exists in dense sand during shearing before reaching the critical state. Figure 9 depicts the concept mentioned above.

3.1. Hardening Parameter H

The equation for the hardening parameter H in the MCC model and in Yao et al. [14] is shown in Equations (7) and (8), respectively. In Equation (7), λ is a constant representing the slope of the straight NCL and a variable $c_p$ is defined as $c_p$= (λ − κ)/(1 + e₀). In Equation (8), despite using the straight NCL in [14], the varied slope of the NCL $\overline{\lambda }$ is assumed in such a way that ${\overline{c}}_p$ = ($\overline{\lambda }$ − κ)/(1 + e₀) = [(λ − κ)/(1 + e₀)](Mc/Mf), where Mc and Mf are the stress ratio at the critical state and stress ratio during shearing. As mentioned in the early section, loose sand has no peak, so Mf = Mc, resulting in Equation (8) becoming Equation (7) of the MCC model. On the other hand, for dense sand, Mf ≥ Mc during shearing. In this paper, a new value of $\overline{\lambda }$ can be calculated directly from the RCL without any assumption using the secant slope of the RCL in Equation (6). Therefore, the final version of H in the proposed model is shown in Equation (9).

$$H=\int dH=\int \frac{1}{c_p}d{\epsilon }_v^p ;(c_p=\frac{\lambda -\kappa }{1+e_0})$$

(7)

$$H=\int dH=\int \frac{1}{\overline{c_p}}\frac{M_c^4}{M_f^4}\frac{M_{f-}^4{\eta }^{*4}}{M_c^4-{\eta }^{*4}}d{\epsilon }_v^p ;({\overline{c}}_p=\frac{\overline{\lambda }-\kappa }{1+e_0}=\frac{\lambda -\kappa }{1+e_0}\frac{M_c^4}{M_f^4})$$

(8)

where ${\eta }^{*}$ is the maximum stress ratio during shearing.

$$H=\int dH=\int \frac{1}{\overline{c_p}}\frac{M_{f-}^4{\eta }^{*4}}{M_c^4-{\eta }^{*4}}d{\epsilon }_v^p ;({\overline{c}}_p=\frac{{\overline{\lambda }}_s-\kappa }{1+e_0} ; {\overline{\lambda }}_s calculated by Equation (6))$$

(9)

3.2. Yield Function

The yield function of the MCC model is represented by an ellipse in p-q plane, as shown in Equation (10). To account for the behavior of loose sand, Yao et al. [14] introduced the initial state parameter χ ($0\le \chi \le 1$) into the MCC model, resulting in a Lemniscate-shaped yield function [30], as shown in Equation (11). The value of χ is zero when sand is on the RCL, and χ increases with distance above the RCL, i.e., χ links to both the shape of the yield function and the distance above the RCL. Hence, χ can describe the degree of looseness of sand. Figure 10a shows the yield functions of Equations (10) and (11).

$$f=\mathit{ln}\left(\frac{p}{p_0}\right)+\mathit{ln}\left(1+{\eta }^2/M^2\right)=\int dH$$

(10)

$$f=\ln \left(\frac{p}{p_0}\right)+\ln \left(1+\frac{{\eta }^2/M^2}{1-\chi {\eta }^2/M^2}\right)=\int dH$$

(11)

According to [14], the original value of χ is derived by the stress path of a constant mean effective stress ($p=p_0$), as expressed in Equation (12). The χ modified by incorporating the yield function in Equation (11) and the new hardening in Equation (9) using the same method as [14] is shown in Equation (13).

$$\chi =\frac{EXP\left(\frac{e_0-e_{RCL}}{\lambda -\kappa }\right)-2}{EXP\left(\frac{e_0-e_{RCL}}{\lambda -\kappa }\right)-1} ; \chi \ge 0$$

(12)

$$\chi =\frac{EXP\left(\frac{e_0-e_{RCL}}{{\overline{\lambda }}_t-\kappa }\right)-2}{EXP\left(\frac{e_0-e_{RCL}}{{\overline{\lambda }}_t-\kappa }\right)-1} ; \chi \ge 0$$

(13)

where $e_{RCL}$ is the void ratio on the RCL at the same mean stress of the initial state, $p_0$. In Equation (13), ${\overline{\lambda }}_t$ is calculated using the tangential slope of the RCL in Equation (5) at the initial mean effective stress $p_0$.

