Modeling the Hydro-Mechanical Coupling Behavior of Unsaturated Geotechnical Materials Based on Non-Equilibrium Thermodynamic Theory
Modeling the Hydro-Mechanical Coupling Behavior of Unsaturated Geotechnical Materials Based on...
Yang, Guangchang;Liu, Yang;Chen, Peipei
2020-08-15 00:00:00
applied sciences Article Modeling the Hydro-Mechanical Coupling Behavior of Unsaturated Geotechnical Materials Based on Non-Equilibrium Thermodynamic Theory 1 , 1 2 Guangchang Yang *, Yang Liu and Peipei Chen Department of Civil Engineering, University of Science and Technology Beijing, Beijing 100083, China; yangliu@ustb.edu.cn School of Science, Beijing University of Civil Engineering and Architecture, Beijing 102616, China; chenpeipei@bucea.edu.cn * Correspondence: yang3ang@ustb.edu.cn; Tel.: +86-010-62332957 Received: 19 July 2020; Accepted: 13 August 2020; Published: 15 August 2020 Abstract: A new hydro-mechanical model for unsaturated geotechnical materials based on the non-equilibrium thermodynamic theory is presented in this paper. Common concepts, such as yield criterion and flow rules, are not involved in the constitutive relationships, and are replaced with the thermodynamic concepts of granular temperature, granular entropy, migration coecients, and energy functions. The dissipation system and the migration coecient relationships are theoretically determined, and the constitutive relations of non-elastic deformation and granular temperature are obtained by dissipation relations and thermodynamic identity. Thus, the relationship between dissipation mechanism and macro mechanical behavior can be established by migration coecients and energy functions. The model can reflect the complex hydro-mechanical coupling behavior of unsaturated geotechnical materials subjected to various mechanical paths. The validity of the model is verified by comparing the modeling results with experimental data, and reasonable agreement is achieved. Keywords: unsaturated geotechnical materials; hydro-mechanical coupling; non-equilibrium thermodynamic; constitutive model 1. Introduction The development of hydro-mechanical models of unsaturated geotechnical materials plays an important role in solving the problems in geotechnical engineering [1–3]. Numbers of elastoplastic models were developed for unsaturated geotechnical materials based on critical state theory [4–8]. For example, Alonso et al. [4] proposed a Barcelona basic model (BBM) based on the modified Cam-clay model [9] and derived a loading collapse yield function (LC curve) to describe the collapse phenomenon when wetting. Wheeler and Sivakumar [5] proposed an elastoplastic model similar to BBM while using a dierent LC yield function. As a classical model, BBM has been widely quoted and modified by researchers [10–12]. In addition, Sheng et al. [13] presented a volumetric model, called SFG (Sheng-Fredlund-Gens), for describing the volume change of unsaturated soils during the wetting and drying processes. These models adopt the double-stress variables, mean net stress, and suction, but ignore the eect of saturation on strength and deformation. Lots of experimental data shows that geotechnical materials with dierent saturation present dierent mechanical properties, even when the suction and mean net stress are identical. Some authors introduce saturation into the constitutive model to describe the eect of saturation on unsaturated geotechnical materials characteristics [14–16]. In addition, soil-water characteristic curves (SWCC) have obvious hysteresis characteristics, which has Appl. Sci. 2020, 10, 5668; doi:10.3390/app10165668 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 5668 2 of 13 been considered when establishing the hydro-mechanical coupling models of unsaturated geotechnical materials [13,17–19]. Many existing coupling models of geotechnical materials are based on classical elastoplastic theory, which is also the most widely used constitutive theory. However, most of the constitutive equations used in these models are empirical equations obtained by fitting the experimental results [20], which lack a strict theoretical basis. With the help of thermodynamic theory, some authors established the corresponding hyper-plasticity constitutive models by defining the free energy function and the dissipative potential function to overcome the theoretical defects of the classical elastic-plastic model [21–24]. Although these models still follow the theoretical framework of the classical elastoplastic model, they provide a clear physical meaning for the concepts of yield criteria and flow rules. Recently, based on the physical conservation laws and non-equilibrium thermodynamic theory, the granular solid hydrodynamics modeling method has been successfully applied to the constitutive model of geotechnical materials [25–27]. The constitutive relations of geotechnical materials are established by constructing thermodynamic identity and combining free energy function and energy dissipation law. The dissipative system, which causes the system to deviate from the equilibrium state, can be determined theoretically. Energy dissipation has a great influence on the mechanical behavior of geotechnical materials. It is most convenient and reliable to describe these contents with entropy and thermodynamic language, and it can also ensure that the model does not conflict with the basic physical principles. For multi-scale geotechnical materials, this method is simple and eective and it can grasp the essential characteristics of multi-scale behavior of materials from the perspective of energy dissipation. In this paper, a new hydro-mechanical coupling model for unsaturated geotechnical materials is established based on the non-equilibrium thermodynamic theory. The dissipative system and thermodynamic relationships are determined theoretically, and the constitutive relationships of non-elastic strain and granular temperature are obtained by