applied sciences

Article Dilatancy Characteristics and Constitutive Modelling of the Unsaturated Based on Changes in the Mass

Xiao Xu 1, Guoqing Cai 2,*, Zhaoyang Song 2, Jian Li 2, Chongbang Xu 1, Hualao Wang 1 and Chengang Zhao 2,3

1 Bridge and Research Center, Research Institute of Highway Ministry of Transport, Beijing 100088, China; [email protected] (X.X.); [email protected] (C.X.); [email protected] (H.W.) 2 Key Laboratory of Urban Underground Engineering of Ministry of Education, Beijing Jiaotong University, Beijing 100044, China; [email protected] (Z.S.); [email protected] (J.L.); [email protected] (C.Z.) 3 School of , Guilin University of Technology, Guilin 541004, China * Correspondence: [email protected]; Tel.: +86-188-1026-2621

Abstract: Most theories are limited to strain hardening and shrinkage under high compressive stresses, and there are some shortcomings in the selection of suction or degree of saturation as the water content state varies in the constitutive models of unsaturated soil. Based on the triaxial shear tests of unsaturated compacted soil (a of high plasticity) with different water content and confining pressure (low-confining), a shear dilatancy model of unsaturated soil based on the mass water content is proposed in this paper. The influence of the water content on the shear deformation characteristics of the unsaturated soil is analysed. The –dilatancy relationship and the prediction equation of the minimum dilatancy rate of the unsaturated soil under different   water content and different confining pressure are provided. Selecting the mass water content as the state variable, a constitutive model suitable for the dilatancy of unsaturated soil is established. Citation: Xu, X.; Cai, G.; Song, Z.; Li, J.; Xu, C.; Wang, H.; Zhao, C. The method of determining model parameters based on the mass water content is analysed. The Dilatancy Characteristics and applicability of the model is verified by comparisons between the predicted and experimental results. Constitutive Modelling of the Unsaturated Soil Based on Changes Keywords: soil mechanics; unsaturated soil; mass water content; stress–dilatancy relationship; in the Mass Water Content. Appl. Sci. minimum dilatancy rate; constitutive modelling 2021, 11, 4859. https://doi.org/ 10.3390/app11114859

Academic Editor: Fernando M. S. 1. Introduction F. Marques Existing theories of and soil mechanics majorly focus on the strength and defor- mation characteristics of soil under high compressive stresses. The descriptions of stresses Received: 19 April 2021 and deformations during are limited to strain hardening and shrinkage. However, Accepted: 20 May 2021 the soil is often subjected to low stress and tensile stress in engineering problems [1–3], and Published: 25 May 2021 the stresses and deformations during shearing often exhibit peaks and dilatancy. In such cases, the soil mechanics theories that are based on the effect of a high compressive stress Publisher’s Note: MDPI stays neutral will no longer be applicable. Thus, it is necessary to establish an appropriate stress–strain with regard to jurisdictional claims in published maps and institutional affil- relationship model to describe the actual working conditions with an improved accuracy. iations. The that are involved in practice are mostly over- consolidated or compacted unsaturated soils [4]. Presently, the main research ideas on the stress–strain relationship of unsaturated soils—starting from the Barcelona basic model (BBM) proposed by Alonso et al. [5]—which are mostly based on the critical state theoretical framework, can be unified as the Cambridge-type model. It is generally challenging to Copyright: © 2021 by the authors. accurately describe the high peak stress ratio and dilatancy behaviour of the soil using this Licensee MDPI, Basel, Switzerland. type of model, and the research on an appropriate stress–strain model is still lacking. This article is an open access article distributed under the terms and An appropriate dilatancy prediction equation (stress–dilatancy relationship) is the conditions of the Creative Commons key to establishing a stress–strain constitutive model of soil [6]. Initially, the researchers Attribution (CC BY) license (https:// expressed the dilatancy rate as various functional relationships of the current effective 0 creativecommons.org/licenses/by/ stress ratio η and the critical state constant M. An experimental comparison then 4.0/). revealed that the dilatancy equation should also consider the state variables

Appl. Sci. 2021, 11, 4859. https://doi.org/10.3390/app11114859 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 4859 2 of 16

that reflect the influences of the void ratios or the confining pressures. In the past two decades, experimental studies on the mechanical properties of the unsaturated soils have shown that the dilatancy is also a function related to the change in the water content under the unsaturated state [7,8]. However, little attention has been paid to the stress–dilatancy relationship of the unsaturated soil, and the function of the saturated soil is mostly directly adopted when deriving the constitutive model. To better describe the dilatancy characteristics of soil, researchers have introduced multi-mechanism models, such as the boundary surface model [9] and the sub-loading surface model [10], in which the prediction method of the minimum dilatancy rate during the shear deformation process was introduced. Initially, based on the summary of the experimental patterns, Bolton [11], as as Jefferies and Shuttle [12], expressed the minimum dilatancy rate Dmin through their newly defined state parameters (stress state and compacted state), IR (relative dilatancy index), and ψct (volume deformation state parameter), respectively, as follows.

Dmin = α · IR (1)

and Dmin = X · ψct (2) where α and X represent the dilatancy coefficients under the corresponding state param- eters (they can also be referred to as the minimum dilatancy coefficient). Moreover, the minimum dilatancy coefficient has been obtained as a function of the soil fabric, as demon- strated in previous studies [11–13]. The changes in the water content in unsaturated soil will induce changes in the soil fabric, such as the transition from a single-pore distribution to a dual-pore fabric under aggregation [14–16]. However, there are relatively few studies on the relationship between the minimum dilatancy rate and change in the water content. At the same time, the multi-mechanism models of unsaturated soils that have been attempted to be established in recent years have mostly adopted either suction or degree of saturation as the state variable for the water content [14,17]. Degree of saturation couples the impacts of the compaction state (such as the or confining pressure) and water content state simultaneously. Experimental studies have shown that the impact paths of the compaction state and water content state on the dilatancy of the soil are different, or even the opposite. Thus, they should be distinguished in the description. In addition, it is complicated to obtain the suction of the soil at the engineering site. The suction is often high when the soil is dry (specifically when the adsorption effect is dominant) [18], which could reach tens or even hundreds of MPa. In this case, it will no longer be applicable to use suction to characterise the water content state in the constitutive model. Based on triaxial shear tests of the unsaturated compacted soils with different water contents and different confining pressures (low-confining), the variation in shear deforma- tion characteristics of the unsaturated soils under different water contents and different compaction states, as well as the expression of the stress–dilatancy relationship, are ex- plored in this study. The prediction equation of the minimum dilatancy rate based on the compaction state and mass water content during the shear process is provided. A constitu- tive model suitable for describing the shear deformation characteristics of the unsaturated compacted soil is established based on the mass water content. The model can properly simulate the peak stress and dilatancy characteristics of the unsaturated compacted soils with different water contents and confining pressures.

