Chin. Sci. Bull. (2014) 59(26):3314–3324 DOI 10.1007/s11434-014-0411-6

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Article

Civil & Architectural Engineering

A model considering solid-fluid interactions stemming from capillarity and adsorption mechanisms in unsaturated expansive clays Jian Li • Chenggang Zhao • Guoqing Cai Yanxin Guo

•

Received: 31 October 2013 / Accepted: 8 January 2014 / Published online: 30 May 2014 Ó Science China Press and Springer-Verlag Berlin Heidelberg 2014

Abstract Solid-fluid interactions in unsaturated expansive clays can be divided into capillarity and adsorption effects based on their physical mechanisms. Most constitutive models for unsaturated soils are proposed on the basis of the capillarity mechanism, ignoring the contributions of the adsorption effect to mechanical and hydraulic behaviors. For expansive clays, however, the adsorption effect which leads to more complex behavioral characteristics than that in low plasticity clays cannot be ignored. In the light of this, a new binary-medium model for unsaturated expansive clays is proposed, involving a consideration of the solid-fluid interactions stemming from the capillary and the adsorption mechanisms at the same time. Firstly, we assume that expansive clay is a mixture of two ideal parts, i.e. the ideal capillarity part and the ideal adsorption part, and then an ideal capillarity model and an ideal adsorption model, each of which is available for the corresponding ideal part, are established. Furthermore, a participation function is used to reflect the degrees of capillarity effect and adsorption effect. Finally, predictions are performed on the results of the consolidation tests and the cyclical controlled-suction tests published in literature. After comparing predicted results with test results, it is illustrated that the established model can quantitatively predict mechanical and hydraulic behaviors in expansive clays.

J. Li  C. Zhao (&)  G. Cai  Y. Guo School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China e-mail: [email protected] C. Zhao College of Civil Engineering and Architecture, Guilin University of Technology, Guilin 541004, China

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Keywords Unsaturated expansive clays  Binarymedium model  Adsorbed water  Capillarity phenomenon  Physicochemical phenomenon

1 Introduction Expansive clay is a type of soil usually experiencing large swelling strain when wetted because of its mineralogical composition. Gens and Alonso [1] stated that the physicochemical phenomena which is related to the properties of the clay minerals is obvious, which leads to the complex mechanical and hydraulic behaviors of unsaturated expansive clays. Previous research [2, 3] has indicated that it is necessary to use matric suction and net stress as independent stress state variables to describe behaviors in an unsaturated soil adequately. It is worth noting that matric suction (or potential) consists of two components, i.e. a capillary component and an adsorptive component [4, 5]. The former one is termed capillary suction and is given by ðua  uw Þ, in which ua and uw are pore air pressure and pore water pressure, respectively. The capillary suction is commonly associated with the capillarity phenomenon arising from the surface tension of water. Capillary water comprised of bulk water arises from the air-water surface tension which is the result of the intermolecular forces acting on molecules in the contractile skin [6]. The adsorptive component is associated with the physicochemical phenomenon arising from the interactions between solid matrix and pore water, such as electrostatic force, van der Waals force, hydration force, etc. Adsorbed water comprised of bound water arises from these forces. In fact, matric suction represents the degree of adsorption from the water to the solid phase [7]. Therefore, a large suction value refers to a large potential of water

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immediately adjacent to the solid surface, which, however, should not be viewed as pressures in the sense of conventional bulk thermodynamic. In general, the influences of physiochemical phenomenon on mechanical and hydraulic behaviors in low plasticity soils are not obvious. For expansive clays, however, these influences may be significant and the proportion of the adsorptive component to matric suction increases with the decrease of water content, which results in the variety of mechanical and hydraulic behaviors in unsaturated expansive clays. On the one hand, at high value of water contents, the solid-fluid interactions are dominated by the capillarity phenomenon. In this state, the mechanical and hydraulic behaviors of expansive clays are similar to that of low plasticity soils. The following is a summary of experimental observation. (1) The soil deformation mainly depends on the values of net stress, matric suction and degree of saturation. Particularly, a reduction in matric suction for a given confining stress may induce an irrecoverable volumetric compression, depending on the stress level. (2) As stated by Alonso et al. [8], lots of research has figured out the hydraulic hysteresis behavior in unsaturated soils. In addition, the soil-water characteristics curve is sensitive to mechanical actions. These are also observed in unsaturated expansive clays at high water content levels [9–11]. On the basis of these, lots of researchers established the elasto-plastic models for unsaturated soils which involve the mutual effect between the hydraulic and mechanical behaviors [12–15]. These models can also quantitatively predict the mechanical and hydraulic behaviors in unsaturated expansive clays at high water content levels. On the other hand, at low value of water contents, the solid-fluid interactions are dominated by the physicochemical phenomenon. As a result, the mechanical and hydraulic behaviors of expansive clays are different from that at high water content levels. The differences are as follows. (1) Lu and Likos [16] pointed out that interparticle force of unsaturated soils are mainly composed of soil skeleton force, physicochemical force, surface tension force, cementation force and the force originating from pore water pressure. At low water content levels, however, the surface tension force and the force arising from negative pore-water pressure drastically diminish or cease to exist, and the physicochemical force approaches a constant value for clayey soils. Meanwhile the interparticle force could reach to several hundred kPa and remain fairly constant. (2) The evolutions of pore size density functions of compacted clayey soils during drying and wetting cycle were analyzed by lots of researchers [17, 18]. The results of these tests have shown that aggregates in the compacted clay swell with an increase of water content or shrink with