constructing thermodynamic identity and dissipative relations. The relationship between dissipation mechanism and macro mechanical behavior is established by migration coecients and energy functions. The parameter calibration method is also described, and the ability of the model to capture the hydro-mechanical coupling behavior of unsaturated geotechnical materials is verified by laboratory experimental data. 2. Non-Equilibrium Thermodynamic-Based Model Framework 2.1. Thermodynamic State Variables From the point of view of material structure, geotechnical materials are significantly dierent to ordinary fluids or solids, such as crystals and metals [28]. In addition to macro and micro space levels, geotechnical materials also have a granular level, also called the mesoscopic level. The interaction at this level is the contact force between particles. There is also irregular motion at the mesoscopic level, such as particle sliding, collision, and rolling, which is also called the granular fluctuation motion. Condensed-matter physicists use the concept of granular entropy S or granular temperature T to describe the fluctuation motion at the mesoscopic level by analogy with the irregular motion of molecules at the micro level [29]. Additionally, solids, including geotechnical materials, have a special feature of spatial translational symmetry breaking [30], which will cause the material to add a state variable in relation to deformation, i.e., elastic strain " . It is worth noting that the total strain " is not ij ij directly used as the state variable, because it may contain plastic strain " and cannot correspond to ij the thermodynamic state. n o Thus, , m, S," , S can be taken as thermodynamic state variables for geotechnical materials, ij where , m, and S are density, momentum, and entropy, respectively, which are the common state variables for ordinary fluids and solids. Since unsaturated geotechnical materials are a mixture of solid a a (S), liquid (L), and gas (G), its thermodynamic state variables can be written as bulk density = n , a a a S L G momentum m = v , total entropy S = n S + n S + n S , granular entropy S , and elastic strain L g S G i i Appl. Sci. 2020, 10, 5668 3 of 13 e a a " , where , n , and v are intrinsic density, porosity, and velocity, respectively, and a =[S, L, G]. ij i n o Additionally, its corresponding conjugate variables are , v , T, , T , where is the chemical ca g ca ij potential, T is the temperature, T is thee granular temperature, and is the interaction force between g ij particles, respectively [31]. 2.2. Thermodynamic Relationships The thermodynamic total dierential form of unsaturated geotechnical materials is obtained as follows: . . . . . . e a a w = " + T S + TS + + v m , a 2 [S, L, G], (1) g g ca ij ij i a=S,L,G where w is the total energy density. According to the Equation (1), some relationships can be obtained, @w @w such as = , T = . ij e g @" @S ij Since mass , momentum m, and energy w are conserved quantities, the total mass, the total momentum, and the total energy of a unit of unsaturated geotechnical materials remain constant when there is no heat exchange with the outside world. The dissipative system of the unsaturated geotechnical materials can be obtained by S =0 and S < 0, based on the maximum entropy principle. Therefore, it can be obtained that the dissipation forces of unsaturated geotechnical materials . . L G vS vL vG are , T ,r T," , , and the corresponding dissipation flows are " , I , q /T, , , , ij g i ij g k ij ij ij ij ij ij . . L G va where " , , and are the strain rate, " is the plastic strain rate of solid phase, is the viscous ij ij ij ij ij stress, and q is the heat conduction. The Onsager relationship [32] indicates that the entropy production rate R can be written as the sum of the product of dissipative forces and dissipation flows. . . p k vS vL L vG G R = " + I T + r T + " + + (2) g g ij i ij ij ij ij ij ij ij Moreover, the dissipative flow can be written as a linear function of the dissipative force [33], followed by: 2 . 3 2 3 2 3 6 7 6 0 0 0 0 0 7 ijkl 6 ij 7 6 7 6 kl 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 0
0 0 0 0 6 7 6 7 6 7 I 6 7 g T 6 7 6 7 6 g 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 0 0 0 0 0 6 q 7 6 7 ik 6 7 6 i 7 6 7 r T 6 k 7 6 7 6 7 6 7 6 7 6 7 = S 6 . 7 (3) vS 6 7 6 7 0 0 0 0 0 6 7 6 7 6 7 6 " 7 6 7 6 7 ijkl 6 kl 7 ij 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 L L vL 6 7 6 7 6 7 0 0 0 0 0 6 7 6 7 6 7 6 7 6 7 6 ijkl 7 ij kl 6 7 6 7 6 7 4 5 6 7 6 7 4 vG 5 4 5 0 0 0 0 0 kl ijkl ij The intermediate matrix is a positive definite matrix, also known as the transfer coecient matrix, and should satisfy the reciprocity of Onsager and the Curie principle. However, the interaction between each dissipative flow in Equation (3) is not obvious, so the non-diagonal phases are considered to be 0. 2.3. Energy Dissipation at a Mesoscopic Level Entropy is similar to conserved quantities and has superposition and non-zero properties. However, the dierence is that it can be produced, also known as entropy production. It is found that the behavior of entropy is also eective when deviating from thermodynamic equilibrium [30]. Thus, the total entropy balance equation can be written as: a L LS G GS # = R/T +r f v r # v r # (4) a k k k L k G k k a=S,L,G where # = S/ is specific entropy. Appl. Sci. 2020, 10, 5668 4 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 13 where ϑ=S ρ is specific entropy. Dierent to the irregular motion of micro molecules, the interaction between particles (fluctuation Different to the irregular motion of micro molecules, the interaction between particles motion) is generally non-elastic, which will lead to energy dissipation [34]. If there is no sustained (fluctuation motion) is generally non-elastic, which will lead to energy dissipation [34]. If there is no excitation, the fluctuation motion of particles will decay in the form of