2. Material and Experimentation 2.1. Materials In this study, the test soil sample was prepared artificially by mixing commercial kaolin and river at a dry weight ratio of 70 and 30%, respectively. The prepared soil that was obtained using this method contains rich minerals. The kaolin has a high purity (over 96%), a uniform grain-size distribution, an average grain size of approximately Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 20

2. Material and Experimentation 2.1. Materials In this study, the test soil sample was prepared artificially by mixing commercial ka‐ olin and river sand at a dry weight ratio of 70 and 30%, respectively. The prepared soil that was obtained using this method contains rich clay minerals. The kaolin has a high purity (over 96%), a uniform grain‐size distribution, an average grain size of approxi‐ mately 20.3 μm , and stable properties and structures. In addition, the clay mineral com‐ position in the soil sample is kaolin. As there are no expansive minerals, the influence of expansiveness can be neglected. Figure 1 illustrates the grain–size distribution of sand. Due to the high liquid limit of pure kaolin, the diffusion of water is relatively slow. There‐ fore, river sand is mixed to achieve an improvement, which is a common approach in engineering practices. The selection of this prepared soil can ensure a suitable repeatabil‐ ity and representativeness of the tests. Test material is classified as a silt of high plasticity or MH according to the Unified System (USCS). The physical properties of the tested soil are summarised in Table 1.

Table 1. Physical properties of the tested soil. Appl. Sci. 2021, 11, 4859 3 of 16 Property Index Value

Specific gravity 2.72

20.3Liquidµm, limit and (%) stable properties and structures.57 In addition, the clay mineral composition in the soil sample is kaolin. As there are no expansive minerals, the influence of expansiveness canPlastic be neglected.limit (%) Figure1 illustrates the grain–size38 distribution of sand. Due to the high liquid limit of pure kaolin, the diffusion of water is relatively slow. Therefore, river sand is mixedPlasticity to achieve index an improvement, which is a19 common approach in engineering practices. The selection of this prepared soil can ensure a suitable repeatability and representativeness Maximum dry density (g/cm3) 1.3 of the tests. Test material is classified as a silt of high plasticity or MH according to the UnifiedOptimum Soil water Classification content (%) System (USCS). The36 physical properties of the tested soil are summarised in Table1.

100

80

60

40

Percentage finer, (%) 20

0 0.01 0.1 1 10 Particle size, (mm) FigureFigure 1. 1. Grain-sizeGrain‐size distributiondistribution ofof sand.sand.

Table2.2. Triaxial 1. Physical Test properties under Low of Confining the tested soil.Pressures In the test, the dry density of the soil sample is controlled at 1.2 g/cm3, and the water Property Index Value contents is set at 4, 8, 12, 16, 20, 24, 28, 32, and 36%, respectively. Based on the soil water characteristicSpecific gravity curve (SWCC) measured by the wetting 2.72 contact filter paper method given Liquid limit (%) 57 in the Xiao et al. [18] test (used the same sample, shown in Figure 2,  in the figure Plastic limit (%) 38 d Plasticity index 19 Maximum dry density (g/cm3) 1.3 Optimum water content (%) 36

2.2. Triaxial Test under Low Confining Pressures In the test, the dry density of the soil sample is controlled at 1.2 g/cm3, and the water contents is set at 4, 8, 12, 16, 20, 24, 28, 32, and 36%, respectively. Based on the soil water characteristic curve (SWCC) measured by the wetting contact filter paper method given in the Xiao et al. [18] test (used the same sample, shown in Figure2, ρd in the figure represents the dry density of the sample), the suction of the samples in different water content states can be obtained [19]; the range of suction corresponding to the initial state of the specimen (sample preparation) is 86 to 8248 kPa (corresponding to the water content range from 36 to 4%). The triaxial samples with a diameter and height of 39.1 and 80.0 mm, respectively, are used, and the samples are prepared by implementing the layered compaction method. During the sample preparation process, the prepared wet soil samples with controlled water contents are measured according to the controlled dry density, and they are compacted in eight layers. Each layer is controlled to a height of 10 mm to ensure the uniformity of the sample. The triaxial shear test is conducted by applying GDS (Global Digital Systems Limited) unsaturated triaxial testing of soil (UNSAT), where the volume deformations of the samples are measured by the GDS-HKUST system (Global Digital Systems Limited—Hong Kong University of Science and Technology) [20]. During the test, a hard plastic film is used to separate the ceramic disk and sample to ensure that Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 20

represents the dry density of the sample), the suction of the samples in different water content states can be obtained [19]; the range of suction corresponding to the initial state of the specimen (sample preparation) is 86 to 8248 kPa (corresponding to the water content range from 36 to 4%). The triaxial samples with a diameter and height of 39.1 and 80.0 mm, respectively, are used, and the samples are prepared by implementing the layered compaction method. During the sample preparation process, the prepared wet soil sam‐ ples with controlled water contents are measured according to the controlled dry density, and they are compacted in eight layers. Each layer is controlled to a height of 10 mm to ensure the uniformity of the sample. The triaxial shear test is conducted by applying GDS (Global Digital Systems Limited) unsaturated triaxial testing of soil (UNSAT), where the volume deformations of the samples are measured by the GDS‐HKUST system (Global Digital Systems Limited—Hong Kong University of Science and Technology) [20]. During the test, a hard plastic film is used to separate the ceramic disk and sample to ensure that the test is performed under a constant water content. The top of the sample is connected to the atmosphere through the pore pressure control pipeline, and the pore‐air pressure is maintained at 0 (relative atmospheric pressure) during the test. For the samples with Appl. Sci. 2021, 11, 4859 different water contents, four sets of triaxial shear tests are performed under different4 of 16 confining pressures set to 0 (unconfined), 25, 50, and 100 kPa, respectively. To simulate the failure characteristics of the soils in a similar shallow environment under a constant water content, combined with the Xiao et al. [21] analysis, the confining pressure is applied at a ratethe testof 25 is kPa/20 performed min during under the a constant test. Once water the confining content. The pressure top of has the reached sample the is connected target value,to it the remains atmosphere stable throughfor 10 min the to pore ensure pressure that the control pore‐air pipeline, pressure and in the sample pore-air can pressure be completelyis maintained dissipated at 0 in (relative time; that atmospheric is, the compacted pressure) sample during is thefully test. consolidated For the samples and ex with‐ hausted.different Subsequently, water contents, the shear four test sets is of performed, triaxial shear and teststhe shear are performed rate is set to under 0.1%/min different duringconfining the test pressures [21]. The setshear to 0test (unconfined), is terminated 25, when 50, and the 100 axial kPa, strain respectively. of the sample To simulate reaches the 20%.failure After characteristicsthe shear test is of completed, the soils in the a similar sample shallow is removed environment and its failure under aform constant can be water observed.content, The combined mass change with of the the Xiao sample et al. is [21 weighed,] analysis, and the the confining water content pressure of the is applied sample at a is examinedrate of 25 layer kPa/20 by layer. min duringThe error the does test. not Once exceed the confining 0.5%. pressure has reached the target value, it remains stable for 10 min to ensure that the pore-air pressure in the sample can beConsolidated completely dissipateddrained triaxial in time; tests that of is,the the saturated compacted samples sample are is also fully conducted consolidated sim‐ and ultaneouslyexhausted. for Subsequently,comparison. The the dry shear density test is and performed, water content and the are shear controlled rate is set at 1.2 to 0.1%/ming/cm3 and during36%, respectively, the test [21]. to The prepare shear the test samples is terminated in the when test. The the axialsample strain preparation of the sample method reaches is the20%. same After as that the for shear the test unsaturated is completed, triaxial the test. sample The is tests removed are performed and its failure after vacuum form can be saturationobserved. and The back mass pressure change saturation. of the sample Compared is weighed, with and the the unsaturated water content triaxial of the tests, sample two issets examined of higher layer confining by layer. pressures, The error 300 does and not 400 exceed kPa, are 0.5%. added in the saturated state.