The state parameter χ, in Equation (13), is constant during shearing and depends on both the initial mean effective stress $p_0$ and the initial void ratio $e_0$. For sand on the RCL, the ${\overline{\lambda }}_t$ and ${\overline{\lambda }}_s$ in Equations (5) and (6), respectively, are applicable. For loose sand, the $e_0$ is located above the RCL, the ${\overline{\lambda }}_t$ can be assumed by the interpolation function as shown in Equation (14). Figure 10 illustrates the effect of the initial parameter χ to the yield function and stress–strain curves from the drained triaxial compression simulation. Figure 11 shows the comparison between the test results and the predictions from the MCC, Sand model by Yao et al. [14] and the proposed model. For the analysis, seven soil parameters for loose sand are listed in

Table 1. Soil parameters for drained analyses.

Soil Model	Γ	α	θ	${\mathit{p}}_{\mathit{r}\mathit{e}\mathit{f}}$	λ	κ	${\mathit{M}}_{\mathit{c}}$	ν
MCC and Sand Model [14]	1.97	-	-	-	0.141	0.009	1.305	0.3
Proposed Model	0.864 *	0.0795 *	0.016 *	7.5 *	-	0.009	1.305	0.3

* Fitting parameters from existing data of the NCL (see Figure 7).

[11].

$${\overline{\lambda }}_t={\overline{\lambda }}_{\left(RCL\right)}+\chi (2\alpha -{\overline{\lambda }}_{\left(RCL\right)})$$

(14)

In Figure 11, the predictions from both the MCC model and sand model by Yao at al. [14] overestimate the shear strength of sand. However, the results from [14] provides a better result than the MCCs because of the improvement of the plastic strain calculation through the parameter χ. For the proposed model with both the initial parameters χ and $\overline{\lambda }$, the prediction agrees well with test results.

3.3. Peak Strength of Dense Sand

In [14], the yield function of the MCC model can be used for dense sand without the modifications, i.e., $\chi =0$ in Equations (11) and (12). The goal then becomes identifying the peak in the stress–strain relationship of dense sand. A beautiful derivation for another state parameter ${\chi }_2$ controlling $M_f$ during shearing is introduced to the MCC model in Yao et al. [14], as shown in Equation (15). The additional soil parameter $M_{fmax}$ and $N_d$ are required to work with its state parameter. Although this formulation works, it required the densest condition for sand in the calculation. Furthermore, Yao et al. [14] assumed the slope of the densest condition to be the same value of the slope of the unloading-reloading curve with a constant slope κ. The computation for $M_f$ needs to be changed in order to simplify the model for sand and silty sand.

Been and Jeffries [31] introduced the effective state parameter for sand $\Psi =e-e_{cs}$, where $e$ and $e_{cs}$ are the void ratio and the critical state void ratio at the same mean effective stress, respectively. In [32,33], the relationships of $M_f$ are proposed relating to $\Psi =e-e_{cs}$. Some researchers have hypothesized that the ratio $e/e_{cs}$ could function as a state parameter, e.g., [34]. However, other researchers, such as [35,36], favor using the mean effective stresses ratio $p/p_{cs}$ where $p$ and $p_{cs}$ are the current and critical state mean effective stress. According to a comprehensive review by Lashkari [37], two of the most effective state variables which can provide a reasonable peak strength $M_f$ of granular soil are $\Omega =I_D\mathrm{l}\mathrm{n}(p_{cs}/p)$ and $\Psi =e-e_{cs}$, where $I_D=(e_{max}-e)/(e_{max}-e_{min})$ is the relative density, $e_{max}$ the maximum void ratio and $e_{min}$ the minimum void ratio. The expression for $M_f$ from Ω and Ψ are shown in Equations (16) and (17), respectively. In this research, we utilized the equation of Manzari and Dafalias [34] in Equation (17) in order to achieve the basic model without requiring many extra parameters. The parameter $n_b$ can be determined from the traiaxial compression test of the dense sand.

$$M_f={(M}_{fmax}-M_c)\sqrt{{\chi }_2}+M_c$$

(15)

$$M_f=M_c[1+n_b{I}_D\mathrm{l}\mathrm{n}(p_c/p\left)\right]$$

(16)

$$M_f=M_c[1+n_b〈-\Psi 〉]$$

(17)

Before moving forward to the next section of modeling silty sand, the predictive capability of clean sand for both loose and dense conditions are demonstrated by the comparison between the model prediction and two sets of the test results from Verdugo [38] and Verdugo and Ishihara [39]. Adding $n_b$ to the model, eight soil parameters listed in

Table 2. Soil Parameters for undrained analyses.

