Combined Influences of Water Content and Coarse Grain Content on Shear Strength of Unsaturated Soil Mixture

An interlayer existed between the ballast layer and subgrade in the conventional railway substructure. Considering that the shear strength τ of the interlayer soil was influenced by the changes in the ballast grain content and water content, this aspect was explored in the present study. Monotonic triaxial tests were fulfilled, which considered five coarse grain contents fv and three water contents of fine soil wf. The results showed that the growth in fv contributed to an increment in τ of the soil mixture under both saturation and unsaturation. Conversely, in previous studies, the growth of fv induced an increment in τ under saturation, but a decline in that under unsaturation. This was explained by the competing influences of fv and suction ψ: in previous studies, increasing fv induced a decline in the dry density of the fine soil fraction ρd–f, which contributed to a decline in ψ. When the negative influence of declining ψ outweighed the positive influence of the incrementing fv, the τ of the soil mixture decreased. Meanwhile, modelling of the τ–ψ relationship in the soil mixture with varying fv was performed. This proposed model was examined using the test results from both the present and previous studies, which shows its reasonably good performance.


Introduction
Over the years of trains moving loads, interpenetration between the ballast layer and subgrade occurred, leading to the formation of an interlayer in the conventional railway track (Trinh [1]).This interlayer soil plays an important role for the railway track in (a) spreading the traffic loadings from the ballast layer to the subgrade to avoid failure because of excessive deformation, and (b) eliminating the effect of rainwater seepage in the subgrade (Trinh et al. [2]).Field observation shows a decline in ballast grain content with the incrementing depth of the interlayer.Due to the unstable water content in the field, the water content of the interlayer soil varied frequently.In this circumstance, the shear strength τ of the interlayer soil was significantly affected by the changes in the ballast grain content and water content, which was of great importance for the stability and safety of the railway track.To ensure the good performance of railway track, it appears important to study this aspect in depth.
Numerous studies have examined the influence of water content on τ of the soil mixture (Duong et al. [3]; Qi et al. [4,5]; Wan et al. [6]; Bian et al. [7]).In the saturated state, the pore water pressure was accumulated under the influence of train moving loads, resulting in a decline in τ.With declining water content, the shear strength τ increased owing to an increase in suction.The influence of coarse grain content on the shear strength τ of the mixture was explored in previous studies.Vallejo et al. [8] explored the τ of a fine/coarse soil mixture using two glass beads, with a size of 0.4 and 5 mm.They found that increasing the coarse grain content induced a transition in the soil structure from a fine-grain-supported structure to a coarse-grain-supported structure.Seif EI Dine et al. [9] carried out triaxial tests to explore the τ of a mixture with a gravel and sandy soil matrix.From these tests, an increment in the gravel content from 0% to 35% contributed to a small increment in the τ of the mixture.That is because in this range of gravel content, the τ of the soil mixture was mainly dominated by the sandy matrix.Wang et al. [10] explored the τ of a soil mixture with a wide range of coarse grain content f v (defined by the volumetric proportion of the coarse grains to the mixture), from 0% to 45%.They found a characteristic f v-cha value: τ incremented with the incrementing f v slightly when f v < f v-cha , but largely when f v > f v-cha .Correspondingly, two fabrics in the mixture were identified: a fine-grainsupported fabric when the f v < f v-cha and a coarse-grain-supported fabric when f v > f v-cha .It is noted that the influence of the water content w on the τ of the mixture with these two fabrics was not investigated by Wang et al. [10].Duong et al. [3] explored the influence of w and f v on the τ of the interlayer soil with varying f v = 50~56% using large-scale triaxial tests.From these tests, the coupling effects of w and f v on the τ were observed: increasing f v resulted in an increment in τ at saturation, while a reverse trend was obtained during unsaturation.One should pay attention to the fact that the interlayer soil with f v = 50~56% only has the coarse-grain-supported fabric, without the fine-soil-supported fabric.Up to now, the influences of w and f v on the τ of a soil mixture with two varying fabrics have not been investigated yet.
Different models were proposed for describing the τ of unsaturated soils.Abramento and Carvalho [11] investigated the τ and suction ψ of residual soil from natural slopes, and developed an exponential function between τ and ψψ.Vanapalli et al. [12] predicted the τ of unsaturated soil as a function of suction ψ using the soil water retention curve (SWRC), which was verified by the experimental results on a glacial till.Vilar [13] proposed an empirical hyperbolic formulation for describing the τ-ψ relationship, with one set of measured experimental data required for its application.Han and Vanapalli [14] developed a normalized function to describe the τ-ψ relationship, which incorporated the SWRC.This model was successfully applied to a given soil, such as coarse-grained sands and expansive clays.However, no model describes the τ-ψ relationship for a soil mixture with varying f v values.
This study explored the combined influence of w and f v on the τ of a soil mixture.Monotonic triaxial tests were fulfilled, which considered the f v and water content of fine soil w f values.The experimental results show the variation in the τ of a soil mixture according to the f v and w f .Afterwards, modelling of the shear strength τ of the soil mixture was performed by incorporating the SWRC, which was examined using both the test results from the present study and those from previous studies.