100 90 3 d=1.3 g/cm (%) r 80 3 S d=1.1 g/cm 70 3 d=1.2 g/cm 60 Fitted modified VG model 50 40 30

Degree of saturation, 20 10 0 1 10 100 1,000 10,000 100,000 Suction, s (kPa) FigureFigure 2. Soil 2.‐Soil-waterwater characteristic characteristic curve. curve.

3. Test ResultsConsolidated and Discussion drained triaxial tests of the saturated samples are also conducted simul- 3 3.1. taneouslyShear Behaviours for comparison. with Different The Water dry density Contents and under water Low content Confining are Pressures controlled at 1.2 g/cm and 36%, respectively, to prepare the samples in the test. The sample preparation method is the same as that for the unsaturated triaxial test. The tests are performed after vacuum saturation and back pressure saturation. Compared with the unsaturated triaxial tests, two sets of higher confining pressures, 300 and 400 kPa, are added in the saturated state.

3. Test Results and Discussion 3.1. Shear Behaviours with Different Water Contents under Low Confining Pressures The test results when the water content is 16% and the confining pressure is 50 kPa can be observed in Figure3, which representatively illustrates the shear behaviours of the soil samples with different water contents during the triaxial test. Figure3a shows the deviatoric stress; Figure3b shows the volume strain; Figure3c shows the variation of the stress ratio q/p in the shearing process with the dilatancy rate dεv/dεd. The following parameters are indicated in the figure: w is the water content of the sample (%), σ3 is the confining pressure that is applied during the triaxial test, ε1 is the axial strain during shearing, q is the deviatoric stress, p is the average net stress (the pore-air pressure is ua = 0 during the test), εv is the volume strain, and εd is the deviatoric strain. During the triaxial test, the sample shows volume shrinkage in the beginning (point I in Figure3 indicates the state point of maximum volume shrinkage), and then it transitions to dilatancy (point II corresponds to the peak stress state point). After passing the peak stress, the sample displays softening until the shearing stabilises. When the water content is high, sample softening is not evident. In a saturated state, the sample primarily shows shrinkage, shown Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 20 Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 20

The test results when the water content is 16% and the confining pressure is 50 kPa The test results when the water content is 16% and the confining pressure is 50 kPa can be observed in Figure 3, which representatively illustrates the shear behaviours of the can be observed in Figure 3, which representatively illustrates the shear behaviours of the soil samples with different water contents during the triaxial test. Figure 3a shows the soil samples with different water contents during the triaxial test. Figure 3a shows the deviatoric stress; Figure 3b shows the volume strain; Figure 3c shows the variation of the deviatoric stress; Figure 3b shows the volume strain; Figure 3c shows the variation of the stress ratio q p in the shearing process with the dilatancy rate d v dd . The following stress ratio q p in the shearing process with the dilatancy rate d v dd . The following parameters are indicated in the figure: w is the water content of the sample (%),  3 is parameters are indicated in the figure: w is the water content of the sample (%),  3 is the confining pressure that is applied during the triaxial test, 1 is the axial strain during the confining pressure that is applied during the triaxial test, 1 is the axial strain during shearing, q is the deviatoric stress, p is the average net stress (the pore‐air pressure is shearing, q is the deviatoric stress, p is the average net stress (the pore‐air pressure is ua  0 during the test),  v is the volume strain, and  d is the deviatoric strain. During ua  0 during the test),  v is the volume strain, and  d is the deviatoric strain. During the triaxial test, the sample shows volume shrinkage in the beginning (point I in Figure 3 the triaxial test, the sample shows volume shrinkage in the beginning (point I in Figure 3 indicates the state point of maximum volume shrinkage), and then it transitions to dila‐ indicates the state point of maximum volume shrinkage), and then it transitions to dila‐ tancy (point II corresponds to the peak stress state point). After passing the peak stress, tancy (point II corresponds to the peak stress state point). After passing the peak stress, Appl. Sci. 2021, 11, 4859 the sample displays softening until the shearing stabilises. When the water content is 5high, of 16 the sample displays softening until the shearing stabilises. When the water content is high, sample softening is not evident. In a saturated state, the sample primarily shows shrink‐ sample softening is not evident. In a saturated state, the sample primarily shows shrink‐ age, shown in Figure 4 [22]. In this study, the development and change process of the age, shown in Figure 4 [22]. In this study, the development and change process of the stress and deformation during shearing are discussed. stressin Figure and4 deformation[ 22]. In this during study, shearing the development are discussed. and change process of the stress and deformation during shearing are discussed.

1000 1000  900  3 900  3 800  800  Expansion Ⅱ 700 Ⅱ  Expansion Ⅱ 700 

600 Ⅱ  2

q : w = 16 %, =50 kPa  Ⅰ

600 3  : w=16 %,  =50 kPa 2 (kPa) 500  v 3 Ⅰ q : w = 16 %, =50 kPa p

q 3   : w=16 %,  =50 kPa (kPa) 500 v 3 / p (%) q

q 400  v   / (%)

400 q v

300   Ⅰ 

300 1 Ⅰ  1 200  w = 16 %, =50 kPa

200  Ⅱ 3 Ⅰ w = 16 %, =50 kPa

100  Ⅱ 3 100 Ⅰ 0  0246810 Shrinkage 0 0           0246810  (%) 0246810Shrinkage 0       d / d      (%) 0246810 v d 1 (%) d / d v d 1 (%) (a) (b) (c) (a) (b) (c)

Figure 3. Shear behaviour of the triaxial tests: (a) deviatoric stress; (b) volume strain; (c) stress ratio with the dilatancy rate. FigureFigure 3. 3. ShearShear behaviour behaviour of of the the triaxial triaxial tests: tests: ( (aa)) deviatoric deviatoric stress; stress; ( (bb)) volume volume strain; strain; ( (cc)) stress stress ratio ratio with with the the dilatancy dilatancy rate. rate. 8 8 -4 -4 7  = 200kPa 7  3 = 200kPa Shrinkage 3 -3 Shrinkage 6 -3 6  = 200kPa  3 = 200kPa 5 3 kPa) 5 -2 2 kPa) -2 2 4  = 100kPa (10  = 100kPa 4 3  3 = 100kPa (10  = 100kPa

(%) 3

q 3 v -1 (%)

q 3  v - -1  = 50kPa 3  3  = 50kPa -  = 50kPa  3 = 50kPa 3 2 3 2 0 1 0 Expansion  = 25kPa 1  = 25kPa 3 3 Expansion  = 25kPa  = 25kPa 3 0 3 1 002468101214161820 1 02468101214161820 02468101214161820 02468101214161820   / %    / % (a) Deviatoric stress (b) Volume strain (a) Deviatoric stress (b) Volume strain Figure 4. Shear behaviour at saturation. Figure 4. Shear behaviour at saturation. Figure 4. Shear behaviour at saturation. 3.2. between the Soil Particles in Unsaturated State The effective stress between the unsaturated soil particles can be divided into external normal stress and inter-particle tensile stress caused by internal suction, which can be expressed using the average normal stress:

p0 = p − σs (3)

where p0 denotes the effective stress in the unsaturated state; p can be directly obtained from the force that is applied on the sample surface. As for the inter-particle tensile stress σs (suction stress [23], negative), because the adsorption part is difficult to quantify, the uniaxial tensile strength σtu that is directly measured in the Xiao et al. [18] test (used the s s same sample, shown in Figure5) is applied to approximate the σ (σtu = −σ ). Lu et al. [24] explained that this approximation method can be applied within a certain error range. Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 20