Soil Parameter	Γ	α	θ	${\mathit{p}}_{\mathit{r}\mathit{e}\mathit{f}}$	κ	${\mathit{M}}_{\mathit{c}}$	ν	${\mathit{n}}_{\mathit{b}}$
Proposed Model	0.864 *	0.0795 *	0.004 *	3.06 *	0.02	1.265	0.25	1.34

* Fitting parameters from existing data of the CSL.

from [37,38,39] are used in the analyses of dense sand.

Figure 12 and Figure 13 show the comparison of the test results and the calculation of the proposed model. In Figure 12, the results of the undrained triaxial tests at the initial mean effective stress of 490 kPa and three different initial void ratios ($e_0$) of 0.861, 0.883, and 0.910 are demonstrated. Figure 12a depicts the stress path, while Figure 12b illustrates the stress–strain curves. The peak can be observed in the stress–strain curves of $e_0$ = 0.883 and 0.910 on both the test results and the calculation. Figure 13 shows the results of the undrained triaxial tests for a fixed value of the initial void ratio $e_0$ = 0.833 and four different values of the initial mean effective stresses ($p_0)$ of 3000, 2000, 1000, and 100 kPa. The peak occurred on the stress–strain curves of $p_0$ = 3000, 2000, 1000 kPa. The result demonstrates agreement between the peak from the calculation and the test. It is important to notice that when shearing reached the critical state, all stress–strain curves in Figure 13b tended to the same shear strength.

4. Critical State Model for Silty Sand

4.1. Impact of Fine Particles on Sand Properties

Due to the partial participation of fines in the void between sand grains. The equivalent intergranular void ratio (e*) suggested by Thevanayagam and Martin [29] was employed in place of the global void ratio to account for the effect of fines in sand. The equation of e* is expressed as follows:

$$e^{*}=\frac{e+\left(1-\beta \right)FC}{1-\left(1-\beta \right)FC}$$

(18)

where $\beta$ represents the fines influence factor and FC represents the amount of fines content. The reasonable value of $\beta$ is $0\ll \beta \ll 1$. When $\beta =0$, β = 1, and $0<\beta <1$ fines take no action, full action, and partial action in the load bearing mechanism, respectively. The dependence of β on the particle size ratio SR = (D10 of host sand)/(d50 of silt) was discovered using experimental data from Ni et al. [40]. The value of D10 can be evaluated by the grain size distribution test of host sand, while the value of d50 can be determined by a photo of fine grain. The empirical equation for β, presented by Lashkari [9], takes into account the impacts of the particle size ratio SR as well as the roundness of the soil grains and are stated as follows:

$$\beta ={\beta }_0\times \left(FC\right){\times SR}^{-0.2}$$

(19)

where

$${\beta }_0=(1.93+0.04{〈r-1〉}^2)(1+3.2{〈r-1〉}^{2}EXP(-22FC\left)\right)$$

(20)

and $r$ is the ratio of the average roundness of sand (rc) and fines (rf), respectively.

It is generally accepted that the CSL in the e-p space gradually moves downward when FC increases, e.g., [34,41,42,43,44,45]. However, when drawing in terms of the equivalent intergranular void ratio e*, the CSL are close together in a narrow range. As a result, for different FC, a specific curve of the e*-p can be established. In this paper, the e* in Equation (18) is used to model the silty sand. Therefore, the state parameter for silty sand becomes ${\Psi }^{*}=e^{*}-e_{cs}$ and the $M_f$ for silty sand expressed as follows:

$$M_f=M_c[1+n_b〈-{\Psi }^{*}〉]$$

(21)