macro energy dissipation until sustained excitation, the fluctuation motion of particles will decay in the form of macro energy the fluctuation disappears and reaches equilibrium again. As shown in Figure 1, the dissipation dissipation until the fluctuation disappears and reaches equilibrium again. As shown in Figure 1, the process of geotechnical materials will produce granular entropy S and thermal entropy S. Since the dissipation process of geotechnical materials will produce granular entropy Sg and thermal entropy interaction between particles is non-elastic, entropy production will be generated along with it and S. Since the interaction between particles is non-elastic, entropy production will be generated along finally granular entropy S will decay to thermal entropy S. with it and finally granular entropy Sg will decay to thermal entropy S. Figure 1. The two-stage irreversible dissipation processes of the granular system. Figure 1. The two-stage irreversible dissipation processes of the granular system. The granular entropy balance equation can be obtained by analogy in Equation (4) The granular entropy balance equation can be obtained by analogy in Equation (4) S S # = R /T I (5) ρ g ϑ =R g T g -I g (5) gg g g where ϑ =S ρ is specific granular entropy, Rg is granular entropy production, and Ig is the where # = S / is specific granular entropy, R is granular entropy production, and I is the granular g gg g g g entropy decay rate, which is the entropy production of the system caused by the granular temperature. granular entropy decay rate, which is the entropy production of the system caused by the granular For unsaturated geotechnical materials containing water and gas, the change of saturation will temperature. change the attraction force and cementation between particles, which will cause the recombination For unsaturated geotechnical materials containing water and gas, the change of saturation will movement between particles [17,35], such as the wetting collapsibility phenomenon. Therefore, change the attraction force and cementation between particles, which will cause the recombination the change rate of saturation is also a driving force of particle fluctuation, which should be attributed movement between particles [17,35], such as the wetting collapsibility phenomenon. Therefore, the to the dissipation mechanism at the mesoscopic level. Thus, the change rate of saturation s can be change rate of saturation is also a driving force of particle fluctuation, which should be attributed to regarded as a dissipative force, and the suction s is the corresponding dissipative flow. Considering the dissipation mechanism at the mesoscopic level. Thus, the change rate of saturation s can be the viscosity at the mesoscopic level and the dissipation caused by saturation change, the granular regarded as a dissipative force, and the suction s is the corresponding dissipative flow. Considering entropy production rate R is expressed as: the viscosity at the mesoscopic level and the dissipation caused by saturation change, the granular entropy production rate Rg is expressed as: . vg R = " + sT S (6) g ij g r ij vg R=σε +sT s (6) gij ij g r where sT S is the dissipation at the mesoscopic level due to saturation changes, and when T =0, g r g where sT s is the dissipation at the mesoscopic level due to saturation changes, and when Tg=0, gr . that is, no dissipation occurs at the mesoscopic level, sT S also disappears (multiplied by T to reflect g r g sT s that is, no dissipation occurs at the mesoscopic level, also disappears (multiplied by Tg to gr this phenomenon). reflect this phenomenon). Similarly, the viscous dissipation relationship at the mesoscopic level can also be expressed as: Similarly, the viscous dissipation relationship at the mesoscopic level can also be expressed as: vg g vg g = T " (7) kl σ =Tλε ij ijkl (7) ij g ijkl kl vg g vg g where σ is the viscous stress and λ is the migration coefficient at the mesoscopic level. where is ij the viscous stress and is the ijmigration kl coecient at the mesoscopic level. Additionally, ij ijkl when Addition T =ally 0, ther , when e is no Tg=0, t dissipation here is no at the dissipation mesoscopic at the meso level and scop theic viscous level and dissipation the viscou disappears. s dissipation disappears. The energy density associated with the fluctuation motion of the particles w is expressed as The energy d S ensity associated with the fluctuation motion of the particles wg is expressed as w = b T /2, where b is the material parameter [29]. According to Equation (1), there is g g w=bρ T2 , where b is the material parameter [29]. According to Equation (1), there is () gg @w S S # = = bT (8) g g ∂w SS @T ρρ ϑ == bT gg (8) ∂T g Appl. Sci. 2020, 10, 5668 5 of 13 Thus, # = bT can be obtained. Similarly, defining the tensor as g g ijkl sg = + (9) sg vg ij il jk lk ijkl where and are the migration coecients at the mesoscopic level. sg vg Combining Equations (5)–(9), the motion equation of granular temperature can be obtained: . . . . . " " " " ss gg s s v v r T = c c + c c + c c (10) gg 2 5 3 5 4 5 S S S S 1/ 1/ 1/ 1/ where T = ( ) T , c = ( ) /
, c = ( ) /
, c = ( ) /b, c =
