# The Effect of the Nonplastic Fines on the Cyclic Resistance of the Saturated Sand-Fines Mixtures

Authors: Ke Cheng
Journal: Advances in Materials Science and Engineering
 (2023)

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## Abstract

A series of undrained cyclic triaxial tests were carried out to study the nonplastic fines on the cyclic behaviour of saturated sand-fines mixtures. The influences of the fines content (FC), global void ratio (e), and relative density (Dr) were considered. The results show that the cyclic resistance ratio (CRR) of the mixtures firstly decreases and then increases, finally stabilizing with increasing FC when the samples are prepared at a constant Dr. The CRR reaches its minimum when the FC is 20%. However, the CRR decreases and then increases with increasing FC when the samples are prepared at a constant e. The CRR reaches its minimum when the FC is at 30%. The CRR of the mixtures increases with increasing Dr when the FC is given. A single Dr or e cannot describe the dense state of the mixture effectively. A new index, Dr/e, is proposed in this study to describe the dense state of the mixtures. A semiempirical model is proposed to evaluate the CRR of the mixtures using the parameters of Dr/e and FC of the mixture based on the back analysis method. The applicability of the model is verified by the test data from this study and other scholars. In addition, the parameters of Dr/e and FC of the mixture can be obtained from basic geotechnical tests, which is convenient for the application of the model.


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## Body

## 1. Introduction

Liquefaction is a special phenomenon wherein the excess pore pressure of a mass of soil increases and the shear resistance decreases when subjected to a monotonic, cyclic, or dynamic load. The mass undergoes very large unidirectional shear strains; it appears to flow until the shear stress is as low as or lower than the reduced shear resistance [1]. Such a phenomenon leads to the lateral spreading of gently sloping ground, densification and vertical ground settlements, and slope instability. The loss of life and damage to facilities and infrastructure due to liquefaction were very significant in past earthquakes, such as those in Kobe, Wenchuan, and east Japan [2–4]. For these reasons, soil liquefaction is one of the most important and complex issues studied in geotechnical earthquake engineering. Scholars have devoted themselves to the phenomenon’s investigations, and many significant results have been achieved [5–11]. For many years, the phenomenon of liquefaction was thought to be limited to sand. Finer-grained soils were considered incapable of generating high pore pressures, which were commonly associated with soil liquefaction. Therefore, most of the previous research work on soil liquefaction has been focused on relatively clean sands [12–14]. However, natural sandy soils, though, contain fines (passing sieve no. 200, particle size less than 0.074 mm) more or less. Natural sands are always mixed with fines. Such combinations are called silty sands or sand-fines mixtures. Historically, many cases of earthquake-induced liquefaction were observed to occur in silty sands. Field performance data during earthquakes indicated that liquefaction-related failures had also occurred at sites with silty sands and sandy silts. For example, liquefaction occurred in the sands induced by the Chi-Chi earthquake was with very high fines content (FC) [15]. Therefore, the effect of FC on the liquefaction behavior of sand-fines mixtures has elicited particular interest from