Reconstituted Fine Soil and Micro-Ballast
The interlayer soil was substituted by a mixture of reconstituted fine soil and micro-ballast in the laboratory study.Figure 1 presents a comparison of the grain size distribution (GSD) curves between in situ fine soil and reconstituted fine soil, which shows good agreement.The basis soil properties of the reconstituted fine soil are as follows: the specific gravity Gs = 2.68, the liquid limit wL = 32%, the plasticity index Ip = 20%, the optimum water content of the fine soil wopt-f = 13.7% and the maximum dry density of the fine soil ρdmax-f = 1.82 Mg/m 3 , respectively.The basis soil properties of the reconstituted fine soil are as follows: the specific gravity G s = 2.68, the liquid limit w L = 32%, the plasticity index I p = 20%, the optimum water content of the fine soil w opt-f = 13.7% and the maximum dry density of the fine soil ρ dmax-f = 1.82 Mg/m 3 , respectively.
The GSD curve for the micro-ballast was obtained from that of ballast based on the parallel similitude method (Figure 1), which was consistent with Wang et al. [10].In accordance with Wang et al. [10,15], the parameter f v was employed for quantifying the volume of coarse grains in the soil mixture.
Note that during the compaction, the fine soil fraction was controlled at a constant w opt-f = 13.7% and ρ dmax-f = 1.82 Mg/m 3 .As a result, the target dry density of the mixture ρ d increased with increasing f v .The as-compacted samples with a constant w opt-f = 13.7% and varying f v values were then either dried to w 1-f = 7.0% and w 2-f = 10.6% or wetted to w 3-f = 17.6%, with reference to the protocols developed by Su et al. [16] and Han and Vanapalli [17].In the drying procedure, 1 h of air drying was adopted each time, followed by an equilibration time of at least 7 h, which minimizes the development of cracks and fissures.In the wetting procedure, an increment of 10 g water was employed each time, with the same equilibration time adopted.The drying/wetting procedure adopted was based on the consideration that after compaction at optimum water content, the in situ soil mixture was dried to a lower or wetted to a higher water content and attained an equilibrium state with the external environment (Yang et al. [18,19]).Figure 2 depicts a unique soil water retention curve (SWRC) for a mixture with an unchanged ρ dmax-f = 1.82 Mg/m 3 and different f v = 0%, 20% and 35% (Su et al. [20]).The SWRC for the mixture was described by the van Genuchten [21] model with the parameters a = 4.500 × 10 −4 , n = 1.250, m = 0.570.Note that the parameter a approximates the inverse of the air-entry pressure, the parameter n related to the pore size distribution of the soil and the parameter m controlled the symmetry of the soil water retention curve.More details on the water retention properties of the mixture can be found in Su et al. [20].