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a decrease of water content. (3) At low water content levels, a majority of pore water attaches to the surfaces of solid matrix, thus the hydraulic properties mostly depends on the specific surface area of the clay mineral, however not affected by the dry densities of soils [9, 10]. In fact, the capillarity and the adsorption effects exist simultaneously for unsaturated expansive clays in the most conditions. However, their mechanisms are almost not distinguished in the existing constitutive models for unsaturated expansive clays [11, 19–23]. Hence, it is necessary to develop a model considering each of them. In this paper, we attempt to propose a new binary-medium model for unsaturated expansive clays according to the method of establishing the disturbed models [24, 25], the damage models [26, 27] and the binary-medium models [28, 29]. Firstly, on the basis of the characters of unsaturated expansive soil structure, we assume that expansive clay is a mixture of two parts, i.e. an ideal capillarity part and an ideal adsorption part, each of which is in the corresponding ideal condition. In the ideal capillarity condition, physiochemical phenomenon has no influences on the mechanical and hydraulic behaviors, however, the behaviors depend on the capillarity effect and others factors; in the ideal adsorption condition, the solid-fluid interactions are stemmed only from adsorption mechanism. Such division is clear to consider the influences of two kinds of solid-fluid interactions to the behaviors in unsaturated expansive clays. Secondly, an ideal capillarity model and an ideal adsorptive model are proposed, each of which is available for a corresponding ideal condition. Thirdly, the expression of participation function is given, which is used to reflect the degrees of capillarity effect and adsorption effect. Finally, the binary-medium model is used to simulate the consolidation tests and the cyclical controlled-suction tests published in literature.

2 The choice of constitutive variables Identifying constitutive variables is the first step to deduce an elasto-plastic model for unsaturated soils. On the basis of the thermodynamics theory, stress variables can be deduced from the equilibrium equation for unsaturated soil [30, 31], besides, energy-conjugate variables can be chosen from the expression of input work of unsaturated soil [32–36]. Baker and Frydman [37] stated that compacted clays are composed of aggregates (ag), capillary water (c), adsorbed water (ad) and air (a) phases. Typically, the adsorbed water in the form of thin films attaches to the aggregates surfaces and the capillary water concentrates at the contact position of the adjacent aggregates. Furthermore, the aggregates consist of a solid matrix and adsorbed water. The pores between the aggregates will be termed macropores and the

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pores inside the aggregates will be termed micropores. On the basis of this and the thermodynamics theory, Li et al. [36] deduced the input work of unsaturated expansive clays with double porosity which is expressed as c 0a a 0ad ad a M a a a W ¼  u0c ;i wi  u;i wi  u;i wi þ u n Sr q_ =q ad M þ ðrij  Sar ua dij  Scr uc dij  Sad r u dij Þe_ij  nM sc S_ cr  nM ðua  uad ÞS_ ad r  w c c w ad ad þ pe_m  u c  u c q ; =q v

ð1Þ

where the excess pore pressure gradient in a phase is a a defined as u0a ;i ¼ u;i  q gi ða ¼ c,a,adÞ, and a represents capillary water, adsorbed water or air phases (the same as below); ua is the pore pressure in a phase; the comma notation is used to indicate differentiation with respect to a spatial coordinate; qa is the density of a phases and we assume that the densities of capillary and adsorbed water are the same, i.e. qc ¼ qad ¼ qw ; gi is the gravitational acceleration vector; wai is the artificial seepage velocity of a phase and is defined as wai ¼ nM Sar ðfia  vi Þ; nM is the macroporosity and is defined as nM ¼ V M =V, in which V M and V are the volume of macropores and the total volume, respectively; the total porosity n is the sum of the macroand microporosity, i.e. n ¼ nM þ nm , in which nm is defined as nm ¼ V m =V and V m is the volume of micropores; Sar is the degree of saturation of a phase and is defined as Sar ¼ V a =V M , in which Va is the volume of a phase; the macrostructural degree of saturation is defined as a c ad SM r ¼ Sr þ Sr ; vi and fi are the average velocities of the aggregates and a phase, respectively; rij is the total stress; m dij is unit tensor of second-order; eM ij and ev are the macrostructural strain tensor and the microstructural volumetric strain, respectively, and the total strain is the sum of the deformations of macro- and microstructure,  m i.e. eij ¼ eM ij þ ðev 3Þdij ; sc is the capillary suction and is defined as sc ¼ ua  uc ; p is the average stress and is defined as p = rii/3; cc is the mass exchange rate between the capillary water phase and other phases; cad is the mass exchange rate between the adsorbed water phase and other phases. Here we ignore the phase transformation between the water and the air phases caused by temperature change. In addition, we define the compressive stress and the shrinkage strain are positive. In the derivation process of Eq. (1), it has been assumed that the micropores are filled with water. When the assumption is ignored, the expression of input work is modified as c 0a a 0ad ad a a c c W ¼ u0c ;i wi  u;i wi  u;i wi þ ðrij  u Sr dij  u Sr dij M M c _c M a ad _ ad  uad Sad r dij Þe_ij  n s Sr  n ðu  u ÞSr  w c c w a a a ad ad q ; þ pe_m ð2Þ v  u c =q  u c =q  u c