scholars in the past few decades, and plenty of test data are available in the technical literature. However, many data points and conclusions in the technical literature are seemingly contradictory, and the subject of the effect of FC on the liquefaction behavior of sand-fines mixtures seems to be associated with confusion.For the laboratory test, Amini and Qi [16], and Chang et al. [17], among others, found that the cyclic resistance ratio (CRR) of the sand-fines mixtures increased with the increase of the FC, which meant the FC exerts a positive effect on the liquefaction resistance of the mixtures. However, Troncoso and Verdugo [18], Vaid [19], Chien et al. [20], and Papadopoulou and Tika [21], among others, found that the CRR decreased with the increase of the FC, which meant the FC exerts a negative effect on the liquefaction resistance of the mixtures. In addition, Dash et al. [22], Polito and Martin [23], Xenaki and Athanasopoulos [24], and Thevanayagam [25, 26], among others, held the view that the CRR decreased firstly and then increased with the FC increasing from low to high, which meant the FC exerts a negative effect on the liquefaction resistance of the mixtures in the low FC but exerts a positive effect in the high FC. It is believed that the FC plays an important role in determining the sand structure and the consequent extreme void ratios. These, in turn, have significant influences on the compressibility and liquefaction potential of a sand deposit [27]. There exists a threshold value of fines content, FCth, which is not unique but it may depend on the characteristics of the coarse and fine grains, as well as on the value of the global void ratio (e) of the mixtures. Cabalar et al. [28] studied the compressional behavior of gravel and clay mixture and believed that the FCth is the boundary between fine-domination and coarse-domination of a mixture. Thevanayagam [26] proposed a conceptual framework by introducing intergranular and interfine void ratio to describe the contact state of the mixtures rather than e. Rahman et al. [29, 30] proposed a semiempirical formula to calculate FCth. Cabalar and Hasan [31] and Monkul and Ozden [32] studied the compressional behavior of a sand and clay mixture. The intergranular void ratio was used to express the compressive response of the mixture. Wei and Yang [33, 34] proposed a critical state constitutive model for the silty sands to describe the mechanical behavior and studied the cyclic behavior and liquefaction resistance of silty sands with the presence of initial static shear stress. Furthermore, Yang et al. [35] believed that e was a better parameter to describe the contact and mechanical behavior of the sand-fines mixtures than the skeleton void ratio. It was logically inconsistent with the assumption underlying the concept of the skeleton void ratio that fines make no contribution to the force transfer.With respect to the conflicting conclusions in the technical literature and the confusion about the effect of the FC on the liquefaction behaviour of sand-fines mixtures, a series of undrained cyclic triaxial test investigations were carried out to understand the cyclic behaviours of saturated sand-fines mixtures, especially the CRR. The influences of FC,e, and Dr were considered. A new index, Dr/e, is proposed in this study for the dense state of the sand-fines mixtures. A semiempirical model is proposed to evaluate the CRR of the sand-fines mixtures using the parameters of Dr/e and FC of the fines based on the back analysis method. The test results and proposed model are good references for engineering practice.