Monotonic Triaxial Tests
Monotonic triaxial tests were fulfilled for the determination of the shear strength τ of the mixture with five fv values (0%, 10%, 20%, 35% and 45%) and three wf values (7.0%, 10.6% and 17.6%).The values of the confining pressure σ3 were 30, 60 and 120 kPa based The SWRC for the mixture was described by the van Genuchten [21] model with the parameters a = 4.500 × 10 −4 , n = 1.250, m = 0.570.Note that the parameter a approximates the inverse of the air-entry pressure, the parameter n related to the pore size distribution of the soil and the parameter m controlled the symmetry of the soil water retention curve.More details on the water retention properties of the mixture can be found in Su et al. [20].

Monotonic Triaxial Tests
Monotonic triaxial tests were fulfilled for the determination of the shear strength τ of the mixture with five f v values (0%, 10%, 20%, 35% and 45%) and three w f values (7.0%, 10.6% and 17.6%).The values of the confining pressure σ 3 were 30, 60 and 120 kPa based on the consideration of the traffic loadings and the interlayer soil's depth, which were consistent with those in Wang et al. [10].In the case of w f = 17.6% (S r = 100%), the confining pressure σ 3 was applied overnight, allowing the dissipation of the pore water pressure.This was followed by the shearing process until the end of the tests.In the case of w f = 7.0% and 10.6% (S r = 40% and 60%), the same consolidation time overnight was adopted prior to shearing.All the tests were fulfilled with a low shear rate of 0.1 mm/min.These tests were performed with reference to the protocol adopted by Wang et al. [10].The test ended with either a peak deviator stress that presented when the axial strain ε a < 15%, or when the axial strain ε a = 15% (ASTM D7181-11 [22]).