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where ca is the mass exchange rate between the air phase and other phases. The first three terms on the right side of Eq. (2) are the work due to the seepages of the pore fluids relative to the aggregates; the fourth term is the work due to the macrostructural deformation; the fifth and sixth terms are the work due to the volume changes of capillary water and adsorbed water, respectively; the seventh term is the work due to the microstructural deformation; the eighth to tenth terms are the work due to the mass exchanges between the pore fluids and other phases. Here, it has been assumed that the air phase in macropores is continuous, and then the pore air pressure is consistent with atmospheric pressure. Thus the work caused by the compression of the air phase is omitted in Eq. (2). The mass conservation expressions of water and air phases in micropores should be expressed as (  m m wm wm oðnm Sm =qw ; r Þ ot þ div ðn Sr fi Þ ¼ c  m m am am a o½nm ð1  Sm r Þ ot þ div ½n ð1  Sr Þfi  ¼ c =q ; ð3Þ where Sm r is the microstructural degree of saturation and is wm defined as Sm =V m , in which V wm and V m are the r ¼ V volume of water in micropores and the volume of micropores, respectively; fiwm and fiam are the average velocities of the water phase and the air phase in micropores, respectively; cwm is the mass exchange rate between the water phase in micropores and other phases, and the constrain, i.e. cwm þ cc þ cad ¼ 0, should be satisfied; cam is the mass exchange rate between the air phase in micropores and other phases, and the constrain, i.e. cam þ ca ¼ 0, should be satisfied. Here, it has been assumed that the densities of the water and air phases in micropores are constant. Equation (3) can be further expressed as ( m m _m wm  e_m =qw ; v Sr þ n Sr ¼ c ð4Þ  e_m ð1  Sm Þ  nm S_ m ¼ cam =qa : v

r

r

Here, it has been assumed that the average velocities of the water and air phases in micropores are equal to the average velocities of aggregates [36], in addition, the values of nm and Sm r are constant [32]. Substitute Eq. (4) to (2) and then obtain the work input to unsaturated expansive clays with double porosity as c 0a a 0ad ad a a c c W ¼  u0c ;i wi  u;i wi  u;i wi þ ðrij  u Sr dij  u Sr dij M M _c M a ad _ ad  uad Sad r dij Þe_ij  n sc Sr  n ðu  u ÞSr  m  ðuc  uad Þccad qw þ ½ðp  ua Þ þ ðua  uad ÞSm r e_v  ðua  uad Þnm S_ m ; ð5Þ r

ccad

where is the mass exchange rate between the capillary water phase and the adsorbed water phase in macropores.

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In order to choose the energy-conjugate variables to establish the ideal capillarity condition and the ideal adsorption condition, Eq. (5) needs to be further simplified for the two ideal conditions, respectively. The work caused by the seepages is omitted in the following simplified expressions. In the ideal capillarity condition, the mechanical and hydraulic behaviors of unsaturated expansive clays are not affected by the physiochemical phenomenon, however, they rely on the capillarity effect and others factors. Therefore, the terms which are the work caused by physiochemical phenomenon, i.e. the sixth to the ninth terms in Eq. (5) are omitted. Then, in isotropic stress state, the simplified Eq. (5) is expressed as  _c W ¼ p e_M v  s c Sr ;

ð6Þ

where p* is the interparticle stress and is defined as p ¼ p0 þ ðua  uc ÞScr , in which the term ðuad Sad r Þ corresponding to the physicochemical effect is removed; p0 is net stress and is defined as p0 ¼ p  ua ; sc is the modified capillary suction and is defined as sc ¼ nM sc . The first term in the right side of Eq. (6) indicates that the quantity which * is conjugate to e_M v is the stress variable p . The expression of interparticle stress, in which the effective stress parameter is the capillary degree of saturation, is the same with Bishop’s stress. The second term demonstrates that the strain-like quantity which is work conjugate to sc is the increment of capillary degree of saturation S_ cr . In the ideal capillarity condition, the adsorptive component of matric suction is negligible. Thus it is reasonable to replace sc with the modified matric suction sm which is defined as sm ¼ nM sm , where sm is the matric suction. In the ideal adsorption condition, solid-fluid interactions are occupied by physiochemical mechanism, thus the terms which are the work caused by capillarity phenomenon, i.e. the fifth and the seventh terms in Eq. (5) are omitted and the term corresponding to capillarity effect in the interparticle stress is deleted. In isotropic stress state, the simplified Eq. (5) is expressed as M M a ad _ ad W ¼ ½p0 þ ðua  pad ÞSM r e_v  n ðu  u ÞSr þ ½p0 þ ðua  pad ÞSm e_m  nm ðua  uad ÞS_ m r