3.2. Effective Stress between the Soil Particles in Unsaturated State The effective stress between the unsaturated soil particles can be divided into exter‐ nal normal stress and inter‐particle tensile stress caused by internal suction, which can be expressed using the average normal stress:

p  p  s (3)

where p denotes the effective stress in the unsaturated state; p can be directly ob‐ tained from the force that is applied on the sample surface. As for the inter‐particle tensile stress  s (suction stress [23], negative), because the adsorption part is difficult to quan‐

tify, the uniaxial tensile strength  tu that is directly measured in the Xiao et al. [18] test s s Appl. Sci. 2021, 11, 4859 (used the same sample, shown in Figure 5) is applied to approximate the  ( tu6 of 16). Lu et al. [24] explained that this approximation method can be applied within a certain error range.

40 (kPa) (kPa) tu 30  Uniaxial tensile strength

20

10

Uniaxial tensile strength , 0 0 10203040 Mass water content, w (%) FigureFigure 5. 5. TensileTensile strength strength characteristic characteristic curve curve as as a a function function of of mass mass water water content. content.

3.3.3.3. ChangeChange ofof thethe SampleSample StateState duringduring thethe ShearingShearing ProcessProcess and and the the Shear Shear Steady Steady State State TheThe tests tests showed showed that that the the water water content content has has a significanta significant impact impact on on the the shear shear deforma- defor‐ tionmation characteristics characteristics of the of soils the soils [25–27 [25–27].]. In the In saturated the saturated state, state, the soil the particles soil particles are dispersed are dis‐ andpersed completely and completely wrapped wrapped by water. by During water. During the shearing the shearing process, process, the particles the particles are nearly are slidingnearly sliding on the sameon the plane, same plane, and there and isthere almost is almost no macroscopic no macroscopic volume volume change. change. As the As waterthe water begins begins to drain, to drain, specifically specifically when when the the water water content content falls falls below below the the plastic plastic limit,limit, the cohesive particles agglomerate, the pore-water shrinks and wraps in the agglomerate, the cohesive particles agglomerate, the pore‐water shrinks and wraps in the agglomerate, or it forms a liquid meniscus at the contact point between the agglomerates. During the or it forms a liquid meniscus at the contact point between the agglomerates. During the shearing process, the relatively large agglomerates or sand particles roll over each other, shearing process, the relatively large agglomerates or sand particles roll over each other, and the soil exhibits a relatively strong dilatancy macroscopically. The soil begins to exhibit and the soil exhibits a relatively strong dilatancy macroscopically. The soil begins to ex‐ properties of coarse-grained soils [28], such as when the water content is 36%. As the water hibit properties of coarse‐grained soils [28], such as when the water content is 36%. As the content continues to decrease, this phenomenon tends to be more evident, such as when water content continues to decrease, this phenomenon tends to be more evident, such as the water content is 16%. Thereafter that, as the water content continues to drop, the ag- when the water content is 16%. Thereafter that, as the water content continues to drop, glomerates begin to lose the adsorbed water and disperse. Subsequently, the agglomerates the agglomerates begin to lose the adsorbed water and disperse. Subsequently, the ag‐ tend to return to the original grain size of kaolin (completely dried). In this case, the shear glomerates tend to return to the original grain size of kaolin (completely dried). In this behaviour is similar to the fully saturated state; that is, the soil is sheared under its own case, the shear behaviour is similar to the fully saturated state; that is, the soil is sheared particle size. However, because it is difficult to reach a completely dry state (water content ofunder 0), the its soilown only particle exhibits size. However, characteristics because that it approximate is difficult to thereach saturated a completely state, dry such state as when(water the content water of content 0), the is soil 4%. only exhibits characteristics that approximate the saturated state,During such as the when shearing the water process, content the mass is 4%. water content of the soil remains unchanged, but the compaction state (specific volume) continues to change. Figure6 shows the changing processes of the sample states on the e − ln p0 plane during the shearing process at three water contents of 4, 16, and 36%, in which e represents the void ratio of the sample. The control state points of the shearing process are clearly labelled in the figure: the maximum volume shrinkage state point, namely the phase transition state point; the peak state point, namely the maximum deviatoric stress point; and the strain localisation state point, which can be referred to as the shear steady state point. At the shear steady state point, an obvious shear plane is formed, and the shearing of the sample tends to be stable. In other words, the stress tends to be constant, and the deformation continues to steadily develop. By comparing with the critical state definition, changes in the sample states during shearing indicate that the shear development is approaching the critical state. However, due to the occurrence of strain localisation, the sample deformation does not meet the critical state in the end; instead, it reaches a shear steady state. Moreover, due to the non-uniformity of the sample deformation, it is difficult to reach the strict critical state in the test. The shear steady state is chosen herein to approximate the critical state. Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 20

During the shearing process, the mass water content of the soil remains unchanged, but the compaction state (specific volume) continues to change. Figure 6 shows the chang‐ ing processes of the sample states on the e ‐ ln p plane during the shearing process at three water contents of 4, 16, and 36%, in which e represents the void ratio of the sample. The control state points of the shearing process are clearly labelled in the figure: the max‐ imum volume shrinkage state point, namely the phase transition state point; the peak state point, namely the maximum deviatoric stress point; and the strain localisation state point, which can be referred to as the shear steady state point. At the shear steady state point, an obvious shear plane is formed, and the shearing of the sample tends to be stable. In other words, the stress tends to be constant, and the deformation continues to steadily develop. By comparing with the critical state definition, changes in the sample states during shear‐ ing indicate that the shear development is approaching the critical state. However, due to the occurrence of strain localisation, the sample deformation does not meet the critical Appl. Sci. 2021, 11, 4859 state in the end; instead, it reaches a shear steady state. Moreover, due to the non‐uniformity7 of 16 of the sample deformation, it is difficult to reach the strict critical state in the test. The shear steady state is chosen herein to approximate the critical state.

strain localisation state 1.6 strain localisation state peak state 1.50 peak state maximum volume shrinkage state maximum volume shrinkage state

w = 16 %, =100 kPa w = 4 %, =100 kPa 3 1.45 3 w = 4 %, =50 kPa 1.5 w = 16 %,3 =50 kPa 3 w = 4 %, =25 kPa

e 3 e w = 16 %,3 =25 kPa 1.40

1.4 1.35 Void ratio, ratio, Void 1.30 Void ratio, Void ratio, 1.3 1.25

1.20 1.2 3.5 4.0 4.5 5.0 5.5 6.0 6.5 3.5 4.0 4.5 5.0 5.5 6.0 lnp (kPa) lnp (kPa) (a) (b)

1.50 strain localisation state peak state maximum volume shrinkage state 1.45

w = 36 %,3 =100 kPa 1.40 w = 36 %,3 =50 kPa e

w = 36 %,3 =25 kPa 1.35

Void ratio, 1.30

1.25

1.20 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 lnp (kPa) (c)