Experimental Results
Figure 3 shows the shear behaviors of the soil mixture with different f v and w f values under a constant σ 3 = 120 kPa.For the deviator stress q-axial strain εa curves at fv = 0% (Figure 3a), under a given wf, the q increased with an increment in the εa until a peak value, prior to its decline.With increasing wf, the q decreased significantly.This could be attributed to the decrease in suction ψ.Similar observations can be made when fv = 10%-45% (Figure 3b-e).It was found that under a constant wf, no matter whether the conditions were saturated and unsaturated, an increase in fv from 0% to 45% led to an increase in q (Figure 3a-e), which was For the deviator stress q-axial strain ε a curves at f v = 0% (Figure 3a), under a given w f , the q increased with an increment in the ε a until a peak value, prior to its decline.With increasing w f , the q decreased significantly.This could be attributed to the decrease in suction ψ.Similar observations can be made when f v = 10-45% (Figure 3b-e).It was found that under a constant w f , no matter whether the conditions were saturated and unsaturated, an increase in f v from 0% to 45% led to an increase in q (Figure 3a-e), which was due to the reinforcement effect of coarse grains.A similar observation was made by Wang et al. [10].
For further analysis, the deviator stress q at failure was selected, which was defined as the peak deviator stress or the deviator stress at ε a = 15% (ASTM D7181-11 [22]).Figure 4 shows the failure envelops of the soil mixture with different f v = 0%, 10%, 20%, 35% and 45% in the q-p plane, where p is the mean stress.At fv = 0% (Figure 4a), an increase in the wf induced a decrease in the slope and intercept of the failure envelops.Similar observations were made for fv = 10~45% (Figure 4b-e).Correspondingly, the cohesion c and friction angle φ of the soil mixture with different fv and wf were determined, as conducted by Trinh et al. [2] and Wang et al. [10].
Table 1 presents the cohesion c and friction angle φ of the soil mixture with varying wf and fv values.It can be observed that under a given wf, increasing the fv induced a de- and (e) f v = 45% in the q-p plane.
At f v = 0% (Figure 4a), an increase in the w f induced a decrease in the slope and intercept of the failure envelops.Similar observations were made for f v = 10~45% (Figure 4b-e).Correspondingly, the cohesion c and friction angle ϕ of the soil mixture with different f v and w f were determined, as conducted by Trinh et al. [2] and Wang et al. [10].
Table 1 presents the cohesion c and friction angle ϕ of the soil mixture with varying w f and f v values.It can be observed that under a given w f , increasing the f v induced a decline in c, which was owing to the decrease in the fine soil content.With an increase in w f , the c decreased due to the decrease in suction.It was found that under a constant w f , increasing the f v resulted in an increase in ϕ, due to more coarse grains involved during the shearing.When w f decreased, the ϕ was observed to increase.That could be explained by the suction-induced aggregation of the fine soil: according to the findings by Delage et al. [23], a fine matrix fabric was formed on the wet side of the optimum (e.g., w 3-f = 17.6% > w opt-f = 13.7%), while a fine aggregate fabric was obtained on the dry side (e.g., w 1-f = 7.0% and w 2-f = 10.6% < w opt-f = 13.7%).The decrease in w f from 17.6% to 10.6% and 7.0% contributed to a change in the fabric of the fine soil fraction from a fine matrix fabric to a fine aggregate fabric.This change in the fabric resulted in an increase in ϕ.Similarly, suction-induced aggregation of unsaturated loess was observed by Ng et al. [24] using scanning electron microscopy; who reported that an increase in suction up to 40 MPa gave rise to a notable increase in soil stiffness.Figure 5 shows the shear strength τ of the soil mixture with various f v = 50% and 56% and a water content of mixture w = 4% and 12%, which were obtained from Duong et al. [3]. the c decreased due to the decrease in suction.It was found that under a constant wf, increasing the fv resulted in an increase in φ, due to more coarse grains involved during the shearing.When wf decreased, the φ was observed to increase.That could be explained by the suction-induced aggregation of the fine soil: according to the findings by Delage et al. [23], a fine matrix fabric was formed on the wet side of the optimum (e.g., w3-f = 17.6% > wopt-f = 13.7%), while a fine aggregate fabric was obtained on the dry side (e.g., w1-f = 7.0% and w2-f = 10.6% < wopt-f = 13.7%).The decrease in wf from 17.6% to 10.6% and 7.0% contributed to a change in the fabric of the fine soil fraction from a fine matrix fabric to a fine aggregate fabric.This change in the fabric resulted in an increase in φ.Similarly, suctioninduced aggregation of unsaturated loess was observed by Ng et al. [24] using scanning electron microscopy; who reported that an increase in suction up to 40 MPa gave rise to a notable increase in soil stiffness.Figure 5 shows the shear strength τ of the soil mixture with various fv = 50% and 56% and a water content of mixture w = 4% and 12%, which were obtained from Duong et al. [3].It is noted that in situ fine soil and ballast were employed by Duong et al. [3], which were different from the reconstituted fine soil and micro-ballast in the present study (Figure 1).For the fine soil fraction, the GSD curve for the reconstituted fine soil was consistent with that of the in situ fine soil (Figure 1).For the coarse grain fraction, according to the findings of Qi et al. [5], micro-ballast can be adopted as a substitute for the ballast when studying the mechanical behaviors of interlayer soil.Figure 5 shows that with increasing fv, the τ decreased under the unsaturated condition (w = 4%, Sr = 32%) but increased under the saturated condition (w = 12%, Sr = 100%).These observations were different from those in the present study, which found that the increasing fv induced an increase in the shear strength τ under both unsaturated and saturated conditions.Note that It is noted that in situ fine soil and ballast were employed by Duong et al. [3], which were different from the reconstituted fine soil and micro-ballast in the present study (Figure 1).For the fine soil fraction, the GSD curve for the reconstituted fine soil was consistent with that of the in situ fine soil (Figure 1).For the coarse grain fraction, according to the findings of Qi et al. [5], micro-ballast can be adopted as a substitute for the ballast when studying the mechanical behaviors of interlayer soil.Figure 5 shows that with increasing f v , the τ decreased under the unsaturated condition (w = 4%, S r = 32%) but increased under the saturated condition (w = 12%, S r = 100%).These observations were different from those in the present study, which found that the increasing f v induced an increase in the shear strength τ under both unsaturated and saturated conditions.Note that the τ was taken as the deviator stress at failure in Figure 4.This was explained as follows: in Duong et al. [3], a stable dry density of the mixture ρ d = 2.01 Mg/m 3 was adopted.In this case, the growth of f v from 50% to 56% induced a decline in the dry density of the fine soil fraction ρ d-f from 1.33 to 1.17 Mg/m 3 , and hence a decline in suction ψ.
While the negative influence of the declining ψ outweighed the positive influence of the incrementing f v , the τ was observed to decrease.On the contrary, a stable ρ d-f = 1.82 Mg/m 3 was adopted in the present study.This gave rise to a constant suction ψ in the mixture with increasing f v , as shown in Figure 2 (Su et al. [20]).In this case, the increasing f v contributed to an increase in the τ of the soil mixture under both saturation and unsaturation, due to its only positive reinforcement influence.
Comparisons between Duong et al. [3] and the present study show that the τ of the soil mixture was notably influenced by the f v , w and ρ d-f .The combined influence of the w and ρ d-f on the τ could be reflected by the influence of the ψ.The SWRC was incorporated into different models to describe the effect of the ψ on the τ, such as the models by Vanapalli et al. [12] on glacial till and Han and Vanapalli [14] on cohesionless or cohesive soil.To date, no model describes the τ-ψ relationship of the soil mixture with different f v by incorporating the SWRC.