The amount of FC also affects the stress ratio at the critical state $M_c$. In the work of Lashkari [9], the value of $M_c$ for the specific FC can be determined by the concept of threshold fines content (${FC}_{th}$) expressed in Equation (22). The stress ratio at the critical state for FC ($M_c\left(FC\right)$) can be calculated by Lashkari [9] in Equation (23) with two more parameters, i.e., $M_c\left(0\right)$ where the $M_c$ at zero fines content and ${∆M}_c=M_c\left({FC}_{th}\right)-M_c\left(0\right)$. The equations for the ${FC}_{th}$ by Rahman [27], $M_c\left(FC\right)$ and by Lashkari [9] are presented in Equations (22) and (23).

$${FC}_{th}=0.4\left(\frac{1}{1+EXP(0.50-0.13\times SR)}+\frac{1}{SR}\right)$$

(22)

$$M_c\left(FC\right)=M_c\left(0\right)+\frac{3}{2}\left({∆M}_c\right){\left(\frac{FC}{{FC}_{th}}\right)}^2-\frac{1}{2}\left({∆M}_c\right){\left(\frac{FC}{{FC}_{th}}\right)}^3$$

(23)

As described above, five additional parameters are required in the analysis of silty sand, i.e., rc, rf, SR, $M_c\left(0\right)$, and ${∆M}_c$. The value of $M_c$, $M_c\left(0\right)$, $M_c\left(FC\right)$, and ${∆M}_c$ can be determined from the triaxial compression test of sand.

4.2. Mathematical Modeling of Stress–Strain Behavior in Silty Sands

The proposed model of silty sand was developed from the based sand model of Yao et al. [14] and combined to the silty-sand principles of Lashkari [9] described in the previous sections. The main features of the proposed model compared to the model of Yao et al. are as follows:

(1)

A new curve of the CSL which destroyed the restriction of the straight CSL employed in the based model, Equation (1), and removed the inaccuracy from the CSL in the form of a power function, which are frequently used in the model of granular soil in the literature, Equation (2).

(2)

A modification in the state parameter, Equation (17), that determines the peak of the stress–strain curve during shearing.

By include these variables, Yao et al.’s MCC-based model is able to more accurately represent the behavior of sand and expand its applicability to silty sand with minimal soil parameters. The mathematical description for the elastoplastic modeling can be summarized as follows.

4.2.1. Transformed Stress Tensor ${\tilde{\sigma }}_{ij}$

A transformed stress tensor ${\tilde{\sigma }}_{ij}$ is applied instead of the ordinary stress tensor ${\sigma }_{ij}$ to combine the MCC model with the SMP criterion for clay [13] and sand [14]. Once the ${\sigma }_{ij}$ is given, the ${\tilde{\sigma }}_{ij}$ can be calculated as follows:

$${\tilde{\sigma }}_{ij}=\tilde{p}{\delta }_{ij}+{\tilde{s}}_{ij}=p{\delta }_{ij}+\frac{R}{\sqrt{s_{kl}{s}_{kl}}}{s}_{ij}$$

(24)

where ${\delta }_{ij}$ is the Kronecker’s delta, $p$ and $s_{ij}$ are the mean stress and deviatoric stress tensor calculated from the ordinary stress tensor, respectively, $\tilde{p}$ and ${\tilde{s}}_{ij}$ are the mean stress and deviatoric stress tensor calculated from the transformed stress tensor, respectively, which can be written as

$$\tilde{p}=\frac{1}{3}{\tilde{\sigma }}_{ii}$$

(25)

$${\tilde{s}}_{ij}={\tilde{\sigma }}_{ij}-\tilde{p}{\delta }_{ij}$$

(26)

and $R$ is the radius of the Extended Mises cone which can be calculated using

$$R=\sqrt{\frac{2}{3}}\frac{2I_1}{3\sqrt{\left(I_1{I}_2-I_3\right)\left(I_1{I}_2-9I_3\right)-1}}$$

(27)

where $I_1$, $I_2$ and $I_3$ are the first, second, and third stress invariants, respectively.