## 2. Undrained Cyclic Triaxial Tests

### 2.1. Testing Materials

Fujian sand was used as the host sand material. Fujian sand is one type of clean sand in China, which has been widely used in recent studies. Stone powder was used as nonplastic fines. The mixtures of Fujian sand and stone powder are called FS for convenience. The FC of FS mixtures are 10%, 20%, 30%, 40%, and 50%. The particle size distribution curves of FS mixtures under various FC are schematically illustrated in Figure1. The basic properties of the sand and fines are listed in Table 1.Figure 1
Grain size distribution curves of the sand and soil mixtures.Table 1
Basic properties of the sand and fines used in the test.
TypeGsρmax (g/cm3)ρmin (g/cm3)emaxeminCuCcFujian sand2.641.8881.6160.6340.3985.3680.523Stone powder2.651.6261.2151.1810.630——Note.Gs means specific gravity; Cu means coefficients of uniformity; Cc means coefficients of curvature; emax and emin mean maximum and minimum global void ratio, respectively. ρmax and ρmin mean maximum and minimum dry density, respectively.


### 2.2. Testing Apparatus and Method

The undrained cyclic triaxial tests were carried out using a DYNTTS-type cyclic triaxial test system manufactured by GDS Instruments, UK. The full view of the apparatus is shown in Figure2. The axial force, displacement, confining pressure, and back pressure of the test system can be independently controlled by the equipment, and monotonic or cyclic loads can be applied according to the test requirements. The maximum dynamic load that can be applied in the test system is 10 kN and the frequency is 2 Hz. The axial force and displacement are controlled by a servo motor in the base, which can apply sine, cosine, and other forms of loads or displacements with an accuracy of 0.1 kPa and 0.001 mm. The confining pressure is applied using water. The volume precision of the confining pressure and back pressure controller can reach 1 mm3 and the pressure precision can reach 0.1 kPa (maximum value of 2 MPa). The overall accuracy of the test system is high enough to meet the accuracy requirements of the tests.Figure 2
GDS dynamic triaxial apparatus.The cylindrical specimen has a diameter of 50 mm and height of 100 mm. The specimen was prepared using a dry tamping method, which has previously been used by several scholars to test granular material [36, 37]. The well-mixed sand and fines were tamped into a cylindrical specimen preparation device for four layers in the dynamic triaxial apparatus using the dry tamping method. The presaturation was conducted after the specimen preparation. The presaturation consists of 3 steps: (1) permeating the specimen with CO2 for 30 minutes; (2) flushing with deaired water for 60 minutes; (3) and flushing all water lines. After the presaturation, the back pressure saturation was initiated. Back pressure was gradually applied and the Skempton B-value was checked until it was greater than 0.95, which guaranteed the saturation of the tested sample. The saturated sample was consolidated under an effective target confining pressure until the variable quantity of the back pressure volume was smaller than 5 mm3 every 5 minutes. The sample was tested thereafter. The undrained cyclic triaxial tests were performed in accordance with the ASTM D5311D/5311M standard [38].The influences of the FC,e, and Dr were considered to study the cyclic behaviours of saturated sand-fines mixtures, especially the CRR. The undrained cyclic triaxial tests were conducted at various cyclic stress ratios (CSRs). The CSR of the sample in an isotropic consolidation state is defined in equation (1) [39](1)CSR=σd2σ0′,where σd is the applied sinusoidal cyclic axial stress amplitude.The maximum and minimum global void ratios of FS with various FCs are shown in Figure3. The values were determined according to the ASTM D4253 and D4254 standards [40, 41].Figure 3
Maximum and minimum global void ratios of FS with various FCs.A series of undrained cyclic triaxial test investigations were carried out to understand the cyclic behaviours of saturated sand-fines mixtures, especially the CRR. The influences of the FC,e, and Dr were considered. The detailed undrained cyclic triaxial test conditions are shown in Table 2. The effective confining pressure (σ0′) and loading frequency (f) of the test are 100 kPa and 0.1 Hz, respectively.Table 2
Detailed undrained cyclic triaxial test conditions.
MixtureSandFinesFC (%)Dr (%)eσ0′ (kPa)f (Hz)FSFujian sandStone powder0/10/20/30/40/5050—1000.1FSFujian sandStone powder10/20/30/40/50—0.4741000.1FSFujian sandStone powder2040/50/60—1000.1


## 2.1. Testing Materials

Fujian sand was used as the host sand material. Fujian sand is one type of clean sand in China, which has been widely used in recent studies. Stone powder was used as nonplastic fines. The mixtures of Fujian sand and stone powder are called FS for convenience. The FC of FS mixtures are 10%, 20%, 30%, 40%, and 50%. The particle size distribution curves of FS mixtures under various FC are schematically illustrated in Figure1. The basic properties of the sand and fines are listed in Table 1.Figure 1
Grain size distribution curves of the sand and soil mixtures.Table 1
Basic properties of the sand and fines used in the test.
TypeGsρmax (g/cm3)ρmin (g/cm3)emaxeminCuCcFujian sand2.641.8881.6160.6340.3985.3680.523Stone powder2.651.6261.2151.1810.630——Note.Gs means specific gravity; Cu means coefficients of uniformity; Cc means coefficients of curvature; emax and emin mean maximum and minimum global void ratio, respectively. ρmax and ρmin mean maximum and minimum dry density, respectively.