Figure 6. Sample state changes during shearing, w = : (a) 4%; (b) 16%; (c) 36%. Figure 6. Sample state changes during shearing,w = : (a) 4%; (b) 16%; (c) 36%. Figure7 shows the critical state test points and the fitted critical state lines on q-p0 and e − ln p0 planes for the different mass water contents. The results show that the q-p0 and e − ln p0 relationships are almost parallel for the different mass water content states. Therefore, the critical state lines of the unsaturated soils with different mass water contents on the q-p0 and e − ln p0 planes have the same slopes of M and λ as the soil in the saturated state. The critical state lines of the unsaturated soils at the different mass water content states can be expressed as follows:

q = Mp0 + κ(w) (4)

e = Γ(w) − λ ln p0 (5) where M and κ(w) represent the slope and intercept of the critical state line on the q-p0 stress plane, respectively; and λ and Γ(w) represent the slope and intercept of the critical state line of the soils with different water contents on the e − ln p0 plane, respectively. Figure7 lists the detailed parameters of the fitted critical state lines and correlation coefficients, from which it can be observed that the correlation coefficients are relatively high. In Figure7b, the critical state line of the soil first moves upwards (saturated to 36, 16%) and Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 20

Figure 7 shows the critical state test points and the fitted critical state lines on q ‐ p and e − ln p planes for the different mass water contents. The results show that the q ‐ p and e − ln p relationships are almost parallel for the different mass water content states. Therefore, the critical state lines of the unsaturated soils with different mass water contents on the q ‐ p and e − ln p planes have the same slopes of M and  as the soil in the saturated state. The critical state lines of the unsaturated soils at the different mass water content states can be expressed as follows:

q  Mp  w (4)

e  w   ln p (5) where M and  w represent the slope and intercept of the critical state line on the q ‐ p stress plane, respectively; and  and w represent the slope and intercept of the critical state line of the soils with different water contents on the e − ln p plane, respec‐ Appl. Sci. 2021, 11, 4859 8 of 16 tively. Figure 7 lists the detailed parameters of the fitted critical state lines and correlation coefficients, from which it can be observed that the correlation coefficients are relatively high. In Figure 7b, the critical state line of the soil first moves upwards (saturated to 36, 16%)then and downwards then downwards (16 to 4%) (16 withto 4%) the with decrease the decrease in the waterin the water content; content; this is this due is to due the tosignificant the significant impact impact of the of water the water content content on the on soilthe soil shear shear deformation deformation characteristics characteristics that thatwere were previously previously mentioned. mentioned.

1400 1.6

Saturated state 1200 Saturated state w = 4% 1.5 w = 4% e w = 36% w = 36% 1000 w = 16% w = 16% fitted critical state lines 1.4 Fitted critical state lines 800 1.3 (kPa)

q 600 1.2 w critical state lines correlation coefficients w critical state lines correlation coefficients 400 2 2 / % slope intercept R ratio, void state Critical / % slope intercept R Saturated 1.46 17.26 0.9998 saturated -0.092 1.677 0.923 4 1.46 79.03 0.983 1.1 200 4 -0.092 1.778 0.871 16 1.46 107.64 0.965 16 -0.092 1.823 0.870 36 1.46 109.58 0.997 36 -0.092 1.800 0.978 0 1.0 0 100 200 300 400 500 600 700 800 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 p (kPa) (a) (b) lnp (kPa) 0 0 FigureFigure 7. Critical 7. Critical state state for the for thedifferent different water water contents: contents: (a) (qa)‐qp-p ;;( (b)) ee−− ln p ..

3.4.3.4. Stress–Dilatancy Stress—Dilatancy Relationship Relationship In the axisymmetric stress space of the triaxial test, the dilatancy rate can be expressed In the axisymmetricp p stressp spacep of the triaxial test, the dilatancy rate can be ex‐ as D = dε /dε ,p wherep dε and dpε representp the plastic volume strain increment and pressed as Dv  dd d , wherev d dand d represent the plastic volume strain incre‐ plastic deviatoricv straind increment, respectively.v d The critical state theory shows that the peak mentstrength and plastic of the soil deviatoric can be describedstrain increment, by two parts,respectively. the dilatancy The critical and critical state theory state strength; shows thatthat the is, peak it can strength be expressed of the assoil a can stress–dilatancy be described equation.by two parts, The the triaxial dilatancy test results and critical in this statestudy strength; (Figure that3) show is, it that can thebe expressed peak stress as during a stress–dilatancy shearing corresponds equation. toThe the triaxial minimum test results in this study (Figure 3) show that the peak stress during shearing0 corresponds to dilatancy rate. Based on this observation, the relationship between η and Dmin is taken max  theas minimum the starting dilatancy point to explorerate. Based the stress–dilatancy on this observation, equation the ofrelationship the unsaturated between soils  inmax the anddifferent Dmin volumeis taken statesas the andstarting water point content to explore states. the stress–dilatancy equation of the un‐ saturatedFigure soils8 showsin the different the relationships volume states between and thewater maximum content states. effective stress ratio and minimum dilatancy rate for the varying water contents and confining pressures. It can be 0 observed from the figure that η max and Dmin approximate a linear relationship, which can be expressed as follows: 0 η max = ηr + µ · Dmin (6)

where ηr and µ represent the friction and dilatancy parameters, respectively. In Figure8, for the peak state, the representative fitted lines with the mass water contents of 4, 16, and 36% indicate that ηr remains unchanged at the critical state stress ratio M (same value for the saturated and unsaturated states) when the water content changes; moreover, µ is a function of the mass water content. In addition, at the same mass water content, the linear correlation of the test points under the different confining pressures implies that the value of µ is the same when the confining pressure changes. Therefore, the maximum effective stress ratio and minimum dilatancy rate in the unsaturated state develops along the same slope as the confining pressure changes, and the slope changes accordingly as the water content changes. Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 20

Figure 8 shows the relationships between the maximum effective stress ratio and minimum dilatancy rate for the varying water contents and confining pressures. It can be

observed from the figure that max and Dmin approximate a linear relationship, which can be expressed as follows:

max r    Dmin (6)

where r and  represent the friction and dilatancy parameters, respectively. In Figure 8, for the peak state, the representative fitted lines with the mass water contents of 4, 16,

and 36% indicate that r remains unchanged at the critical state stress ratio M (same value for the saturated and unsaturated states) when the water content changes; moreo‐ ver,  is a function of the mass water content. In addition, at the same mass water con‐ tent, the linear correlation of the test points under the different confining pressures im‐ plies that the value of  is the same when the confining pressure changes. Therefore, the Appl. Sci. 2021, 11, 4859 maximum effective stress ratio and minimum dilatancy rate in the unsaturated state9 of de 16‐ velops along the same slope as the confining pressure changes, and the slope changes accordingly as the water content changes.

2.8 Legend identification under different water content（%） 4 8 12 16 20 24 28 32 36 2.6

w = 24% w = 28% 2.4 25kPa

2.2 50kPa  max  2.0 Fitted lines: Saturated 3 =100kPa w = 36%  =25kPa w = 16% 3 1.8 w = 4% Critical state stress ratio 50kPa

1.6 100kPa M 1.4            D min Figure 8. Relationship between the maximum effective stressstress ratioratio andand minimumminimum dilatancydilatancy rate.rate.