v

r

ð7Þ

¼ p0 e_v þ ðua  uad Þe_w ; where ew is the volumetric strain of the water phase and its increment is defined as e_w ¼ e_w =ð1 þ eÞ; ew is the water ratio and is defined as ew ¼ Sr e; Sr is the degree of satum m ration and is defined as Sr ¼ ðeM =eÞSM r þ ðe =eÞSr ; the total void ratio e is the sum of the macro- and microstructural void ratio, i.e. e ¼ eM þ em ; eM and em are defined as eM ¼ V M =V s and em ¼ V m =V s , respectively, in which V s is the solid volume. The first term in the right side

of the second equals sign in Eq. (7) indicates that the quantity which is conjugate to e_v is the stress variable p0 . The second term demonstrates that the strain-like quantity which is work conjugate to ðua  uad Þ is the increment of total volumetric strain of the water phase e_w . Houslby [32] also deduced a similar expression. As a variable, water ratio is more suited to describe the adsorbed water content than degree of saturation at low water content levels on the basis of the previous experimental studies [9, 10] which pointed out that there is a one-to-one correspondence between water ratio and matric suction during the drying or wetting path for the same type of expansive clay at the same levels. The relationship between water ratio and matric suction is related to the inherent properties of clay rather than the soil density. However, there is not the similar relationship between degree of saturation and matric suction, which is due to the definitions of water ratio and degree of saturation. The water ratio, as a variable, is define as the volume ratio of the pore water to solid matrix, and the degree of saturation is define as the volume ratio of the pore water to pore. The latter one contains the information of pore volume, therefore the relationship between degree of saturation and matric suction is complex. On the other hand, it is not feasible to choose ðua  uad Þ as a constitutive variable due to the limitations of the measurement devices. In the ideal adsorption condition, however, the capillary component of matric suction is negligible. Thus it is reasonable to replace the stress variable ðua  uad Þ with matric suction in the ideal adsorption model. Both of them are able to reflect the degree of the physicochemical effect in unsaturated expansive soils.

3 The ideal capillarity model Previously, Bishop’s stress and the modified suction have been used to develop elasto-plastic models for unsaturated soils based on the capillary mechanism [11–15]. These models are able to capture observed key features in unsaturated soils. We choose the constitutive model established by Liu et al. [14] as the prototype of the ideal capillarity model. In this paper, pore air pressure is supposed to be constant so that the air hardening effect considered in the model proposed by Liu et al. [14] is omitted. A summary of the simplified model formulation, consisted with the one proposed by Wheeler et al. [12], is as follows. Modelling soil deformation and variation for degree of saturation is under an elasto-plastic framework. A strain c increment (deM v and dSr ) consists of elastic and plastic components. The elastic increment of macrostructural volumetric strain is expressed as

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deMe v ¼

j dp ; vp

ð8Þ

where v is the specific volume; j is the gradient of the unloading-reloading line on the ( ln p*, v) plane. The elastic relationship of the capillary water is jw dSce ds ; ð9Þ r ¼   sm þ pa m where jw is the gradient of the scanning line on the ðln sm ; Sce r Þ plane; pa is a reference stress which is used in case of the term in the right side of Eq. (9) is meaningless in the condition of the value of suction being 0. Two yield functions are constructed to reflect the mechanical and hydraulic behaviors. The first is the loading-collapse (LC) curve describing the rearrangement of the aggregates in soils. The second yield function is the suction change (SI/SD) curve describing the variation of degree of saturation. The formulation of LC yield curve is defined as fLC ¼ p  pc ¼ 0;

ð10Þ

where pc is the pre-consolidation stress. The LC yield curve is a line vertical to p* axis on the ðp ; sm Þ plane. The SI/SD yield curve could be expressed as ( fSI ¼ sm  smI ¼ 0; ð11Þ fSD ¼ smD  sm ¼ 0; where smI and smD are the hardening parameters and they are intersections of scanning curve with primary drying and wetting curves, respectively. For the LC yield curve, the hardening parameter is pc . Considering the coupled movement of the LC yield curve caused by the yielding on the SI/SD curve, the increment of pc is given by dpc pc

¼

v deMp v kj



ksw dScp r kw  j w

smI

þ pa

¼

dsmD smD

þ pa

¼

dScp r k w  jw

þ

kws v deMp v kj

;

ð13Þ

where kws is the other coupling parameter, which reflects the amounts of the shift in the primary drying and wetting lines due to deMp v .