## 2.2. Testing Apparatus and Method

The undrained cyclic triaxial tests were carried out using a DYNTTS-type cyclic triaxial test system manufactured by GDS Instruments, UK. The full view of the apparatus is shown in Figure2. The axial force, displacement, confining pressure, and back pressure of the test system can be independently controlled by the equipment, and monotonic or cyclic loads can be applied according to the test requirements. The maximum dynamic load that can be applied in the test system is 10 kN and the frequency is 2 Hz. The axial force and displacement are controlled by a servo motor in the base, which can apply sine, cosine, and other forms of loads or displacements with an accuracy of 0.1 kPa and 0.001 mm. The confining pressure is applied using water. The volume precision of the confining pressure and back pressure controller can reach 1 mm3 and the pressure precision can reach 0.1 kPa (maximum value of 2 MPa). The overall accuracy of the test system is high enough to meet the accuracy requirements of the tests.Figure 2
GDS dynamic triaxial apparatus.The cylindrical specimen has a diameter of 50 mm and height of 100 mm. The specimen was prepared using a dry tamping method, which has previously been used by several scholars to test granular material [36, 37]. The well-mixed sand and fines were tamped into a cylindrical specimen preparation device for four layers in the dynamic triaxial apparatus using the dry tamping method. The presaturation was conducted after the specimen preparation. The presaturation consists of 3 steps: (1) permeating the specimen with CO2 for 30 minutes; (2) flushing with deaired water for 60 minutes; (3) and flushing all water lines. After the presaturation, the back pressure saturation was initiated. Back pressure was gradually applied and the Skempton B-value was checked until it was greater than 0.95, which guaranteed the saturation of the tested sample. The saturated sample was consolidated under an effective target confining pressure until the variable quantity of the back pressure volume was smaller than 5 mm3 every 5 minutes. The sample was tested thereafter. The undrained cyclic triaxial tests were performed in accordance with the ASTM D5311D/5311M standard [38].The influences of the FC,e, and Dr were considered to study the cyclic behaviours of saturated sand-fines mixtures, especially the CRR. The undrained cyclic triaxial tests were conducted at various cyclic stress ratios (CSRs). The CSR of the sample in an isotropic consolidation state is defined in equation (1) [39](1)CSR=σd2σ0′,where σd is the applied sinusoidal cyclic axial stress amplitude.The maximum and minimum global void ratios of FS with various FCs are shown in Figure3. The values were determined according to the ASTM D4253 and D4254 standards [40, 41].Figure 3
Maximum and minimum global void ratios of FS with various FCs.A series of undrained cyclic triaxial test investigations were carried out to understand the cyclic behaviours of saturated sand-fines mixtures, especially the CRR. The influences of the FC,e, and Dr were considered. The detailed undrained cyclic triaxial test conditions are shown in Table 2. The effective confining pressure (σ0′) and loading frequency (f) of the test are 100 kPa and 0.1 Hz, respectively.Table 2
Detailed undrained cyclic triaxial test conditions.
MixtureSandFinesFC (%)Dr (%)eσ0′ (kPa)f (Hz)FSFujian sandStone powder0/10/20/30/40/5050—1000.1FSFujian sandStone powder10/20/30/40/50—0.4741000.1FSFujian sandStone powder2040/50/60—1000.1


## 3. Test Results and Analysis

### 3.1. Liquefaction Criterion

With respect to the liquefaction failure of the soil mass under cyclic load, Seed and Lee [5] deemed that the effective stress and shear strength of the soil mass were 0 when liquefaction occur. However, Casagrande [42] focused on the flow characteristics of liquefied soil and suggested that liquefied soil had a steady-state shear strength and that damage was mainly manifested as excessive deformation, displacement, or strain. For the undrained cyclic triaxial laboratory test, the liquefaction criteria of clean sand in an isotropic consolidation state are divided into two types. The first type is the pore pressure criterion. The pore pressure increment (∆u) of the sample is equal to σ0′, which means that the excess pore pressure ratio (Ru=∆u/σ0′) under cyclic load is 1. The second type is the deformation criterion. The single amplitude strain (εSA) of the sample reaches 2%∼3%, and the double amplitude strain (εDA) reaches 5% under cyclic load. Considering that the specimen cannot be completely saturated and the excess pore pressure ratio cannot be 1, εDA = 5% was selected as the initial liquefaction criteria in this study.