4.2.2. Stress–strain Modelling for Elastoplastic Constitutive Model

The total strain increment is decomposed into elastic and plastic components as follows:

$$d{\epsilon }_{ij}=d{\epsilon }_{ij}^e+d{\epsilon }_{ij}^p$$

(28)

According to the Hook’s law, the elastic component $d{\epsilon }_{ij}^e$ can be written as

$$d{\epsilon }_{ij}^e=\frac{1+\nu }{E}d{\sigma }_{ij}-\frac{\nu }{E}d{\sigma }_{kk}{\delta }_{ij}$$

(29)

where ν is Poisson’s ratio and the elastic modulus E is given by

$$E=\frac{3(1-2\nu )(1+e_0)}{\kappa }p$$

(30)

The flow rule is used to explain the plastic component $d{\epsilon }_{ij}^p$ as follows

$$d{\epsilon }_{ij}^p=\Lambda \frac{\partial g}{\partial {\tilde{\sigma }}_{ij}}$$

(31)

where Λ is the proportionality constant and $\frac{\partial g}{\partial {\tilde{\sigma }}_{ij}}$ is the gradient of the plastic potential function in the transformed stress space.

The plastic potential function $g$ and the yield function $f$ of the proposed model are the same as in [14] and are shown in Equations (32) and (33), respectively.

$$f=\ln \frac{\tilde{p}}{{\tilde{p}}_0}+\ln \left(1+\frac{\frac{{\tilde{\eta }}^2}{{\mathrm{M}}_{\mathrm{c}}^2}}{1-\chi \frac{{\tilde{\eta }}^2}{{\mathrm{M}}_{\mathrm{c}}^2}}\right)-\tilde{H}=0$$

(32)

$$g=\ln \frac{\tilde{p}}{{\tilde{p}}_0}+\ln \left(1+\frac{{\tilde{\eta }}^2}{{\mathrm{M}}_{\mathrm{c}}^2}\right)=0$$

(33)

The hardening parameter $\tilde{H}$ incorporating the new curve of the CSL for a single sub-step between the points of the mean effective stress $p_1$ and $p_2$ on the CSL can be calculated as

$$H=\int dH=\int \frac{1}{\overline{c_p}}\frac{M_{f-}^4{\eta }^{*4}}{M_c^4-{\eta }^{*4}}d{\epsilon }_v^p$$

(34)

$${\tilde{\eta }}^{*}=\left\{\begin{array}{c}{ \tilde{\eta }}^{*} for d\tilde{\eta }\ge 0\\ {{ \tilde{\eta }}^{*}}_{max.} for d\tilde{\eta }<0\end{array}\right.$$

(35)

$${\overline{c}}_p=\frac{{\overline{\lambda }}_s-\kappa }{1+e_0}$$

(36)

$${\overline{\lambda }}_s=\alpha +\left\{\sqrt{\mathrm{\theta }+{\left(\alpha \mathit{ln}\left(\frac{p_1}{p_{ref}}\right)\right)}^2}-\sqrt{\mathrm{\theta }+{\left(\alpha \mathit{ln}\left(\frac{p_2}{p_{ref}}\right)\right)}^2}\right\}/ln\left(\frac{p_2}{p_1}\right)$$

(37)

where α, θ and $p_{ref}$ are the fitting parameters of the CSL.

The proportionality constant Λ, $M_c$, $M_f$ and the stress gradient $\frac{\partial g}{\partial {\tilde{\sigma }}_{ij}}$ are expressed as

$$\Lambda =\frac{{\overline{\lambda }}_s-\kappa }{1+e_0}\frac{M_c^4-{{\tilde{\eta }}^{*}}^4}{{M_f}^4-{{\tilde{\eta }}^{*}}^4}\frac{M_c^2+{{\tilde{\eta }}^{*}}^2}{M_c^2-{{\tilde{\eta }}^{*}}^2}\left[\frac{\left(M_c^4-\left(1+2\chi \right){\tilde{\eta }}^2{M}_c^2-{\chi (1-\chi )\tilde{\eta }}^4\right)d\tilde{p}+2\tilde{\eta }{M}_c^{2}d\tilde{q}}{(M_c^2-{{\chi \tilde{\eta }}^{*}}^2)(M_c^2+\left(1-\chi \right){\tilde{\eta }}^2)}\right]$$

(38)

$$M_c=\left\{\begin{array}{c} 6sin\varnothing /(3-sin\varnothing ) for FC=0 \\ M_c\left(FC\right) in Equation \left(22\right) for FC>0\end{array}\right.$$