/b. gg s g 2 s sg 3 s vg 4 s 5 2.4. Expression of Stress and Strain The elastic potential energy density function w is generally written as a function of elastic strain, such as the linear elastic model (generalized Hooke’s law). In addition, according to the influence of average eective stress on elastic bulk modulus K and elastic shear modulus G , some authors [29,36] e e constructed the elastic potential energy density functions, which have similar forms to the linear elastic model. By analogy with Hertz contact, Jiang and Liu [29] obtained the elastic potential energy density function of granular materials. 2 1 0.5 2 2 e e e w = B(" ) (" ) + (" ) (11) v v s e e where " is the elastic volume strain, " is the elastic shear strain, is the material parameters, = 5/3 v s for dry sand, and B describes the hardness of the material, the same dimension as the stress. For unsaturated geotechnical materials, the cementation and the matrix suction has a great influence on the contact between particles, and cohesion between particles aects the elastic shear modulus. Following the method of describing the thermal expansion eect of solid [30], adding the contribution of suction s" into Equation (11), and considering the eect of cohesion, Equation (11) can be modified as follows: e 2.5 e 0.5 e 2 e ( ) ( ) ( ) w = B " + " + c " + s" (12) v v s v where =1/; c is the material parameter to reflect the influence of cohesion on the elastic shear modulus and = s , which is similar to Bishop’s eective stress of unsaturated soil. is regarded as the mean eective stress of unsaturated geotechnical materials and is related to ij the elastic potential energy density w . According to Equation (1), there is h i @w e 1.5 0.5 2 0.5 e e e e e = = B (" ) + 0.5(" + c) (" ) +2(" ) e + ss (13) ij ij ij r ij v v s v e ij @" ij Similar to the linear elastic model, can be expressed as: ij e e ( ) = K " + ss +2G e (14) ij e r ij e ij where the K and G can be expressed as: e e 2 3 6 ( ) 7 6 0.5 0.5 s 7 e e 6 7 K = B6(" ) +0.5(" +c) 7 (15) v v 4 e 5 e 0.5 G = B(" ) (16) v Appl. Sci. 2020, 10, 5668 6 of 13 Additionally, the mean eective stress p and shear stress q can be obtained: h i kk 1.5 0.5 2 0 e e e p = = B (" ) +0.5(" +c) (" ) + ss (17) v v s 0.5 e e q = s s = 6B(" +c) " (18) ij ji v s kk where s = is the deviator stress. ij ij ij It can be seen that B is related to the density, and the greater the density, the greater the B. Thus, B = B exp can be obtained according to the compression curve (e-lnp ) of geotechnical (s) materials [26], where e is the void ratio and (s) is the slope of compression curve. According to the migration coecient matrix, the following relationship can be obtained: " = (19) ijkl kl ij When the granular temperature T = 0, the plastic strain will not occur. For isotropic conditions, can be defined as: ijkl s v s = T + T (20) ijkl g il jk g ij lk 2G 3K 6G e e e where and are migration coecients, is an exponential parameter, and = 0.5 when it is v s assumed that the critical elastic shear strain is independent of the shear strain rate [25]. Combining Equations (14), (19), and (20), there is . ss p r e e " = = T " + + T e (21) ijkl kl v g ij s g ij ij . . e e The expressions of elastic volume strain rate " and elastic shear strain rate " can then obtained: v s . . ss 1/ e " = " 3c T " + , c = / (22) v 1 g 1 v s v s v . . 1/ " = " T " (23) s g s s s 2.5. Soil-Water Characteristic The soil-water characteristic curve (SWCC) generally takes the form of the vG model [37], h i s = 1+ , where a, m, and n are the parameters associated with the curve. Since the saturation is also aected by the void ratio e, some authors [38,39] proposed modified SWCC models by adding the deformation factor into the vG model. For example, Tarantion [39] proposed a modified SWCC model to describe the eect of void ratio on soil-water characteristic behavior. He found that the pore water void ratio e can be described by a power function of suction s, e = a s , where a and b are w w 1 1 1 1/b related to the intercept and slope of the lns lne curve. The factor is added to indicate the eect of deformation, and the modified SWCC model is given as: 8 9 2 3 b /n ! n 1 1/b > > > > 6 7 < 6 7 = 6 7 s = >1+6 s7 > (24) > 4 5 > : ; where a , b , and n are the fitting parameters and m =b /n is equal to the parameter m in vG model. 1 1 1 Figure 2 shows the main drying and wetting surfaces of Speswhite kaolin in e lns s space based on the above modified SWCC model (Equation (24)). When the suction remains constant, the degree of saturation will decrease with the increase of void ratio, and when the degree of saturation Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 13 -b n 1b e s= 1+ s (24) 1 where a1, b1, and n are the fitting parameters and m= b n is equal to the parameter m in vG model. Appl. Sci. 2020, 10, 5668 7 of 13 Figure 2 shows the main drying and wetting surfaces of Speswhite kaolin in e~lns~s space based on the above modified SWCC model (Equation (24)). When the suction remains constant, the degree of saturation will decrease with the increase of void ratio, and when the degree of saturation is constant, the suction will increase with the increase of void ratio. Equation (24) can also be reduced is constant, the suction will increase with the increase of void ratio. Equation (24) can also be reduced to the vG model without considering the change of void ratio. to the vG model without considering the change of void ratio. Figure 2. Typical main drying and wetting surfaces of Speswhite kaolin in e lns s space Figure 2. Typical main drying and wetting surfaces of Speswhite kaolin in e~lns~s space (Tarantion’s model). (Tarantion’s model). 3. Model Characteristics and Calibration 3. Model Characteristics and Calibration The hydro-mechanical coupling behavior of unsaturated geotechnical materials can be described using The hydro the granular -mechanic temperatur al coupe ling motio behav n equation ior of uns(Equation aturated ge(10)), otechni mean cal ma e te ective rials ca str n be ess dequation escribed usin (Equation g the gr (17)), anular temperature motio shear stress equation (Equation n equation (18)), (Equ strain ation evol (10) ution ), me equation an effect(Equations ive stress e(22) quatand ion (Eq (23)), uatand ion (1 SWCC 7)), she equation ar stress(Equation equation (E (24)). quatThis ion (1 model 8)), stra does in evo not lut involve ion equyield ation criterion, (Equation flow (22) rand ule, 23 har ))dening , and SW criterion, CC equor ation other (Eq classical uation (concepts 24)). This mode in elastoplastic l does not mechanics. involve yie Instead, ld criterion, thermodynamic flow rule, harden concepts, ingsuch criterion, as granular