### 3.2. Cyclic Resistance Ratio Analysis

The correlations between the CSRs of the FS mixtures with various FCs and the number of initial liquefaction cycles (NL) are shown in Figure 4. The NL is defined as the number of cycles required to lead to the initial liquefaction of the sample under a certain CSR. All samples were prepared at a Dr of 50%. As shown in Figure 4, FC has significant effects on the CSR of the mixtures. The CSR curves significantly differ with increasing FC. The NL of a mixed sample with a certain FC increases with decreasing CSR. The results show that the relationship between the CSR and NL of sandy soil can be described using the following equation [43]:(2)CSR=αNLβ.Figure 4
Correlations between the CSR andNL of FS mixtures at a constant Dr.The test data for the CSR andNL of various mixtures were fitted using this equation. The fitting curves are shown in Figure 4.The CRR is defined as the CSR that leads to the initial liquefaction of the soil at a certainNL. The NL was selected to be 15 in this study. Therefore, the CRR of various sand-fines mixtures is represented by CRR15 in this study. The correlations between the CRR15 and FC of FS mixtures prepared at a constant Dr are shown in Figure 5.Figure 5
Correlations between the CRR15 and FC of various mixtures at a constant Dr.Figure5 shows that the CRR15s of FS mixtures nonlinearly change with increasing FC. The CRR15s firstly decrease and then increase, finally stabilizing with increasing FC.FCth, the threshold fines content, is 20% when the samples are prepared at a constant Dr.The correlations between the CSR of samples prepared at a constant e with various FCs and theNL are shown in Figure 6. The e of the FS mixtures is 0.474. The curves fitted using equation (2) are also shown in Figure 6. The correlations between the CRR15 and FC of FS mixtures prepared at a constant e are shown in Figure 7.Figure 6
Correlations between the CSR andNL of FS mixtures at a constant e.Figure 7
Correlations between the CRR15 and FC of various mixtures at a constant e.As shown in Figure7, the CRR15s of the FS mixtures nonlinearly change with increasing FC. The CRR15 values decrease and then increase with increasing FC. The FCth of FS mixtures prepared at a constant e is 30%.It can be noted that the CRR15 change trend of the mixture samples prepared at a constant e has both similarities and differences compared with that of the samples prepared at a constant Dr. The CRR15 of the two groups of samples first decreases with increasing FC. The reason for this decrease is that the fines occupy the voids between the coarser particles, smoothen the roughness, and reduce the interlocking shear strength of the coarser particles with a low FC. Therefore, the CRR15 first decreases. The CRR15 increases with increasing FC when the samples are prepared at a constant e. The reason is that the fines become the skeleton to bear the load and the sands suspend in the fines. The fines impede the relative movement of the coarser particles and lead to an increase in the CRR15. However, the e of the samples prepared at a constant Dr increases with increasing FC, which leads to a decrease in the CRR15. The CRR15 of the samples prepared at a constant Dr is stable under increasing e and FC. In addition, the FCth of FS mixtures prepared at a constant Dr is 20%. However, the FCth of FS mixtures prepared at a constant e is 30%. The FCth varies with the sample preparation standard. In a word, the correlations between the CRR15 and FC of mixtures prepared at a constant Dr differ from those of samples prepared at a constant e. It is significant to define a constant Dr or e before the test.The correlations between theCSR values of the FS mixture with an FC of 20% under various Dr values and the NL are shown in Figure 8. The curves fitted using equation (2) are also shown in Figure 8. As shown in Figure 8, the Dr affects the CSR of the mixtures. The CSR-NL curves shift upwards with the increase in the Dr from 40% to 60%, which means that the CRR of the mixtures increases with increasing Dr.Figure 8
Correlations between the CSR andNL of various mixtures for various Dr values.As shown in Figure9, the CRR15s of FS mixtures increases almost linearly with the increasing Dr. The e of the mixtures decreases with increasing Dr, which is not conducive to the development of excess pore pressure. Therefore, the CRR of the mixtures increases with increasing Dr.Figure 9
Correlations between the CRR15 and Dr of various mixtures at a constant FC.