A factor of χψ was widely employed for describing the suction ψ effect on τ (e.g., Equation (2) in Vanapalli et al. [12], Equation (3) in Öberg and Sallfors [25], and Equation ( 6) in Han and Vanapalli [14], Table 2), where χ is the effective stress parameter.According to the findings of Han and Vanapalli [14], the use of χψ can upscale the porescale stress ψ to a macroscopic stress χψ, contributing to the unsaturated shear strength τ.This factor was modified in the present study, with consideration of the following two aspects: (a) χ was considered as the effective degree of saturation S e r , which was consistent with Alonso et al. [27] and Lu et al. [28]; (b) a power relationship between τ and ψ was reported by Abramento and Carvalho [11] and Xu and Sun [26] (Equations ( 1) and (4) in Table 2).In this circumstance, a factor of S e r ψ B was generated and, thus, Equation (8) was obtained below: where A and B are constant parameters.As conducted by Vilar [13] (Equation ( 5)) and Han and Vanapalli [14] (Equation ( 6)), one set of referenced experimental data was adopted in the modelling of unsaturated τ.Substituting a referenced shear strength τ ref , the corresponding effective degree of saturation S e r−ref and suction ψ ref in Equation (8) yields Equation (9): Equation ( 10) was derived by dividing Equation (8) with Equation ( 9), where the parameter A disappears: The empirical model by van Genuchten [21] was employed for the description of the SWRC: where the residual degree of saturation S r−r is taken as zero in the present study, a is a parameter with respect to the air-entry value, and n and m are constant parameters.
Combining Equation (10) with Equation ( 11), Equation ( 12) was deduced: For the application of the proposed Equation ( 12), the information on (i) the shear strength under the saturated condition τ sat , (ii) a referenced shear strength τ ref and the corresponding ψ ref and (iii) the parameters a, n and m with respect to the SWRC were required.To verify the validity of the developed Equation ( 12), the present study, the study by Wang et al. [10] and another three previous studies (Rassam and Williams [29], Khalili et al. [30], Khalili and Zargarbashi [31] in Table 3) were selected.Note that Duong et al. [3] were excluded, due to the fact that only two data points were obtained for a given f v (Figure 5).For each study, the experimental results with varying f v were separated into two groups.The first one was employed for the determination of model parameter B in Equation ( 12) (e.g., f v = 0%, 10% and 35% in the present study and Wang et al. [10]; f v = 37% in Rassam and Williams [29]; Khalili et al. [30]; Khalili and Zargarbashi [31]).On the other hand, the second group was employed for examining the performance of the developed Equation ( 12) with the parameter B previously determined (e.g., f v = 20% and 45% in the present study and Wang et al. [10]; f v = 49% in Rassam and Williams [29]; Khalili and Zargarbashi [31]).
The τ of the soil mixture with varying f v = 0~45% was investigated under different σ 3 = 30, 60 and 120 kPa in the present study (Figure 4) and Wang et al. [10].The experimental results at f v = 0%, 10% and 35% were employed for the determination of parameter B in Equation (12), while those at f v = 20% and 45% were adopted to examine the performance of the proposed Equation (12). Figure 6(a 1 ) compares the measured and the corresponding calculated τ of the mixture with f v = 0%, 10% and 35% under σ 3 = 30 kPa.Equation ( 12) provided satisfactory simulations with the coefficient of determination R 2 = 0.92 using the parameter B = 1.13.Figure 6(b 1 ) presents reasonably good agreement for the measured and calculated τ of the soil mixture at f v = 20% and 45% with the previously determined B = 1.13 (R 2 = 0.90).Similar observations were reported when σ 3 increased to 60 kPa Figure 6 The τ of the soil mixture with varying fv = 0~45% was investigated under different σ3 = 30, 60 and 120 kPa in the present study (Figure 4) and Wang et al. [10].The experimental results at fv = 0%, 10% and 35% were employed for the determination of parameter B in Equation (12), while those at fv = 20% and 45% were adopted to examine the performance of the proposed Equation (12). Figure 6a1 compares the measured and the corresponding calculated τ of the mixture with fv = 0%, 10% and 35% under σ3 = 30 kPa.Equation ( 12) provided satisfactory simulations with the coefficient of determination R 2 = 0.92 using the parameter B = 1.13.Figure 6b1 presents reasonably good agreement for the measured and calculated τ of the soil mixture at fv = 20% and 45% with the previously determined B = 1.13 (R 2 = 0.90).Similar observations were reported when σ3 increased to 60 kPa (Figure 6a2,b2) and 120 kPa (Figure 6a3,b3).Equation ( 12) provided good simulations for the cases where σ3 = 60 and 120 kPa with parameter B = 1.10 and 1.07, respectively.The values of R 2 for these two cases are shown in Figure 6a2,b2 and Figure 6a3,b3, which were at least larger than 0.92.
Rassam and Williams [29] investigated the τ of two unsaturated tailing samples with different fv = 37% and 49% under varying σ3 = 30, 125 and 250 kPa using suction-controlled triaxial tests.The values of  = 0, 20, 60 and 100 kPa were considered.Figure 7 depicts the SWRCs for fv = 37% and 49%, respectively.Figure 8a1 compares the measurements by Rassam and Williams [29] and the calculations using Equation (12) for fv = 37% under σ3 = 30 kPa.Equation ( 12) provided good simulations for the cases where σ 3 = 60 and 120 kPa with parameter B = 1.10 and 1.07, respectively.The values of R 2 for these two cases are shown in Figure 6(a 2 ,b 2 ) and Figure 6(a 3 ,b 3 ), which were at least larger than 0.92.
Rassam and Williams [29] investigated the τ of two unsaturated tailing samples with different f v = 37% and 49% under varying σ 3 = 30, 125 and 250 kPa using suction-controlled triaxial tests.The values of ψ = 0, 20, 60 and 100 kPa were considered.Figure 7 depicts the SWRCs for f v = 37% and 49%, respectively.Equation ( 12) provided good simulations for the cases where σ3 = 60 and 120 k parameter B = 1.10 and 1.07, respectively.The values of R 2 for these two cases are in Figure 6a2,b2 and Figure 6a3,b3, which were at least larger than 0.92.
Materials 2023, 16, x FOR PEER REVIEW 13 of 19 Khalili et al. [30] studied the τ of two unsaturated soil mixtures with varying fv = 16% and 25% using suction-controlled triaxial tests.A constant σ3 = 200 kPa was adopted.The values of  = 0, 100, 200 and 400 were considered.When the fv increased from 16% to 25%, the ρd of the mixture increased from 1.69 to 1.91 Mg/m 3 (Table 3).Figure 9 compares the measurements and the calculations for the SWRCs for fv = 16% and 25% using Equation (11).Note that the parameters a, n and m with respect to the SWRC are presented in Table 3. Figure 10a compares the measurements by Khalili et al. [30] and the simulations using the developed Equation ( 12) for fv = 16%.Equation ( 12) provided satisfactory simulations using the parameter B = 0.77 with R 2 = 0.97.Note that the parameters a, n and m with respect to the SWRC are presented in Table 3. Figure 10a compares the measurements by Khalili et al. [30] and the simulations using the developed Equation ( 12) for f v = 16%.Equation ( 12) provided satisfactory simulations using the parameter B = 0.77 with R 2 = 0.97.   Figure 10b shows the measured and predicted τ for f v = 25% from Equation ( 12) using the aforementioned parameter B = 0.77.The comparison shows good agreement (R 2 = 0.95).Khalili and Zargarbashi [31] investigated the τ of two unsaturated soil mixtures with varying f v = 27% and 51% using multi-stage shear tests.The σ 3 was kept constant at 200 kPa.The values of ψ = 0, 30, 70, 100, 200 and 300 were considered for f v = 27%, while those of ψ = 0, 15, 50, 110, 200 and 300 were considered for f v = 51%.An increment of f v from 27% to 51% induced an increasing ρ d of the mixture from 1.53 to 1.63 Mg/m 3 (Table 3).Figure 11 presents the SWRCs for f v = 27% and 51%, which was fitted to Equation (11) (see the values of parameters a, n, m in Table 3).Figure 12a compares the measurements by Khalili and Zargarbashi [30] and the predictions using the developed Equation ( 12) for fv = 27%.Equation ( 12) provided a good description using the parameter B = 1.10 (R 2 = 0.99).Figure 12a compares the measurements by Khalili and Zargarbashi [30] and the predictions using the developed Equation (12) for f v = 27%.Equation ( 12) provided a good description using the parameter B = 1.10 (R 2 = 0.99).Figure 12a compares the measurements by Khalili and Zargarbashi [30] and the predictions using the developed Equation (12) for fv = 27%.Equation ( 12) provided a good description using the parameter B = 1.10 (R 2 = 0.99).Figure 12b presents satisfactory agreement between the measured and the calculated τ for f v = 51% using Equation ( 12) with the aforementioned parameter B = 1.10 (R 2 = 0.96).
Figure 13 compares the measured and the calculated τ of the soil mixture in all the studies.Good agreement was obtained (R 2 = 0.97).The comparisons show the good performance of the developed Equation ( 12) in describing the τ-ψ relationship of the soil mixture with different f v .Figure 12b presents satisfactory agreement between the measured and the calculated τ for fv = 51% using Equation ( 12) with the aforementioned parameter B = 1.10 (R 2 = 0.96).
Figure 13 compares the measured and the calculated τ of the soil mixture in all the studies.Good agreement was obtained (R 2 = 0.97).The comparisons show the good performance of the developed Equation ( 12) in describing the τ- relationship of the soil mixture with different fv.