or othtemperatur er classical co e, migration ncepts incoe ela stcients, oplastiand c mec ener hanic gy s. functions, Instead, t arh eermod replaced. ynamic The concep dissipation ts, susystem ch as gr and anumigration lar temperat coe ure cients , migrat arion e determined coefficients, an theord etically energ,yespecially functions, are the dissipation replaced. The dissipat at the mesoscopic ion system and migr level. The constitutive ation coefficients relationship are determ of non-elastic ined theoretically deformation, espec is obtained ially the by diss thermodynamic ipation at th identity e mesoscopic level , and through. T the he const migration itutcoe ive relat cients ionsh andip of non-e energy functions, lastic defo the rmat dissipation ion is obtained by t mechanism ish connected ermodynam with ic ident macroscopic ity, and through mechanical the migr behavior ation c . These oefficients mean that anthe d energy model fun cancrtions, eflect tthe he di ener ssipat gy dissipation ion mechanism mechanism is connect and edt he witmacr h maoscopic croscopmechanical ic mechanical pr b operties ehaviorinduced . These m by ean it, tand hat tthe he model can reflect the energy dissipation mechanism and the macroscopic mechanical properties multi field coupling characteristics of unsaturated geotechnical materials in a complex environment induce can be d by described it, and t in h aeunified multi fthermodynamic ield coupling char theor acteetical ristics o framework. f unsaturated geotechnical materials in a complex environment can be described in a unified thermodynamic theoretical framework. The model involves some thermodynamic parameters, including , , , , b, and
. However, v s vg sg it is not The model in necessaryvolves some t to determine hthe ermodynam value of ieach c parof amet the ers above , includin coe g cients, η 、ηbecause 、λ 、 theλ parameters 、b、 and v s vg sg required for the calculation of the model are c , c , c , c , and c , and they can be calibrated through 1 2 3 4 5 . However, it is not necessary to determine the value of each of the above coefficients, because the laboratory tests. For example, c can be calibrated through the stress relaxation test, that is, the stress parameters required for the calculation of the model are c1, c2, c3, c4, and c5, and they can be calibrated applied to the sample decreases under the condition of zero strain, and the value of c can be calibrated through laboratory tests. For example, c5 can be calibrated through the stress relaxation test, that is, by the time of stress relaxation; c and c can be determined by the stress-strain relationship at the the stress applied to the sample decreases under the condition of zero strain, and the value of c5 can critical state; c can be determined by the stress paths and stress-strain relationship of the undrained be calibrated by the time of stress relaxation; c2 and c can be determined by the stress-strain shear test; c is related to the volume deformation of soil, which can be calibrated through the isotropic relationship at the critical state; c1 can be determined by the stress paths and stress-strain relationship compression test, and the value of B can also be calibrated; c can be determined by the wetting or 0 4 of the undrained shear test; c3 is related to the volume deformation of soil, which can be calibrated drying test. through the isotropic compression test, and the value of B0 can also be calibrated; c4 can be determined by the wetting or drying test. 4. Modeling the Hydro-Mechanical Coupling Behavior of Unsaturated Geotechnical Materials Test data are used to verify the eectiveness of the proposed model in describing the hydro-mechanical coupling behavior of unsaturated geotechnical materials, such as the irreversible compression in the drying stages of wetting–drying cycles and the eect of a wetting–drying cycle on subsequent behavior during isotropic loading. The test results of two typical geotechnical materials are compared with the modeling results, including non-expansive clay (Speswhite kaolin) [40] and Sr Appl. Sci. 2020, 10, 5668 8 of 13 highly expansive soil (bentonite/kaolin mixture) [41]. The geotechnical materials and the related model parameters are listed in Table 1, and the SWCC parameters come from the literature [39]. Table 1. Geotechnical materials types and model parameters. Soil-water characteristic curves (SWCC). Thermodynamics Parameters Bentonite/Kaolin Bentonite/Kaolin Speswhite Kaolin Speswhite Kaolin Mixture Mixture 7 6 B 2.192 10 kPa 4.545 10 kPa c 6701.2 8536.4 0 2 0.6 0.6 c 15,000 10,000 c 0.0022 0.0028 c 0.15 0.17 c 0.11 0.05 c 37.3 20.6 1 5 SWCC Parameters Drying Wetting Bentonite/Kaolin Bentonite/Kaolin Speswhite Kaolin Speswhite Kaolin Mixture Mixture a 631 1.46 a 368 0.716 1 1 b 0.963 0.243 b 0.922 0.175 1 1 n 14.8 2.56 n 3.87 2.88 4.1. Speswhite Kaolin (Non-Expansive Clay) Ravenendiraraj [40] carried out a series of experiments to study the hydro-mechanical coupling behavior through various stress path tests, including wetting, drying, isotropic loading, and unloading. Here, using the data of Test A9 for comparison, the stress paths are shown in Table 2. The mean net stress loading rate was 0.033 kPa/min, while the suction change rate was 0.0167 kPa/min in wetting or drying stages. The values of the slope of compression curve and unloading curve were 0.177 and 0.024, respectively. Table 2. Stress paths of Test A9. Stage p (kPa) s (kPa) From To Description A B Isotropic loading 10!200 300 B C Isotropic unloading 200!10 300 C D Wetting 10 300!10 D E Drying 10 10!300 E F Isotropic loading 10!375 300 Figure 3 shows the comparison of modeling results and test data for isotropic loading and unloading stages. The isotropic loading–unloading cycle, A-B-C, performed at constant suction 300 kPa, is followed by a wetting–drying cycle, C-D-E, and then subsequent isotropic reloading, E-F, at the same suction 300 kPa. Due to the eect of hydraulic hysteresis, the saturation degree increases, while the specific volume is almost unchanged during the wetting–drying cycle (see points C and E). Additionally, the results show the degree of saturation changes irreversibly during the mechanical loading and unloading process, A-B-C. In addition, it can be seen that, during the second isotropic loading stage, E-F, yield occurs before the first loading, which can be interpreted as the increase of degree of saturation during the preceding wetting–drying cycle [5]. The modeling results are in good agreement with the test data. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 13 loading stage, E-F, yield occurs before the first loading, which can be interpreted as the increase of loading stage, E-F, yield occurs before the first loading, which can be interpreted as the increase of degree of saturation during the preceding wetting–drying cycle [5]. The modeling results are in good Appl. Sci. 2020, 10, 5668 9 of 13 degree of saturation during the preceding wetting–drying cycle [5]. The modeling results are in good agreement with the test data. agreement with the test data. 0.9 2.2 Model results 0.9 2.2 Model results Test data Model results A Test data Model results Test data 0.8 2.1 Test data 0.8 2.1 2.0 C 0.7 2.0 0.7 1.9 0.6 1.9 0.6 1.8 0.5 10 100 200 400 10 100 200 400 1.8 0.5 ⎯ p /kPa 10 100 200 400 10 ⎯ p /kPa 100 200 400 ⎯ p /kPa ⎯ p /kPa (a) (b) (a) (b) Figure 3. Comparison between modeling results and measured test data for Speswhite kaolin under Figure 3. Comparison between modeling results and measured test data for Speswhite kaolin under isotropic loading and unloading: (a) specific volume; (b) degree of saturation. Figure 3. Comparison between modeling results and measured test data for Speswhite kaolin under isotropic loading and unloading: (a) specific volume; (b) degree of saturation. isotropic loading and unloading: (a) specific volume; (b) degree of saturation. Figure 4 shows the modeling results and test data under the wetting–drying cycle, while the mean Figure 4 shows the modeling results and test data under the wetting–drying cycle, while the net stress remains constant during the process. It can be seen that the change of specific volume was Figure 4 shows the modeling results and test data under the wetting–drying cycle, while the mean net stress remains constant during the process. It can be seen that the change of specific volume almost the same during the wetting and drying process and changed little after a wetting–drying cycle mean net stress remains constant during the process. It can be seen that the change of specific volume was almost the same during the wetting and drying process and changed little after a wetting–drying due to a slight variation in the saturation degree (about 0.05). The modeling results were consistent was almost the same during the wetting and drying process and changed little after a wetting–drying cycle due to a slight variation in the saturation degree (about 0.05). The modeling results were with the test data. cycle due to a slight variation in the saturation degree (about 0.05). The modeling results were consistent with the test data. consistent with the test data. 2.20 0.90 2.20 0.90 Model results 0.85 Test data Model results Model results 2.15 0.85 Test data Test data Model results 0.80 2.15 Test data 0.80 2.10 0.75 2.10 0.75 0.70 2.05 0.70 0.65 2.05 C C 0.65 2.00 0.60 10 100 200 400 10 100 200 400 2.00 0.60 s /kPa 10 100 200 400 10 100 200 400 s /kPa s /kPa s /kPa (a) (b) (a) (b) Figure 4. Comparison between predicted results and measured test data for Speswhite kaolin during the wetting and drying cycle: (a) specific volume; (b) degree of saturation. Figure 4. Comparison between predicted results and measured test data for Speswhite kaolin during Figure 4. Comparison between predicted results and measured test data for Speswhite kaolin during the wetting and drying cycle: (a) specific volume; (b) degree of saturation. 4.2. Bentonite/Kaolin Mixture (Highly Expansive Doil) the wetting and drying cycle: (a) specific volume; (b) degree of saturation. 4.2. Bentonite/Kaolin Mixture (Highly Expansive Doil) The mixture include about 10% Wyoming sodium bentonite and 90% Speswhite kaolin [41], 4.2. Bentonite/Kaolin Mixture (Highly Expansive Doil) and the model parameters are also listed in Table 1. The data of Tests 8/9/10 were used to verify the The mixture include about 10% Wyoming sodium bentonite and 90% Speswhite kaolin [41], and eciency of the model, and these behaviors were well simulated by the proposed model. The stress The mixture include about 10% Wyoming sodium bentonite and 90% Speswhite kaolin [41], and the model parameters are also listed in Table 1. The data of Tests 8/9/10 were used to verify the paths are as follows: the model parameters are also listed in Table 1. The data of Tests 8/9/10 were used to verify the efficiency of the model, and these behaviors were well simulated by the proposed model. The stress efficiency of the model, and these behaviors were well simulated by the proposed model. The stress paths are as follows: Test 8: Wetting–drying at constant mean met stress p = 10 kPa; Suction s(kPa) = 300(a)!20(b)!300(c) paths are as follows: Test 8: Wetting–drying at constant mean met stress p = 10kPa; Test 9: Isotropic loading–unloading at constant suction s = 200 kPa; Mean met stress p (kPa) = Test 8: Wetting–drying at constant mean met stress p = 10kPa; Suction s(kPa) = 300(a)→20(b)→300(c) 10(a)!100(b)!10(c)!250(d)!100(e) Suction s(kPa) = 300(a)→20(b)→300(c) Test 9: Isotropic loading–unloading at constant suction s = 200kPa; Test 10: (1) Isotropic loading–unloading at constant suction s = 200 kPa; Mean met stress p (kPa) = Test 9: Isotropic loading–unloading at constant suction s = 200kPa; Mean met stress p (kPa) = 10(a)→100(b)→10(c)→250(d)→100(e) 10(a)!100(b)!10(c) (2) Wetting–drying at constant mean met stress p = 