### 3.3. Semiempirical Evaluation Model for the CRR

The test results show that the correlations between the CRR15 and FC of sand-fines samples prepared at a constant Dr differ from those of samples prepared at a constant e. The FCth of FS mixtures prepared at a constant Dr is 20%. However, the FCth of FS mixtures prepared at a constant e is 30%. The findings mean a single Dr or e cannot describe the contact state of the mixture effectively. In addition, the CRR15 of the mixture under a certain FC increases with the increasing Dr and decreases with the increasing e. As a result, a new index, Dr/e, is proposed in this study to describe the dense state of the sand-fines mixtures. A semiempirical model is proposed to evaluate the CRR of the sand-fines mixtures using the parameters of Dr/e and FC based on the back analysis method. The model can be described using equation (3), where a, b, c, d, h, and g are the fitting parameters.(3)CRR=a×Dre+b×FC+c×Dre2+d×FC2+h×Dre×FC+g.The test data obtained in this study were fitted using equation (3). The fitting results are shown in Figure 10. The data obtained from the study of Dash et al. [22], Polito and Martin [23], and Sitharam et al. [44] are also fitted using the model to verify its applicability. The fit results of the data from other studies are shown in Figure 10. The fitted parameters for the test data of the study and other references are listed in Table 3.Figure 10
Fitting results based on the semiempirical model. (a) FS. (b) Dash et al. (c) Polito et al. (d) Sitharam et al.
(a)(b)(c)(d)Table 3
Fitting parameters for the test data.
Data originabcdhgGoodness of fit (R2)FS0.01441−0.47054−0.00150.597640.067970.182540.94256Dash et al.0.128920.447390.069843.19421−1.27473−0.028520.93132Polito et al.0.09642−0.38099−0.01038−1.744850.216850.237370.94245Sitharam et al.−0.03493−0.787300.030861.009810.124290.207930.90073It can be seen from Table3 that the goodness of fit for the test data of FS, Dash et al., Polito et al., and Sitharam et al. are 0.94256, 0.93132, 0.94245, and 0.90073, respectively. All the values of the goodness of fit are larger than 0.9, which means the proposed model is applicable for the CRR evaluation of sand-fines mixtures with nonplastic fines. In addition, the parameters of Dr/e and FC of the mixture can be obtained from basic geotechnical tests, which is convenient for the application of the model.


## 3.1. Liquefaction Criterion

With respect to the liquefaction failure of the soil mass under cyclic load, Seed and Lee [5] deemed that the effective stress and shear strength of the soil mass were 0 when liquefaction occur. However, Casagrande [42] focused on the flow characteristics of liquefied soil and suggested that liquefied soil had a steady-state shear strength and that damage was mainly manifested as excessive deformation, displacement, or strain. For the undrained cyclic triaxial laboratory test, the liquefaction criteria of clean sand in an isotropic consolidation state are divided into two types. The first type is the pore pressure criterion. The pore pressure increment (∆u) of the sample is equal to σ0′, which means that the excess pore pressure ratio (Ru=∆u/σ0′) under cyclic load is 1. The second type is the deformation criterion. The single amplitude strain (εSA) of the sample reaches 2%∼3%, and the double amplitude strain (εDA) reaches 5% under cyclic load. Considering that the specimen cannot be completely saturated and the excess pore pressure ratio cannot be 1, εDA = 5% was selected as the initial liquefaction criteria in this study.