Figures9 9 and and 10 10 demonstrate demonstrate the the relationships relationships between between the the dilatancy dilatancy rate rate and and effective effec‐ stresstive stress during during shearing. shearing. Figure Figure9 shows 9 shows the results the results when when the water the water content content is 4, 16, is 4, and 16, 36% and under36% under the different the different confining confining pressures. pressures. At the At same the same mass mass water water content, content, the dilatancy the dila‐ rate–effectivetancy rate–effective stress relationshipstress relationship can be can approximated be approximated with the with same the straight same straight line under line theunder different the different confining confining pressures. pressures. In other In other words, words, the zero the zero dilatancy dilatancy rate pointrate point and and the slopethe slope of the of dilatancythe dilatancy rate rate change change are the are same. the same. Figure Figure 10 shows 10 shows the results the results at a confining at a con‐ pressurefining pressure of 25, 50, of 25, and 50, 100 and kPa 100 under kPa under changing changing water water contents, contents, respectively. respectively. At the At same the Appl. Sci. 2021, 11, x FOR PEER REVIEWconfiningsame confining pressure, pressure, the dilatancy the dilatancy rate can rate be can observed be observed to change to change approximately approximately linearly11 of lin20 ‐ with the effective stress along different slopes under the different water contents, and the early with the effective stress along different slopes under the different water contents, zero dilatancy rate point can be approximated as the same. and the zero dilatancy rate point can be approximated as the same.

  

w = 16%, =25 kPa w = 36%, =25kPa  3  3 w = 36%, =50kPa w = 16%, 3 =50 kPa 3 w = 16%, =100 kPa w = 36%, =100kPa  3  3

d 

d   d

 d /   / d / v

v   / d v 

 = d w = 4%, =25kPa 

= d 3 D D w = 4%, =50kPa = d  3  D  w = 4%, 3 =100kPa

 

   0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 0.00.51.01.52.02.5  = q / p  = q / p  = q / p (a) (b) (c)

Figure 9. Relationship between the dilatancy rate and effective stress ratio for various confining stresses, w = : (a) 16%; Figure 9. Relationship between the dilatancy rate and effective stress ratio for various confining stresses, w = : (a) 16%; (b) 4%; (c) 36%. (b) 4%; (c) 36%. Based on the above analysis, under the different compaction states and water content states, the stress–dilatancy relationship for the unsaturated soil based on the mass water content can be expressed as follows:

M − η0 D = (7) −µ(w)

This is because the change in the water content state of the unsaturated soil has altered the soil fabric, thereby changing the slope of the change in the dilatancy rate with the current effective stress ratio. In other words, it changes the shear deformation Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 20

  

 w = 4%, 3 =100kPa w = 16%, =100kPa   3 w = 36%, =100kPa  3 w = 24%, =100kPa d d  3

 d     / d / d / d v v

 v   w = 4%, =25kPa  3 w = 4%, =50kPa = d = d 3

= d   D D w = 16%, 3 =25kPa D w = 16%, =50kPa  3 w = 36%, 3 =25kPa  w = 36%, 3 =50kPa w = 24%, 3 =25kPa  w = 24%, 3 =50kPa  Appl. Sci. 2021, 11, 4859  10 of 16    0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25  = q / p  = q / p  = q / p (a) (b) (c) characteristics of the soil. However, the compaction state only affects the amplitude of the dilatancy, but not the functional relationship of the dilatancy rate with the effective stress Figure 10. Relationship betweenratio. the The dilatancy above analysisrate and effective indicates stress that ratio the selectionfor various of water the mass contents, water 3 content—a= : (a) 25 kPa; water (b) 50 kPa; (c) 100 kPa. content state parameter that decouples from the compaction state parameter—as the water Appl. Sci. 2021, 11, x FOR PEER REVIEWcontent state variable can deliver a clear meaning and expression. Figure 11 shows12 of the20 dilatancyBased parameters on the aboveµ(w analysis,) for the under different the mass different water compaction contents that states are and obtained water based content on states,the test the results. stress–dilatancy relationship for the unsaturated soil based on the mass water content can be expressed as follows:     w = 4%, =100kPa  3 M  w = 16%, =100kPa  D   3 w = 36%, =100kPa (7)  w  3 w = 24%, =100kPa d d  3

 d     / d / d / d v v

 This is becausev the change in the water content state of the unsaturated soil has al‐ w = 4%, =25kPa  3 w = 4%, =50kPa = d = d 3

= d   D

D w = 16%, =25kPa tered the soil fabric, thereby changing the slope of the change in the dilatancy rate with 3 D w = 16%, 3 =50kPa  w = 36%, =25kPa 3 w = 36%, =50kPa  the current effective stress ratio.3 In other words, it changes the shear deformation charac‐ w = 24%, =25kPa 3  w = 24%, =50kPa  teristics of the soil. However,3 the compaction state only affects the amplitude of the dila‐   tancy, but not the functional relationship of the dilatancy rate with the effective stress 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25  = q / p ratio. The above analysis indicates = q /that p the selection of the mass water = q content—a/ p water (a) content state parameter that(b decouples) from the compaction state parameter—as(c) the wa‐ ter content state variable can deliver a clear meaning and expression. Figure 11 shows the Figure 10. Relationship betweendilatancy the dilatancy parameters rate andw effective for the stress different ratio formass various water water contents contents, thatσ are3 = :obtained (a) 25 kPa; based Figure 10. Relationship between the dilatancy rate and effective stress ratio for various water contents, 3 = : (a) 25 kPa; (b) 50 kPa; (c) 100 kPa. on the test results. (b) 50 kPa; (c) 100 kPa.

 Based on the above analysis, under the different compaction states and water content states, the stress–dilatancy relationship for the unsaturated soil based on the mass water

content  can be expressed as follows:  w = 16%

24%  8% M  4% 28%D  (7)  w 36% This is because the change in the water content state of the unsaturated soil has al‐  -w Test data tered the soil fabric, thereby Polynomial changingfitting (the whole waterthe content slope range) of the change in the dilatancy rate with 2 theDilatancy parameter, current effective stress R =0.99 ratio. In other words, it changes the shear deformation charac‐ Linear fitting of the dilatancy enhancement stage teristics of the soil. However, (water content the decreases compaction from saturation) state only affects the amplitude of the dila‐ 2 tancy, but not the functional R =0.93 relationship of the dilatancy rate with the effective stress  ratio. The0 above 4 8 analysis 1216202428323640 indicates that the selection of the mass water content—a water content state parameter that decouplesMass water contenr, from wthe (%) compaction state parameter—as the wa‐ ter content state variable can deliver a clear meaning and expression. Figure 11 shows the Figure 11. Relationship between the dilatancy parameter µ and mass water content w. dilatancyFigure 11. parameters Relationship  betweenw for thethe dilatancy different parameter mass water  contentsand mass that water are content obtained w . based on4. the Constitutive test results. Modelling 4.1. Yield Function and Compaction State Parameters The above sections analysed the dilatancy characteristics of the unsaturated soils based on the mass water content. Accordingly, the constitutive model of the unsaturated

soils  with the different mass water contents during shearing can be established [29]. Based  w = 16% on the stress–dilatancy relationship (Equation (7)), the yield function can be expressed 24% as follows: 8%  28%" ( + ( )) − ( )# 4% M  p0  1 µ w / µ w F = η0 − 1 + µ(w) · (8) 1 + µ(w) 36% pi 

where µ 6= −1.  -w Test data Polynomial fitting (the whole water content range) In this model, the2 image state points pi (pi represents the stress corresponding to the Dilatancy parameter,  R =0.99 image state) are defined Linear fitting on of each the dilatancy yield enhancement surface stage [ 30 ]; that is, the point on the yield surface (water content decreases from saturation) where the plastic volume R2=0.93 strain increment is zero. In addition, pi is considered as the hardening parameter of the yield surface. 0 4 8 1216202428323640 Mass water contenr, w (%)

Figure 11. Relationship between the dilatancy parameter  and mass water content w .