10 kPa; Suction s(kPa) Mean met stress p (kPa) = 10(a)→100(b)→10(c)→250(d)→100(e) = 200(c)!20(d)!200(e) (3) Isotropic loading–unloading at constant suction s = 200 kPa Mean met stress p (kPa) = 10(e)!250(f)!10(g) r Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 13 Test 10: 1) Isotropic loading–unloading at constant suction s = 200kPa; Test 10: 1) Isotropic loading–unloading at constant suction s = 200kPa; Mean met stress p (kPa) = 10(a)→100(b)→10(c) Mean met stress p (kPa) = 10(a)→100(b)→10(c) 2) Wetting–drying at constant mean met stress p = 10kPa; 2) Wetting–drying at constant mean met stress = 10kPa; Suction s(kPa) = 200(c)→20(d)→200(e) Appl. Sci. 2020, 10, 5668 10 of 13 Suction s(kPa) = 200(c)→20(d)→200(e) 3) Isotropic loading–unloading at constant suction s = 200kPa 3) Isotropic loading–unloading at constant suction s = 200kPa Mean met stress p (kPa) = 10(e)→250(f)→10(g) Mean met stress p (kPa) = 10(e)→250(f)→10(g) Figure 5a shows that no collapse compression occurred in wetting path a→b and the irreversible Figure 5a shows that no collapse compression occurred in wetting path a!b and the irreversible Figure 5a shows that no collapse compression occurred in wetting path a→b and the irreversible shrinkage occurred in drying path b→c due to a greater change in saturation degree (about 0.27), i.e., shrinkage occurred in drying path b!c due to a greater change in saturation degree (about 0.27), i.e., shrinkage occurred in drying path b→c due to a greater change in saturation degree (about 0.27), i.e., a larger effective stress than the previous. Figure 5b provides the clear evidence for the hydraulic a larger eective stress than the previous. Figure 5b provides the clear evidence for the hydraulic a larger effective stress than the previous. Figure 5b provides the clear evidence for the hydraulic hysteresis behavior of the material. hysteresis behavior of the material. hysteresis behavior of the material. 2.4 1.0 2.4 1.0 Model results b c Test data 0.8 Model results b c 2.3 Test data 0.8 2.3 0.6 a a 0.6 2.2 Model results 2.2 0.4 Test data Model results 0.4 2.1 Test data 0.2 2.1 0.2 2.0 0.0 10 100 1000 10 100 1000 2.0 0.0 10 s /kPa 100 1000 10 s /kPa 100 1000 s /kPa s /kPa (a) (b) (a) (b) Figure 5. Comparison between modeling results and Test 8 measured data for bentonite/kaolin mixture Figure 5. Comparison between modeling results and Test 8 measured data for bentonite/kaolin material during the wetting–drying cycle: (a) specific volume; (b) degree of saturation. Figure 5. Comparison between modeling results and Test 8 measured data for bentonite/kaolin mixture material during the wetting–drying cycle: (a) specific volume; (b) degree of saturation. mixture material during the wetting–drying cycle: (a) specific volume; (b) degree of saturation. Figure 6 shows the compressive and rebound characteristics of the material under isotropic Figure 6 shows the compressive and rebound characteristics of the material under isotropic loading–unloading cycles. The saturation degree increased with the increase of mean net stress during Figure 6 shows the compressive and rebound characteristics of the material under isotropic loading–unloading cycles. The saturation degree increased with the increase of mean net stress the loading process, while it changed a little during the unloading process, as shown in Figure 6b. loading–unloading cycles. The saturation degree increased with the increase of mean net stress during the loading process, while it changed a little during the unloading process, as shown in Figure In addition, the small mechanical hysteresis behavior could be seen during the unloading–reloading during the loading process, while it changed a little during the unloading process, as shown in Figure 6b. In addition, the small mechanical hysteresis behavior could be seen during the unloading– loop, and the yield point occurred near p = 100 kPa, corresponding to the maximum mean net stress 6b. In addition, the small mechanical hysteresis behavior could be seen during the unloading– reloading loop, and the yield point occurred near p = 100 kPa, corresponding to the maximum previously applied (Figure 6a). reloading loop, and the yield point occurred near p = 100 kPa, corresponding to the maximum mean net stress previously applied (Figure 6a). mean net stress previously applied (Figure 6a). 2.4 1.0 2.4 1.0 Test data e 2.3 0.8 Model results d f Test data 2.3 0.8 Model results d b 2.2 0.6 d c b 2.2 2.1 0.6 c b Model results 2.1 0.4 Test data Model results 2.0 f 0.4 Test data 2.0 e 0.2 1.9 0.2 1.9 1.8 0.0 1 10 100 1000 1 10 100 1000 1.8 0.0 ⎯ p /kPa ⎯ p /kPa 1 10 100 1000 1 10 100 1000 ⎯ p /kPa ⎯ p /kPa (a) (b) (a) (b) Figure 6. Comparison between modeling results and Test 9 measured data for bentonite/kaolin mixture under isotropic loading–unloading cycles: (a) specific volume; (b) saturation degree. Figure 6. Comparison between modeling results and Test 9 measured data for bentonite/kaolin Figure 6. Comparison between modeling results and Test 9 measured data for bentonite/kaolin mixture under isotropic loading–unloading cycles: (a) specific volume; (b) saturation degree. Test 10 diered from Test 9, only in that the wetting and drying cycle was added between the two mixture under isotropic loading–unloading cycles: (a) specific volume; (b) saturation degree. loading–unloading cycles. The results show that there was no net shrinkage during the wetting–drying cycle (points c and e in Figure 7a), while in the second loading, e!f, the material yielded at a lower value of about 80 kPa due to the wetting–drying cycle. Additionally, the degree of saturation increased with the increasing value of mean net stress, as shown in Figure 7b. Although there is no net volume change, the points c and e were dierent due to the increase of saturation after the wetting–drying cycle. The model can well describe this phenomenon. s s s s r r r r Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 13 Test 10 differed from Test 9, only in that the wetting and drying cycle was added between the two loading–unloading cycles. The results show that there was no net shrinkage during the wetting– drying cycle (points c and e in Figure 7a), while in the second loading, e→f, the material yielded at a lower value of about 80 kPa due to the wetting–drying cycle. Additionally, the degree of saturation increased with the increasing value of mean net stress, as shown in Figure 7b. Although there is no Appl. Sci. 2020, 10, 5668 11 of 13 net volume change, the points c and e were different due to the increase of saturation after the wetting–drying cycle. The model can well describe this phenomenon. 