## 3.2. Cyclic Resistance Ratio Analysis

The correlations between the CSRs of the FS mixtures with various FCs and the number of initial liquefaction cycles (NL) are shown in Figure 4. The NL is defined as the number of cycles required to lead to the initial liquefaction of the sample under a certain CSR. All samples were prepared at a Dr of 50%. As shown in Figure 4, FC has significant effects on the CSR of the mixtures. The CSR curves significantly differ with increasing FC. The NL of a mixed sample with a certain FC increases with decreasing CSR. The results show that the relationship between the CSR and NL of sandy soil can be described using the following equation [43]:(2)CSR=αNLβ.Figure 4
Correlations between the CSR andNL of FS mixtures at a constant Dr.The test data for the CSR andNL of various mixtures were fitted using this equation. The fitting curves are shown in Figure 4.The CRR is defined as the CSR that leads to the initial liquefaction of the soil at a certainNL. The NL was selected to be 15 in this study. Therefore, the CRR of various sand-fines mixtures is represented by CRR15 in this study. The correlations between the CRR15 and FC of FS mixtures prepared at a constant Dr are shown in Figure 5.Figure 5
Correlations between the CRR15 and FC of various mixtures at a constant Dr.Figure5 shows that the CRR15s of FS mixtures nonlinearly change with increasing FC. The CRR15s firstly decrease and then increase, finally stabilizing with increasing FC.FCth, the threshold fines content, is 20% when the samples are prepared at a constant Dr.The correlations between the CSR of samples prepared at a constant e with various FCs and theNL are shown in Figure 6. The e of the FS mixtures is 0.474. The curves fitted using equation (2) are also shown in Figure 6. The correlations between the CRR15 and FC of FS mixtures prepared at a constant e are shown in Figure 7.Figure 6
Correlations between the CSR andNL of FS mixtures at a constant e.Figure 7
Correlations between the CRR15 and FC of various mixtures at a constant e.As shown in Figure7, the CRR15s of the FS mixtures nonlinearly change with increasing FC. The CRR15 values decrease and then increase with increasing FC. The FCth of FS mixtures prepared at a constant e is 30%.It can be noted that the CRR15 change trend of the mixture samples prepared at a constant e has both similarities and differences compared with that of the samples prepared at a constant Dr. The CRR15 of the two groups of samples first decreases with increasing FC. The reason for this decrease is that the fines occupy the voids between the coarser particles, smoothen the roughness, and reduce the interlocking shear strength of the coarser particles with a low FC. Therefore, the CRR15 first decreases. The CRR15 increases with increasing FC when the samples are prepared at a constant e. The reason is that the fines become the skeleton to bear the load and the sands suspend in the fines. The fines impede the relative movement of the coarser particles and lead to an increase in the CRR15. However, the e of the samples prepared at a constant Dr increases with increasing FC, which leads to a decrease in the CRR15. The CRR15 of the samples prepared at a constant Dr is stable under increasing e and FC. In addition, the FCth of FS mixtures prepared at a constant Dr is 20%. However, the FCth of FS mixtures prepared at a constant e is 30%. The FCth varies with the sample preparation standard. In a word, the correlations between the CRR15 and FC of mixtures prepared at a constant Dr differ from those of samples prepared at a constant e. It is significant to define a constant Dr or e before the test.The correlations between theCSR values of the FS mixture with an FC of 20% under various Dr values and the NL are shown in Figure 8. The curves fitted using equation (2) are also shown in Figure 8. As shown in Figure 8, the Dr affects the CSR of the mixtures. The CSR-NL curves shift upwards with the increase in the Dr from 40% to 60%, which means that the CRR of the mixtures increases with increasing Dr.Figure 8
Correlations between the CSR andNL of various mixtures for various Dr values.As shown in Figure9, the CRR15s of FS mixtures increases almost linearly with the increasing Dr. The e of the mixtures decreases with increasing Dr, which is not conducive to the development of excess pore pressure. Therefore, the CRR of the mixtures increases with increasing Dr.Figure 9
Correlations between the CRR15 and Dr of various mixtures at a constant FC.