(39)

where $\tilde{\eta }=\tilde{q}/\tilde{p}$ is the stress ratio and $\tilde{q}=\sqrt{3/2}\sqrt{({\tilde{\sigma }}_{ij}-\tilde{p}{\delta }_{ij})({\tilde{\sigma }}_{ij}-\tilde{p}{\delta }_{ij})}$.

$$M_f=M_c[1+n_b〈-{\Psi }^{*}〉]$$

(40)

where ${\Psi }^{*}=e^{*}-e_{cs}$, $e^{*}$ and $e_{cs}$ can be calculated from Equations (3) and (18), respectively.

$$\frac{\partial g}{\partial {\tilde{\sigma }}_{ij}}=\frac{1}{M_c^2{\tilde{p}}^2+{\tilde{q}}^2}\left[\frac{M_c^2{\tilde{p}}^2-{\tilde{q}}^2}{\left(3\tilde{p}\right)}{\delta }_{ij}+3({\tilde{\sigma }}_{ij}-\tilde{p}{\delta }_{ij})\right]$$

(41)

4.3. Prediction of the Simplified Silty Sand Model

Two sets of silty sand test results [27,45] are compared with the calculation of the proposed model for low FC (FC = 0.15) and high FC (FC = 0.50), as shown in Figure 14 and Figure 15.

In Figure 14, the test results of low FC (FC = 0.15) can be divided into three types [27], i.e., (1) flow ($e_0$ = 0.535 and $p_0$ = 600 kPa), (2) limited flow ($e_0$ = 0.569 and $p_0$ = 1300 kPa), and (3) non-flow ($e_0$ = 0.645 and $p_0$ = 1100 kPa). Although the comparison shows the consistency of the results in the flow case and somewhat consistent in the limited-flow and non-flow cases, the proposed model can still accurately predict the type of the silty sand behavior.

Figure 15 demonstrates the comparison of the test results and the calculation from the proposed model for a high value of FC (FC = 0.50). The results can only be divided into two types, i.e., flow case for the test of ($e_0$ = 0.530, $p_0$ = 500 kPa), and non-flow for the tests of ($e_0$ = 0.630, $p_0$ = 300 kPa) and ($e_0$ = 0.590, $p_0$ = 100 kPa) for both test results and the calculation. The predictions agree well to the test data in the stress–strain curve in Figure 15b, while fairly accurate prediction was found in the stress path in Figure 15a. However, the proposed simplified silty sand model (SSS model) can still capture the correct trending of curves and the correct flow type of silty sand.

5. Conclusions

Adding fines into voids of sand leads to the complex behavior of a whole silty sand. The degree of participation of fines in the load-bearing mechanism can be modeled using the intergranular void ratio e* and the fines influence factor β. Two key factors affecting the value of β are the particle size ratio SR and the roundness of soil grain (both sand and silt). To model the behavior of silty sand precisely, the critical state model has a rise in the complexity leading to twenty or more soil parameters. Therefore, the most critical state for silty sand is not practical in general analysis especially in a limited budget project. Therefore, a simple silty sand model is needed.

The modified Cam-clay model (MCC) including the model in the MCCs family is considered as the simple critical state model in terms of the number of soil parameters. Much modification has been introduced to the MCC to improve its capability, e.g., the SMP yield and failure criterion, effective hardening for both clay and sand, yield function for contractive soil. However, the MCC model and its family lack the link to the curve of the CSL which fits with the behavior of granular soil.

This paper introduces a new equation for the CSL which can be applied to the CSL in the straight-type, the curved type, and the curve connecting by the straight portion. Four fitting parameters are used to control the shape and location of the CSL without extra testing. This feature not only improves the prediction for sand but also extends the capability of the model for silty sand in both drained and undrained analyses. Only eight parameters are required for clean sand (i.e., Γ, α, θ, $p_{ref}$, κ, ν, $M_c$ and $n_b$). Five extra parameters are needed for silty sand, i.e., rc, rf, SR, $M_c\left(0\right)$ and ${∆M}_c$. It is worth to note that the $M_c$ is no longer required for silty sand because $M_c$ can be calculated from $M_c\left(0\right)$ and ${∆M}_c$. Therefore, the total number of soil parameters for silty sand is twelve.

The agreement between the prediction of the proposed model and the existing testing results from distinct researchers confirm that the proposed model is suitable for further analyses of both clean sand and silty sand.