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Substituting Eqs. (12), (13) and (14) into the constituency condition functions, we obtain 8 B1 A22  B2 A12 > > ; < KLC ¼ A11 A22  A12 A21 ð16Þ B1 A21 þ B2 A11 > > : KSI=SD ¼ ; A11 A22  A12 A21 where ofSI=SD ofLC ¼ dp ; B2 ¼ dsm ;  op osm ofLC pc v p v ¼ c ; ¼  kj c kj  ofLC pc ksw SI=SD pc ksw ofSI=SD ¼  ¼  ;  kw  jw m c kw  jw osm kws vðsmI=D þ pa Þ ofSI=SD ¼ ; kj osmI=D

B1 ¼ dp A11 A12 A21

ð12Þ

;

ð14Þ

where KLC and KSI=SD are non-negative loading indexes for the LC and SI/SD curves yielding, respectively. The loading indexes KLC and KSI=SD can be obtained through the plastic consistency conditions. Applying consistency conditions and substituting hardening rules for the LC and SI/SD curves, we obtain 8 ofLC  ofLC  > > > dfLC ¼  dp þ  dpc ¼ 0; < op opc ð15Þ of ofSI=SD  > SI=SD  > > df ¼ ds þ ds ¼ 0: : SI=SD m osm osmI=D mI=D

A22 ¼

where k is the gradient of the normal consolidation line on the ( ln p*, v) plane; kw is the gradient of the main drying or wetting line on the ðln sm ; Sce r Þ plane; ksw is the coupling parameter, which reflects the amount of the shift in the LC yield curve due to dScp r . Similarly, considering the coupled movement of the SI/SD yield curve caused by the yielding on the LC curve, the increments of smI and smD are given by dsmI

The plastic strain increments are given by 8 Mp > < dev ¼ KLC ; ofSI=SD cp > ; :  dSr ¼ KSI=SD osm

 smI=D þ pa ofSI=SD ofSI=SD smI=D þ pa ¼  :  osm kw  jw k w  jw mI=D

Substitute the loading indexes into Eq. (14) to get plastic strain increment. Then the relations between generalized stress increments and generalized strain increments could be expressed as  M   dp dev ¼ ½ C  ; ð17Þ dsm dScr where 2 ½C  ¼

yLC ðkjÞ j vp þ kvpc 4 ySI=SD kws ðkw jw Þ kpc

yLC ksw ðkjÞ  kvðs  þpa Þ mI=D

w  sjþp  a m

ySI=SD ðkw jw Þ kðsmI=D þpa Þ

3 5;

in which k ¼ 1  ksw kws ; the parameter yLC is equal to 1 when the LC curve yields, otherwise 0; the parameter ySI=SD is equal to 1 when the SI/SD curve yields, whereas 0.

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In order to deduce the binary-medium model further, the stress and strain variables in Eq. (17) need to be convert into other forms. Firstly, the generalized total strains are expressed as 8 M m < ev ¼ ev þ ev ; M M m ð18Þ : Sr ¼ e Sc þ ead þ e ; r e e e where eM ad is the macrostructural adsorbed water ratio and is ad M defined as eM ad ¼ Sr e . Then the increments of them are given by    M   M  1  0 dev dev dev ¼ ½D  ¼ : ðSr  Scr Þ n eM =e dSr dScr dScr ð19Þ Here, it has been utilized that the microporosity and the adsorbed water content are constant. Secondly, according to the definition of p* and sm , their increments are expressed as ( )  0  M dp dp dev ½  ½  þ T ; ð20Þ ¼ T a b dsm dScr dsm   0 1 Scr m ½  ¼ and T b sm 1þe 0 nM 1þe According to eqs. (17), (19) and (20), there is    0 dev dp ¼ ½E  ; dSr dsm 

where

½Ta  ¼

 sm . 0

ð21Þ

where ½E ¼ ½D½1  CTb 1 ½C ½Ta .

4 The ideal adsorption model In the ideal adsorption condition, the solid-fluid interactions are occupied by the physiochemical mechanism. As a result, in this state, the mechanical and hydraulic behaviors of expansive clays are different from that in the ideal capillarity condition. Thus it is necessary to establish an ideal adsorption model based on the physicochemical mechanism. In the ideal adsorption model, the work-conjugate variables are  0   p ev ; : ð22Þ sm ew Here the advice that the matric suction is used to substitute the stress variable ðua  uad Þ is adopted. Two yield functions are constructed to describe the mechanical and hydraulic behaviors. The first is the consolidation (CS) curve describing soil deformations. The second is the matric suction change (MSI/MSD) curve representing the change of water content. The formulation of CS curve is defined as

fCS ¼ p0  p0c ¼ 0;