## 3.3. Semiempirical Evaluation Model for the CRR

The test results show that the correlations between the CRR15 and FC of sand-fines samples prepared at a constant Dr differ from those of samples prepared at a constant e. The FCth of FS mixtures prepared at a constant Dr is 20%. However, the FCth of FS mixtures prepared at a constant e is 30%. The findings mean a single Dr or e cannot describe the contact state of the mixture effectively. In addition, the CRR15 of the mixture under a certain FC increases with the increasing Dr and decreases with the increasing e. As a result, a new index, Dr/e, is proposed in this study to describe the dense state of the sand-fines mixtures. A semiempirical model is proposed to evaluate the CRR of the sand-fines mixtures using the parameters of Dr/e and FC based on the back analysis method. The model can be described using equation (3), where a, b, c, d, h, and g are the fitting parameters.(3)CRR=a×Dre+b×FC+c×Dre2+d×FC2+h×Dre×FC+g.The test data obtained in this study were fitted using equation (3). The fitting results are shown in Figure 10. The data obtained from the study of Dash et al. [22], Polito and Martin [23], and Sitharam et al. [44] are also fitted using the model to verify its applicability. The fit results of the data from other studies are shown in Figure 10. The fitted parameters for the test data of the study and other references are listed in Table 3.Figure 10
Fitting results based on the semiempirical model. (a) FS. (b) Dash et al. (c) Polito et al. (d) Sitharam et al.
(a)(b)(c)(d)Table 3
Fitting parameters for the test data.
Data originabcdhgGoodness of fit (R2)FS0.01441−0.47054−0.00150.597640.067970.182540.94256Dash et al.0.128920.447390.069843.19421−1.27473−0.028520.93132Polito et al.0.09642−0.38099−0.01038−1.744850.216850.237370.94245Sitharam et al.−0.03493−0.787300.030861.009810.124290.207930.90073It can be seen from Table3 that the goodness of fit for the test data of FS, Dash et al., Polito et al., and Sitharam et al. are 0.94256, 0.93132, 0.94245, and 0.90073, respectively. All the values of the goodness of fit are larger than 0.9, which means the proposed model is applicable for the CRR evaluation of sand-fines mixtures with nonplastic fines. In addition, the parameters of Dr/e and FC of the mixture can be obtained from basic geotechnical tests, which is convenient for the application of the model.


## 4. Conclusions

A series of undrained cyclic triaxial test investigations were carried out to understand the cyclic behaviours of saturated sand-fines mixtures, especially the CRR. The influences of the FC,e, and Dr were considered. A new index, Dr/e, is proposed in this study to describe the dense state of the sand-fines mixtures. A semiempirical model was proposed to evaluate the CRR of saturated sand-fines mixtures based on the test results of this study and other references. The following conclusions can be drawn:(1)
Both the FC andDr (or e) affect the CRR of the sand-fines mixtures when the effect of the FC on the CRR is considered. It is important to define a constant Dr or e before the test. The CRR of the sand-fines mixtures firstly decreases and then increases, finally stabilizing with increasing FC when the samples are prepared at a constant Dr. The CRR reaches its minimum when the FC is 20%. However, the CRR decreases and then increases with increasing FC when the samples are prepared at a constant e. The CRR reaches its minimum when the FC is at 30%. The CRR of the sand-fines mixtures increases with increasing Dr.(2)
A singleDr or e cannot describe the dense state of the mixture effectively. A new index, Dr/e, is proposed in this study to describe the dense state of the sand-fines mixtures. A semiempirical model is proposed to evaluate the CRR of the sand-fines mixtures using the parameters of Dr/e and FC of the mixture based on the back analysis method. The applicability of the model is verified by the test data from this study and other scholars.



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