Appl. Sci. 2021, 11, 4859 11 of 16

To consider the influence of the change of the compaction state (i.e., specific volume) during shearing in the model, the compaction state parameter based on the image state is defined as follows: ψi = e − ec,i (9)

where ec,i represents the void ratio on the critical state line corresponding to pi.

4.2. Minimum Dilatancy Rate and State Parameters

By combining the compaction state parameter ψi, the minimum dilatancy rate of soil can be predicted, as shown in Figure 12, which leads to the following expression:

Dmin = X(w) · ψi (10)

where X(w) denotes the minimum dilatancy rate coefficient. By fitting the test results in Figure 12, the relationship between X(w) and the mass water content can be obtained, Appl. Sci. 2021, 11, x FOR PEER REVIEWas shown in Figure 13. The changing trend with the water content is consistent with14 of the 20 influence of the water content on the shear deformation characteristics of the soil that are described above.

 w = 16%   =25kPa w = 4% 3 w = 36%

min  w = 24% 50kPa

D Saturated  Dmin-i Fitted curve

 100kPa

 3 =25kPa  50kPa

 100kPa Minimum dilatancy rate,

  =25kPa 100kPa 50kPa 3           Image state parameter, i Figure 12. Minimum dilatancy rate and the corresponding state parameter ψ . Figure 12. Minimum dilatancy rate and the corresponding state parameter i i .

4.3.15 Maximum Yield Surface and Hardening Rule Based on the minimum dilatancy rate, the maximum yield surface can be expressed w = 16% as follows: X µ(w) p  1 + µ(w)  1+µ(w) i,max = 1 + D · (11) p0 24% min M 10 8%

where pi,max4%denotes the image stress that corresponds to the maximum yield surface, i.e., the maximum image stress. The model assumes that the hardening and softening rate of the soil is a first-order function of the distance between the current36% state (expressed by the current image stress pi) 5 and the predictable X -w Test data peak state (expressed by p i,max ), i.e., (pi,max − pi). Thus, the hardening rule can be expressed Polynomial fitting as (the follows whole water [17 content]: range) R2=0.97 Saturated Minimum dilatancy rate coefficient, Linear fitting of the dilatanc. y enhancement stage 0  (water content decreasespi from saturation) η 2 . p = H · M exp 1 − · (pi,max − pi) (12) R =0.993 ε M 0 d 0 1020304050 Mass water contenr, w (%) Figure 13. Relationship between the minimum dilatancy rate coefficientX and the mass water contents w.

4.3. Maximum Yield Surface and Hardening Rule Based on the minimum dilatancy rate, the maximum yield surface can be expressed as follows:

w pi,max  1 w 1w  1 Dmin   (11) p  M 

where pi,max denotes the image stress that corresponds to the maximum yield surface, i.e., the maximum image stress.

The model assumes that the hardening and softening rate of the soil is a first‐order function of the distance between the current state (expressed by the current image stress

pi ) and the predictable peak state (expressed by pi,max ), i.e., pi,max  pi . Thus, the hard‐ ening rule can be expressed as follows [17]: Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 20

 w = 16%   =25kPa w = 4% 3 w = 36%

min  w = 24% 50kPa

D Saturated  Dmin-i Fitted curve

 100kPa

 3 =25kPa  50kPa

 100kPa Minimum dilatancy rate, Appl. Sci. 2021, 11, 4859 12 of 16   =25kPa 100kPa 50kPa 3           where H = Hmin + δHψi represents the hardening modulus, Hmin denotes the mini- Image state parameter,i mum hardening modulus, and δH is the variation coefficient based on the compaction

stateFigure parameter. 12. Minimum dilatancy rate and the corresponding state parameter  i .

15

w = 16% X

24% 10 8%

4%

36% 5 X -w Test data Polynomial fitting (the whole water content range) R2=0.97 Saturated Minimum dilatancy rate coefficient, Linear fitting of the dilatancy enhancement stage (water content decreases from saturation) R2=0.993 0 0 1020304050 Mass water contenr, w (%) FigureFigure 13. 13. Relationship between the the minimum minimum dilatancy dilatancy rate rate coefficient coefficientX X andand the the mass mass water water contentscontentsw w. .

4.4.4.3. Elastic Maximum Properties Yield Surface and Hardening Rule TheBased elastic on the properties minimum are dilatancy expressed rate, as follows the maximum in the model yield [ 17surface]: can be expressed as follows:  p0 n G = A (13) pref w p 1w i,max  1 w  (11)  12(1D+minv)  pK = · GM  (14) 3(1 − 2v) p wherewhereG isi,max the denotes elastic shear the image modulus, stressA thatis the corresponds shear modulus to the constant,maximumn yieldis the surface, shear modulusi.e., the maximum exponent, imagepref is thestress. unit reference pressure, K is the elastic bulk modulus, and v is Poisson’s ratio. The model assumes that the hardening and softening rate of the soil is a first‐order 4.5.function Model of Parameters the distance between the current state (expressed by the current image stress p ) and the predictable peak state (expressed by p ), i.e., p  p . Thus, the hard‐ i The current expression for the effective stress of thei,max unsaturatedi,max soil cannoti accurately describeening rule the can change be expressed in the contact as follows stress [17]: between the soil particles that are caused by the physical and chemical effects [31,32]. Consequently, the inter-particle effective stress parameter σs of the unsaturated soil in the model can be preliminarily determined by applying the uniaxial tensile strength test. Parameters M, λ, Γ(w), µ(w), and X(w) can be determined from the shear tests of the unsaturated soils with the different water contents. In addition, based on the accumulation of the test results, they can be provided directly based on the water content from the fitting relationships with the water content. As demonstrated by the tests in this study, the water content varies in a wide range, from saturated to almost completely dried, involving the whole moisture content variation range; the variations of the parameters Γ(w), µ(w), and X(w) exhibit a unimodal function form, which can be fitted with commonly used polynomials. For example, for the parameters µ(w) (Figure 11) and X(w) (Figure 13), the following fitting functions can be obtained:

µ = −1.14 × 10−3w2 + 0.03w − 0.839 (15)

X = 4.3 × 10−4w3 − 0.043w2 + 0.988w + 5.517 (16) Appl. Sci. 2021, 11, 4859 13 of 16