ð23Þ

p0c

where is the pre-consolidation stress. The proposed CS yield curve is a straight vertical line on the ðlnp0 ; sm Þ plane. However, the effect of plastic increment of water ratio to the pre-consolidation stress p0c is considered by the expressions of deformation and the hardening rule, i.e. the Eqs. (28), (29) and (32), as below. The MSI/MSD curve can be expressed as ( fMSI ¼ sm  smI ¼ 0; ð24Þ fMSD ¼ smD  sm ¼ 0; where smI and smD are the hardening parameters. They are intersections of scanning line with main drying and wetting lines, respectively. The soil deformation and the variation of water content are modeled under an elasto-plastic framework as well. It is supposed that each of plastic strains is resulted from the CS curve yielding and the MSI/MSD curve yielding. This assumption is in accord with that adopted by Chen [11] and Liu [38]. As a result, the strain increments are given by ( dev ¼ deev þ depvðCSÞ þ depvðMSI=MSDÞ ; ð25Þ dew ¼ d epwðCSÞ þ depwðMSI=MSDÞ ; where deev , depvðCSÞ and depvðMSI=MSDÞ are the elastic volumetric strain increment, the plastic volumetric strain increment produced by the CS curve yielding, the plastic volumetric strain increment cause by the MSI/MSD curve yielding, respectively; depwðCSÞ and depwðMSI=MSDÞ are the plastic change of water content cause by the CS and MSI/ MSD curve yielding, respectively. Previous research [9, 10] shows that, for a clayey soil, the main drying and wetting lines are getting closer to each other and even overlap at low water content levels. Therefore, the elastic increment of the volumetric strain of the water phase is omitted in Eq. (25). In other words, the hardening parameters smI and smD are equal at a given state. Elastic volumetric strain, depending on the variations of net stress and matric suction, is expressed as deev ¼

j0 0 j0s dsm ; dp þ 0 vp vðsm þ pa Þ

ð26Þ

where j0 is the gradient of the unloading-reloading line on the (v, ln p0 ) plane; j0s is the gradient of the unloadingreloading line on the ðv; lnðsm þ pa ÞÞ plane. In addition, the deformations of aggregates in compacted clays are reversible [1], so the microstructural volumetric strain is expressed as dem v ¼

j0m 0 j0ms dsm ; dp þ vp0 vðsm þ pa Þ

ð27Þ

where j0m and j0ms are microstructural parameters.

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The plastic strain increments caused by the CS curve yielding are given as ( p devðCSÞ ¼ KCS ; ð28Þ depwðCSÞ ¼ KCS d1 ; where KCS is the non-negative loading index for the CS . curve yielding; d1 is defined as depwðCSÞ depvðCSÞ , indicating direct coupling between the CS and MSI/MSD curves yielding. The plastic strain increments caused by the MSI/ MSD curve yielding are given as 8 p > < devðMSI=MSDÞ ¼ KMSI=MSD d2 ; ð29Þ ofMSI=MSD > : depwðMSI=MSDÞ ¼ KMSI=MSD ; osm where KMSI=MSD is the non-negative loading index for the MSI/MSD curve yielding; d2 is defined as .

   ofMSI=MSD osm depvðMSI=MSDÞ depwðMSI=MSDÞ . According to experimental studies [9, 10], the variation of water content is independent of the volumetric strain increment when the solid-fluid interactions are occupied by adsorption mechanism in expansive clays. Thus the SMI/ SMD curve yielding does not produce plastic change of water content, and then d1 ¼ 0:

ð30Þ

The aggregates created by compaction on the dry of optimum swell with the decrease of matric suction or shrink with the increase of matric suction, which deduces the soils deformation. The double scale model established by Gens et al. [1] and Alonso et al. [19] has taken into account the microstructural deformation and its effects. However, other researchers [11, 21] pointed out that the applicability of double scale model is limited due to the hardness to obtain the quantity of the microstructure. In order to solve the problem, here we consider the influence of the change of water content on soil deformation by the parameter d2, then avoid concerning the microcosmic deformation. The parameter d2 describes the plastic volumetric change caused by the plastic change of ew (i.e. the yielding on the MSI/MSD curve). In terms with experimental results [10, 11], larger net stress tends to result in smaller volume change during wetting and drying paths. This means the closer to CS yield curve is a soil state the smaller swelling upon wetting or smaller contraction upon drying will be. Therefore, a simple expression for d2 is proposed for wetting and drying paths as follows    d2 ¼ l  m 1  p0 p0c ; ð31Þ where l and m are soil parameters.

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For the CS yield curve, the hardening parameter is p0c which defines the size of CS yield curve. Its increment is assumed to be influenced by the plastic volumetric strain, as the following expression dp0c v ¼ 0 dep ; p0c k  j0 v

ð32Þ

where k0 is the gradient of the normal consolidation line. In order to simplify the above equation, the parameter k0 is supposed not to change with matrix suction. However, its function can be given when the relevant test data are sufficient. For the MSI/MSD yield curve, the hardening parameters are smI and smD . Their increments are supposed to be related to the plastic change of water content, as the following expression dsmI dsmD v p ¼ ¼ de ; smI þ pa smD þ pa k0w w