2.4 1.0 Model results a Test data 0.8 2.2 e 0.6 c b 0.4 Model results 2.0 Test data 0.2 0.0 1.8 1 10 100 1000 1 10 100 1000 ⎯ p /kPa ⎯ p /kPa (a) (b) Figure 7. Comparison between modeling results and Test 10 measured data for bentonite/kaolin mixture under isotropic loading–unloading cycles and the wetting–drying cycle: (a) specific volume; Figure 7. Comparison between modeling results and Test 10 measured data for bentonite/kaolin (b) saturation degree. mixture under isotropic loading–unloading cycles and the wetting–drying cycle: (a) specific volume; (b) saturation degree. 5. Conclusions 5. Conclusions Based on the thermodynamic theory, a new hydro-mechanical coupled model for unsaturated geotechnical materials is developed in this paper. This model was quite dierent to the classical Based on the thermodynamic theory, a new hydro-mechanical coupled model for unsaturated elastic-plastic theory model, and common concepts, such as yield criterion and flow rules, were not geotechnical materials is developed in this paper. This model was quite different to the classical involved in the constitutive relations and were replaced with the concepts of granular entropy, granular elastic-plastic theory model, and common concepts, such as yield criterion and flow rules, were not temperature, migration coecients, and energy functions, and dissipation at the mesoscopic level was involved in the constitutive relations and were replaced with the concepts of granular entropy, considered. The dissipative system and thermodynamic relationships were determined theoretically, granular temperature, migration coefficients, and energy functions, and dissipation at the mesoscopic and the constitutive relationships of non-elastic strain and granular temperature were obtained by level was considered. The dissipative system and thermodynamic relationships were determined constructing thermodynamic identity and dissipative relations. The relationship between dissipation theoretically, and the constitutive relationships of non-elastic strain and granular temperature were mechanisms and macro mechanical behavior can be established by migration coecients and energy obtained by constructing thermodynamic identity and dissipative relations. The relationship functions. For multi-scale geotechnical materials, this method was simple and eective, and it could between dissipation mechanisms and macro mechanical behavior can be established by migration grasp the essential characteristics of multi-scale behavior of materials from the perspective of energy coefficients and energy functions. For multi-scale geotechnical materials, this method was simple and dissipation. Therefore, the multi field coupling model of unsaturated geotechnical materials in a effective, and it could grasp the essential characteristics of multi-scale behavior of materials from the complex environment can be established in a unified theoretical framework. perspective of energy dissipation. Therefore, the multi field coupling model of unsaturated Combining with SWCC model, considering deformation eect, the proposed model can well geotechnical materials in a complex environment can be established in a unified theoretical describe the hydro-mechanical coupling behavior of unsaturated geotechnical materials subjected to framework. various mechanical paths, especially the influence of the wetting–drying cycle on subsequent behavior Combining with SWCC model, considering deformation effect, the proposed model can well during isotropic loading and irreversible compression during the drying stages of a wetting–drying describe the hydro-mechanical coupling behavior of unsaturated geotechnical materials subjected to cycle, which have been verified by comparing modeling results with test data. various mechanical paths, especially the influence of the wetting–drying cycle on subsequent behavior during isotropic loading and irreversible compression during the drying stages of a Author Contributions: G.Y. proposed the theoretical model and wrote the manuscript; P.C. collated and analyzed wetting–drying cycle, which have been verified by comparing modeling results with test data. the test data; Y.L. reviewed and edited the manuscript. All authors have read and agreed to the published version of the manuscript. Author Contributions: G.Y. proposed the theoretical model and wrote the manuscript; P.C. collated and Funding: This research was funded by Fundamental Research Funds for the Central Universities, grant number analyzed the test data; Y.L. reviewed and edited the manuscript. All authors have read and agreed to the FRF-TP-20-004A1. published version of the manuscript. Conflicts of Interest: The authors declare no conflict of interest. Funding: This research was funded by Fundamental Research Funds for the Central Universities, grant number FRF-TP-20-004A1. References Conflicts of Interest: The authors declare no conflicts of interest. 1. Hu, R.; Hong, J.-M.; Chen, Y.; Zhou, C.-B. Hydraulic hysteresis eects on the coupled flow–deformation processes in unsaturated soils: Numerical formulation and slope stability analysis. Appl. Math. Model. 2018, 54, 221–245. [CrossRef] 2. Gluchowski, A.; Sas, W. 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