Conclusions
Monotonic triaxial tests were fulfilled to investigate the combined influence of fv and w on the τ of the soil mixture.Meanwhile, modelling of the τ of the soil mixture with varying fv was conducted, which was examined using both the experimental results from the present study and those from previous studies.The results allow the following conclusions to be drawn.
It was found that in the present study, increasing the fv led to an increase in the τ of the soil mixture under both saturation and unsaturation.Conversely, Duong et al. [3] reported that incrementally increasing the fv induced an increment in the τ under saturation, but a decline under unsaturation.This was explained by the combined influence of coarse grain content fv and suction : in Duong et al. [3], the increment in fv contributed to a decline in ρd-f and, thus, a decline in .While the negative influence of the declining  outweighed the positive influence of the increasing fv, the τ of the soil mixture decreased.However, an unchanged ρd-f = 1.82 Mg/m 3 was controlled in the present study.The incrementally increasing fv contributed to an increment in the τ under both saturation and unsaturation, due to its only positive reinforcement influence.
Modelling of the τ of the mixture with various fv was performed by incorporating the SWRC.The combined influence of the ρd-f and w on the τ was reflected by the influence of the .This model was examined using the experimental results from both the present study and previous studies.Comparisons between the measurements and the predictions show the satisfactory performance of the developed model for the description of the τ- relationship of the soil mixture with varying fv.