ð33Þ

where k0w is the gradient of the main drying or wetting line on the ðlnðsm þ pa Þ; ew Þ plane. Moreover, the loading indexes KCS and KMSI=MSD can be obtained through the plastic consistency conditions. Applying consistency condition and substituting hardening rules for yield curves, we obtain 8 ofCS 0 ofCS 0 > > > < dfCS ¼ op0 dp þ op0 dpc ¼ 0; c of ofMSI=MSD > MSI=MSD > > dsm þ dsmI=D ¼ 0: : dfMSI=MSD ¼ osm osmI=D ð34Þ Substitute Eqs. (28), (29), (32) and (33) into the constituency condition functions, and get 8 B01 A022  B02 A012 > > > ; < KCS ¼ A011 A022 ð35Þ > B02 > > ¼ ; K : MSI=MSD A022 where ofMSI=MSD ofCS 0 dp ¼ dp0 ; B02 ¼  dsm ; 0 op osm vp0 ofCS vp0 A011 ¼ 0 c 0 0 ¼  0 c 0 ; k  j opc k j 0 d vp of d2 vp0 2 CS A012 ¼ 0 c0 0 ¼  0 c0 ; k  j opc k j vðs þ p Þ of vðsmI=D þ pa Þ a mI=D MSI=MSD ofMSI=MSD ¼ : A022 ¼ 0 osmI=D osm kw k0w

B01 ¼ 

Substitute the loading indexes into Eqs. (28) and (29) to get the plastic strain increments. Combine the plastic with elastic strain increments and obtain

Chin. Sci. Bull. (2014) 59(26):3314–3324





dev dew

 0 dp ¼ ½G  ; dsm

3321

ð36Þ

where 2 ½G ¼

j0 0 4 vp

0

0

þ yCS ðkvp0j Þ c

j0s vðsm þpa Þ

0

þ

ðyCS þyMSI=MSD Þd2 k0w ofMSI=MSD vðsmI=D þpa Þ osm yMSI=MSD k0w vðsmI=D þpa Þ

dsm

3 5;

the parameter yCS is equal to 1 when the CS curve yields, otherwise 0; the parameter yMSI=MSD is equal to 1 when the MSI or MSD curve yields, whereas 0. In this ideal model, ev and ew are chosen as constitutive variables. In order to deduce the following constitutive model, it is necessary to transform the stain increment fdev ; dew gT into fdev ; dSr gT . According to the definition of water ratio, i.e. ew ¼ e  Sr , the strain variables in Eq. (36) are expressed as        dev dev d ev 0 1 ¼ ½H  ¼ : ð37Þ dew dSr dSr Sr n According to Eqs. (36) and (37), the relations between the generalized stress increments and the generalized total strain increments could be further expressed as follows    0 dev dp ¼ ½I  ; ð38Þ dSr dsm where [I] = [H]-1[G].

5 The binary-medium model In this article, we presume that expansive clay is a mixture of two parts, i.e. the ideal capillarity part and the ideal adsorption part, each of which is in the corresponding ideal condition. And then the ideal capillarity model and the ideal adsorption model are proposed. However, the capillarity and the adsorption effects exist simultaneously for unsaturated expansive clays in most conditions. On the basis of this, we present a participation function n which is the ratio of the volume of the ideal capillarity part to the mixture volume. According to the method adopted by a number of researchers [28, 29], the values of net stress exerting to the two ideal parts could be supposed to be equal, and the same to the matric suction. Thus, the generalized stain increments of the mixture can be expressed as ( c ad c ad deav v ¼ n dev þ ð1  nÞdev þ ðev  ev Þdn; ð39Þ c ad c ad dSav r ¼ n dSr þ ð1  nÞdSr þ ðSr  Sr Þdn; where eav v is the total or average volumetric strain of the mixture; ecv and ead v are the volumetric strains of the ideal capillarity part and the ideal adsorption part, respectively;

Sav r is the total or average degree of saturation of the mixture; Scr and Sad r are the degrees of saturation of the ideal capillarity part and the ideal adsorption part, respectively. Experimental studies [9, 10] show that lower water content results in larger deviation of the observed behaviors from that in the ideal capillarity condition. On account of this, a simple expression for n is proposed as

 max  ew  ew c n ¼ 1  max ; ð40Þ ew  ethre w where emax and ethre w w are the maximal and threshold values of water ratio, respectively; c is a soil parameter; hi is the McCauley bracket such that n is a non-negative function. When the water ratio is equal to emax w , the value of n is 1; when the water ratio is less than or equal to ethre w , the value of n is 0. Substitute Eqs. (21), (38) and (40) into Eq. (39) and get the relation between generalized stress increments and generalized total strain increments for the mixture.