When considering most of the currently available literature on unsaturated soil, en- gineering practice analyses, and some clays with a strong adsorption effect, the research on shear characteristics has mostly only involved (or been expressed in terms of) the en- hancement of dilatancy from saturation to gradual water loss. That is, the stage where the parameter increases with the decrease in the water content. From the test results in this study, it can be observed that the dilatancy enhancement stage of the unsaturated soil—as the water content decreases from saturation—can be fitted with a relatively simple linear form. For example, the parameter µ(w) in Figure 11 and X(w) in Figure 13 can be, respectively, fitted as follows: µ = −0.029w − 0.106 (17) X = −0.376w + 18.750 (18) This type of linear relationship is simpler, satisfying most requirements. The variation of the parameter Γ(w) with the water content, as described in Section 3.3, exhibits similar characteristics as described above. In subsequent studies, by accumulating test results of the different soils, the model parameter prediction equation based on the water content can be further verified or provided, thus facilitating its promotion and application. Parameters Hmin and δH (i.e., hardening modulus H) can be obtained by fitting the deviatoric stress–strain curves in the triaxial tests (that is, trial calculation of the test results). For the elastic parameters, A, n, and pref can be obtained from the unloading-reloading cycle test, and v can be obtained from the local deformation measurement test. In practice, the elastic deformation of the soil during shearing is negligible, and the total deformation can be considered as plastic deformation. Thus, the elastic deformation is neglected in the model calculation. However, starting from the theory, the elastic part is included to complete the model. The initial condition for the calculation—that is, the void ratio e0 of the sample before shearing—can be obtained from the volume deformation record during consolidating and air drainage by applying the confining pressures. Suction or degree of saturation is not directly introduced into the model; instead, the mass water content is used, which is more convenient for practical engineering applications.

5. Model Predictions The capabilities of the proposed model are verified by using the triaxial shear tests of the unsaturated soils under different confining pressures and water contents. For the low-confining pressure triaxial tests performed in this study, the model calculation results and the test results are shown in Figure 14. Table2 lists the model parameters and initial state parameters.

Table 2. Values of the model parameters and initial state parameters.

Mass Water Confining Pressure Interparticle Tensile Stress Initial Condition Critical State Line Content s σ3/kPa σ /kPa e0 λ Γ(w) w/% 4 25 −8 1.248 −0.092 1.778 16 100 −20.4 1.235 −0.092 1.823 36 50 −10 1.236 −0.092 1.800 Stress–Dilatancy Mass Water Confining Pressure Dilatancy Coefficient Hardening Modulus Content Relationship σ3/kPa X(w) w/% M µ(w) Hmin δH 4 25 1.460 −0.723 9.2 300 2 16 100 1.460 −0.629 12.49 140 2 36 50 1.460 −1.216 4.99 160 2 Appl. Sci. 2021, 11, x FOR PEER REVIEW 17 of 20

Appl. Sci. 2021, 11, 4859 14 of 16

    q : w = 16%, =100kPa q:w = 4%, =25kPa 3 3  Experiment   Experiment  Simulation Simulation      

   

(%) v

(kPa)      (%) (kPa) q v    q   v : w = 16%,3 =100kPa  : w = 4%, =25kPa  Experiment   v 3  Simulation Experiment    Simulation      0246810    (%) 024681012 (a)  (b)   (%) 

  q : w = 36%,3 =50kPa  Experiment  Simulation  

  (kPa) (%)

q   v ：  v w = 36%,3 =50kPa  Experiment  Simulation  

 

  0246810  (%) (c)  Figure 14. Comparison between the measured and computed results of the triaxial shear tests for the different confining Figure 14. Comparison between the measured and computed results of the triaxial shear tests for the different confining pressures and water content of the unsaturated soil, (a) w = 4%, σ3 = 25 kPa; (b) w = 16%, σ3 = 100 kPa; (c) w = 36%, pressures and water content of the unsaturated soil, (a) w =4%,  3 =25 kPa; (b) w =16%, 3 =100 kPa; (a) w =36%, 3 =50 kPa. σ3 = 50 kPa. ItIt can be observed that the proposed model can accurately simulate the hardening, maximum shrinkageshrinkage state state point, point, peak peak strength, strength, and and minimum minimum dilatancy dilatancy of the of unsaturatedthe unsatu‐ ratedcompacted compacted soils with soils different with different water contentswater contents and confining and confining pressures. pressures. This is also This the is casealso thefor thecase soil for strength the soil and strength deformation, and deformation, which are more which concerned are more in concerned practice. However, in practice. for However,soil softening for andsoil thesoftening subsequent and the strain subsequent localisation, strain the localisation, model calculation the model results calculation are quite resultsdifferent are from quite the different test points. from This the is test because points. the This is because of the the model—namely foundation of the model—namelystress–dilatancy the relationship stress–dilatancy and the relationship critical state and theoretical the critical methods—are state theoretical all based meth on‐ ods—arethe uniform all deformationbased on the ofuniform the sample deformation during shearing. of the sample The localisation during shearing. of strain The and local the‐ isationshear band of strain phenomenon and the shear in the band shear phenomenon deformation in process the shear should deformation be described process by damage, should bebifurcation, described and by damage, other special bifurcation, theories, and which other are special not covered theories, in thiswhich study. are not covered in this study. 6. Conclusions 6.(1) Conclusions To decouple the effects of the water content state and compaction state and to facilitate engineering applications, the shear deformation characteristics of the unsaturated soils can be simulated based on the mass water content, thereby simplifying and clarifying the problem handling process; Appl. Sci. 2021, 11, 4859 15 of 16

(2) During the shearing process of unsaturated compacted soil under a low confining pressure, obvious dilatancy characteristics can be observed. The soil shows volume shrinkage in the beginning, and then transitions to dilatancy; after passing the peak stress, it displays softening until the shearing stabilises; (3) The influence of the water content on the shear deformation characteristics of the unsaturated soils is analysed. The dilatancy characteristics show a unimodal changing pattern with the water content decreasing from saturation. As the water decreases, the soil particles disperse, agglomerate, and then deagglomerate; the dilatancy char- acteristics macroscopically first move upwards and then downwards; (4) The stress–dilatancy relationship and the prediction equation of the minimum di- latancy rate of the unsaturated soil with different confining pressures and water contents are provided. From two aspects of the peak state point and the whole shear process, it is confirmed that the water content state alters the slope of the change, and the compaction state only affects the amplitude of the dilatancy in the effective stress ratio–dilatancy rate relationship. The minimum dilatancy rate can be directly predicted by the corresponding compaction state and water content state; (5) By selecting the mass water content as the state variable, a constitutive model that is applicable to the dilatancy characteristics of the unsaturated soils is established. The methods for determining the model parameters based on the mass water content are analysed. Considering the stage where the parameter (or dilatancy) increases with the decrease in the water content, which is also the most concerned part of current engineering practice and academic research, the dilatancy parameters can be fitted with a relatively simple linear relationship. Based on the comparison between the triaxial shear test results and the calculation results under different confining pressures and different water contents, the applicability of the established model is verified.

Author Contributions: X.X. conceived and designed the experiments; X.X., G.C. and Z.S. performed the experiments; X.X., J.L. and C.X. analysed the data; H.W. and C.Z. did the analysis and discussion; X.X., G.C. and Z.S. did the modelling and wrote the paper. All authors have read and agreed to the published version of the manuscript. Funding: This work was funded by the National Natural Science Foundation of China (grant number U2034204, 52078031, U1834206), Natural Science Foundation of Beijing (grant number 8202038), and Scientific Foundation of Shanxi Transportation Committee (grant number 2017-1-6). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest.

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