Conclusions
Monotonic triaxial tests were fulfilled to investigate the combined influence of f v and w on the τ of the soil mixture.Meanwhile, modelling of the τ of the soil mixture with varying f v was conducted, which was examined using both the experimental results from the present study and those from previous studies.The results allow the following conclusions to be drawn.
It was found that in the present study, increasing the f v led to an increase in the τ of the soil mixture under both saturation and unsaturation.Conversely, Duong et al. [3] reported that incrementally increasing the f v induced an increment in the τ under saturation, but a decline under unsaturation.This was explained by the combined influence of coarse grain content f v and suction ψ: in Duong et al. [3], the increment in f v contributed to a decline in ρ d-f and, thus, a decline in ψ.While the negative influence of the declining ψ outweighed the positive influence of the increasing f v , the τ of the soil mixture decreased.However, an unchanged ρ d-f = 1.82 Mg/m 3 was controlled in the present study.The incrementally increasing f v contributed to an increment in the τ under both saturation and unsaturation, due to its only positive reinforcement influence.
Modelling of the τ of the mixture with various f v was performed by incorporating the SWRC.The combined influence of the ρ d-f and w on the τ was reflected by the influence of the ψ.This model was examined using the experimental results from both the present study and previous studies.Comparisons between the measurements and the predictions show the satisfactory performance of the developed model for the description of the τ-ψ relationship of the soil mixture with varying f v .