6 Model predictions The capabilities of the proposed model are illustrated with reference to the experimental data for expansive clays carried out by Nowamooz [10]. Nowamooz [10] has carried on a series of tests for compacted clays to investigate soil properties. The consolidation tests and the cyclical controlled-suction tests were selected to verify our model. The soil used in these tests is an artificially prepared mixture of silt and bentonite (40 % and 60 %, respectively). The mixture has a liquid limit of 87 % and a plastic limit of 22 %. The specific gravity is 2.67. All the model parameters used in subsequent simulations are summarized in Table 1. It’s worth noting that the model parameters used to simulate the consolidation tests and the cyclical controlled-suction tests are the same. The binary-medium model includes 15 parameters. The parameters in the ideal capillarity model and in the ideal adsorption model can be determined from the basic experiments of unsaturated soils at high water content levels and at low water content levels, respectively; the parameters in the participation function are selected from simulating tests by trial and error. In addition, the initial values of soil state and hardening parameters for the consolidation tests and the cyclical controlled-suction tests are given in Tables 2 and 3, respectively. In order to simplify the simulations, the adsorbed water content and the microporosity are supposed to be 0 in the ideal capillarity part, and pa is equal to 0. As exhibited in Fig. 1, nonlinear consolidation is well predicted along the loading path in saturation state, which shows that the slope of normal consolidation line declines

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Chin. Sci. Bull. (2014) 59(26):3314–3324

Table 1 Model parameters for the expansive clay 0

Parameter

k

kwD

kwI

j

jw

ksw

kws

k0

Value

0.025

0.090

0.140

0.015

0.010

0.550

0

0.150

0

Parameter

j

Value

0.080

j0s

lD

lI

0

0.65

0.85

k0w 0.082

m

emax w

ethre w

cjconsolidation

cjcontrolledsuction

0

2.00

0.33

1.5

6.0

Table 2 Initial values of soil state and hardening parameters for the consolidation tests eci

cdi ðkN m3 Þ

Specimens

ðpc Þc ðkPaÞ

ead i

ðp0c Þad ðkPa)

a1

9.0

1.973

1.455

5.0

5.0

a2

10.0

1.700

1.455

15.0

5.0

a3

11.1

1.380

1.455

48.5

5.0

Table 3 Initial values of soil state and hardening parameters for the cyclical controlled-suction tests Specimens

cdi ðkN m3 Þ

eci

ead i

ðpc Þc ðMPaÞ

ðp0c Þad ðkPaÞ

Sri ð%Þ

Scri ð%Þ

Sad ri ð%Þ

ðln sI Þc

ðln sD Þc

b1

12.7

1.40

1.09

35.3

0

30.3

20.9

30.3

4.2

3.7

b2

13.5

1.20

0.99

44.3

0

33.3

23.9

33.3

4.4

3.7

with the increase of net stress. This nonlinear consolidation is simulated by a variation of the value of the participation function during the loading path. Model predictions are reasonably consistent with test results. Furthermore, the influences of the value of parameter c on the bending degree of normally consolidated line are discussed, as shown in Fig. 2. Parameter values are 2.0, 1.5, and 1.0, respectively. Through the comparison of the computed results, it illustrates that the bending degree of normally consolidated lines increases with the decline of c.

Fig. 2 The influences of on the normal consolidation curves

Fig. 1 Comparison between the computed and measured results of the consolidation tests

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The comparison between the computed results and the results of the cyclical controlled-suction tests are depicted in Fig. 3. Firstly, Fig. 3a describe that the expansive clays swell with the increase of water content and shrink with the decrease of water content. During the suction cycles, the expansive clays show volumetric shrinkage accumulation. Secondly, the variations of degree of saturation and water ratio during a wetting and drying cycle are illustrated in Fig. 3b, c, respectively. As shown, the capillary hysteresis is predicted in every wetting-drying test. In addition, for a given type of soils with different density, their main drying

Chin. Sci. Bull. (2014) 59(26):3314–3324

3323

(or wetting) curves tend to get closer and the hysteresis loops is getting smaller with matric suctions increasing. It indicates that the influences of the soil compaction degree and stress path on soil-water characteristic curve reduce with the decrease of water content. Model predictions are reasonably consistent with test results.

7 Conclusions In this paper, a binary-medium model for unsaturated expansive clays is presented. The binary-medium model is developed to explore the possibility to capture the influences of solid-fluid interactions dominated by capillary and adsorption mechanisms on the mechanical and hydraulic behaviors in expansive clays. This model is composed of the ideal capillarity model and the ideal adsorption model, each of which is available for the corresponding ideal condition. Furthermore, the participation function is used to reflect the deviations of the observed response of expansive clays from that in the ideal conditions. At last, the predictions are performed on the results of consolidation testes and cyclical controlled-suction testes published in literature. After comparing predicted results with test results, it is illustrated that the established model can quantitatively predict mechanical and hydraulic behaviors in expansive clays. Nevertheless, this model has some shortcomings: (1) the amount of parameters in the model is more than that of other models; (2) it is established in the isotropic stress state, thus is not available in the anisotropic stress state. We need to carry on further work to extend the adaptability of the model as a whole. Acknowledgments This work was supported by the National Natural Science Foundation of China (51278047) and Guangxi Science Foundation (2012GXNSFGA060001).

References

Fig. 3 Comparison between the computed and measured results of the cyclical controlled-suction tests. a Relations between void ratio with matric suction; b relations between degree of saturation with matric suction; c relations between water ratio with matric suction

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