Materials 2023 , 19 Figure 1 .
Figure 1.Grain size distribution curves for fine soil and coarse grains.

Figure 1 .
Figure 1.Grain size distribution curves for fine soil and coarse grains.

Figure 2 .
Figure 2. Measured and calculated SWRC for the soil mixture with different fv values (data from Su et al. [20]).

Figure 2 .
Figure 2. Measured and calculated SWRC for the soil mixture with different f v values (data from Su et al. [20]).

Figure 5 .
Figure 5. Variations in the shear strength with fv under different water contents (after Duong et al. [3]).

Figure 5 .
Figure 5. Variations in the shear strength with f v under different water contents (after Duong et al. [3]).

Figure 7 .
Figure 7. Measured and calculated SWRCs for the soil mixture with different fv values (data from Rassam and Williams [29]).

Figure 7 .
Figure 7. Measured and calculated SWRCs for the soil mixture with different fv values (d Rassam and Williams [29]).

Figure 7 .
Figure 7. Measured and calculated SWRCs for the soil mixture with different f v values (data from Rassam and Williams [29]).

Figure 9 .
Figure 9. Measured and calculated SWRCs for the soil mixture with different fv values (data from Khalili et al. [30]).

Figure 9 .
Figure 9. Measured and calculated SWRCs for the soil mixture with different f v values (data from Khalili et al. [30]).

Figure 11 .
Figure 11.Measured and calculated SWRCs of the mixture with different f v values (data from Khalili and Zargarbashi [31]).

Figure 13 .
Figure 13.Comparisons between measured and calculated shear strength of the soil mixture in all studies.

Author
Contributions: Writing-original draft, Y.S.; Writing-review & editing, J.D.; Formal analysis, B.H. and F.Z.All authors have read and agreed to the published version of the manuscript.

Figure 13 .
Figure 13.Comparisons between measured and calculated shear strength of the soil mixture in all studies.

Table 1 .
Cohesion and friction angle of the soil mixture with varying f v and w f values.

Table 1 .
Cohesion and friction angle of the soil mixture with varying fv and wf values.

Table 2 .
Shear strength equations for unsaturated soils in previous studies.

Table 3 .
Soil properties of the soil mixture in previous studies.