Cylindrical cavity expansion responses in anisotropic unsaturated soils under plane stress condition

Haohua Chen,Xiaolin Weng,Lele Hou,Dean Sun

a Department of Geotechnical Engineering,Tongji University,Shanghai,200092,China

b Department of Civil and Architectural Engineering and Mechanics,University of Arizona,Tucson,AZ,85721,USA

c School of Highway,Chang’an University,Xi’an,710064,China

d Department of Civil Engineering,Shanghai University,Shanghai,200444,China

Keywords:Cylindrical cavity Anisotropic unsaturated soil Plane stress Hydromechanical behavior

ABSTRACT In this paper,an anisotropic critical state model for saturated soils was extended to unsaturated conditions by introducing suction into its yield function.Combining this model with soil-water characteristic curves related to porosity ratio was employed to characterize the coupled hydromechanical behavior of unsaturated anisotropic soil.Based on the plane stress condition,the problem of the cylindrical cavity expansion in unsaturated anisotropic soils was transformed into first-order differential equations using the Lagrangian description.The equations were solved as an initial value problem using the Runge-Kutta algorithm,which can reflect the soil-water retention behavior during cavity expansion.Parametric analyses were conducted to investigate the influences of overconsolidation ratio(OCR),suction,and degree of saturation on the expansion responses of a cylindrical cavity in unsaturated anisotropic soil under plane stress condition.The results show that the above factors have obvious influences on the cavity responses,and the plane strain solution tends to overestimate expansion pressure and degree of saturation but underestimates suction around the cavity compared to the proposed plane stress solution.The theoretical model proposed in this paper provides a reasonable and effective method for simulating pile installation and soil pressure gauge tests near the ground surface of the unsaturated soils.

1.Introduction

Since Bishop et al.(1945) and Hill (1950) firstly applied the cavity expansion theory to the metal indentation problem,the cavity expansion theory was increasingly used to model many problems encountered in geotechnical engineering,such as cone penetration tests and manometer tests (Yu,1990; Chang et al.,2001; Cudmani and Osinov,2001),the pile installation effects and determination of the pile load bearing capacity (Randolph et al.,1979; Potts and Martins,1982; Randolph,2003; Li et al.,2017,2019,2020; Liu et al.,2017; Wang et al.,2020; Chen et al.,2022a;Chen and Zhang,2022; Cui et al.,2022; Jiang et al.,2022; Liang et al.,2022; Shao et al.,2022),prediction of ground responses due to tunnel excavation(Vrakas,2014,2016;Mo and Yu,2016;Yu et al.,2019;Jiang et al.,2021;Chen et al.,2022b;Zhou et al.,2022a),as well as identifying the stability of the wellbore (Chen and Abousleiman,2017).

Over the past decades,on one hand,the cavity expansion theory has been improved by developing new solution techniques(Collins and Stimpson,1994; Cao et al.,2002; Zhao,2011; Chen and Abousleiman,2013; Vrakas,2016; Zhuang and Yu,2019; Su,2020;Pan et al.,2020).On the other hand,numerous scholars have introduced progressive models for different soils in terms of cavity expansion theory (Collins and Yu,1996; Russell and Khalili,2006;Yang and Russell,2015;Li et al.,2016;Mo and Yu,2016;Zou et al.,2017; Sivasithamparam and Castro,2018,2020; Zhou et al.,2018,2021a,b; Liu et al.,2021; Chen et al.,2022c; Chen and Mo,2022).The existing solutions of cavity expansion are mainly for saturated soil under plane strain condition.Nevertheless,some geotechnical problems are more suitable to be modeled by cavity expansion under plane stress condition such as pile installation and pressuremeter tests near the ground surface,as the pressure rather than the deformation in the vertical direction generally keeps constant.Furthermore,due to water evaporation and K0-consolidation condition,the soils encountered in geotechnical engineering are generally unsaturated and show apparent initial stress anisotropy(Liu et al.,2022; Liu et al.,2022; Wang et al.,2022; Zhou et al.,2022b).Although some solutions to the cavity expansion problem in anisotropic saturated soils have been published recently (e.g.Chen et al.,2020a; Chen and Mo,2022),the response of unsaturated soil around a cavity is not well investigated.Different from the cavity expansion of saturated soil under plane strain condition,unsaturated soils near the ground surface have the following characteristics: (1) the hydraulic responses,i.e.the change of the degree of saturation Srand suction s,are involved,indicating that the two variables must be considered; (2) the mechanical and hydraulic behaviors of unsaturated soils are considered as a typical elastoplastic process,and coupling effects between them should be considered; and (3) the vertical deformation is free of restriction and the vertical total stress keeps constant during cavity expansion.

In order to solve the problem of cavity expansion in anisotropic unsaturated soils under the plane stress condition,the wellestablished K0-consolidation anisotropic modified Cam-Clay model (AMCC) was generalized to the unsaturated situation by introducing a so-called load-collapse(LC)yield curve in the Barcelona Basic Model(BBM),and thus the interactive effects between Srand mechanical behavior of unsaturated soils were specified.A soilwater characteristic curve(SWCC)considering the influence of void ratio was introduced to model the elastoplastic hydraulic response,which was coupled with the mechanical response by involving the void ratio e and suction s.Considering the plane stress condition and introducing an auxiliary variable,the problem was transformed into a system of first-order differential equations,which could be solved by a computing program MATLAB based on the Runge-Kutta algorithm as an initial value problem.The results of the present solution with different initial suction and overconsolidation ratio (OCR)values were compared with those under the plane strain condition to underline the characteristic expansion responses of anisotropic unsaturated soils under the plane stress condition.

2.Critical state model for anisotropic unsaturated soils

2.1.Stress and strain variables

Two couples of work conjugate stress and strain variables,the average soil skeleton stress tensor σ*ijand strain εij,the s and Srare applied to express the mechanical and hydraulic behaviors of unsaturated soils.The above stress variables are defined following Sun et al.(2008) as

where σijis the total stress tensor; uaand uwdenote the air and water pressures in pore,respectively;and δijis the Kronecker delta.

It can be seen from Eqs.(1)and(2)that the average skeleton stress considers the hydraulic variables(Sr)and suction(s),and hence the hydromechanical behavior of unsaturated soils could be reproduced.

2.2.Model of hydraulic behavior

The degree of saturation Sris regarded as a strain variable and the s-Srrelation is taken as an elastoplastic process to consider the hydraulic hysteresis of unsaturated soils.As shown in Fig.1,the SWCC considering the effect of void ratio is used to reflect the hydraulic characteristics of unsaturated soils.Based on the experimental data,Sun et al.(2008)proposed a widely used SWCC which was adopted in this study.The SWCC consists of primary wetting/drying curve and scanning curve,which can be mathematically expressed as

Fig.1.The SWCC considering the effect of void ratio.

where e denotes the void ratio; λsrand κsrare the slopes of the primary drying/wetting curve and the scanning curve,respectively;λseis the slope of the Sr-e curve under constant suction; and“D(·)”means the increment of a variable of a soil particle within a specific short time or process,which is a Lagrangian description.

As shown in Fig.1,loading/unloading leads to the change of void ratio,and hence the mechanical behavior has an obvious influence on the hydraulic properties of unsaturated soils.On the contrary,the changes of the hydraulic variables Srand s also result in variation of the average skeleton stress.Therefore,the coupling of the mechanical and hydraulic behaviors in unsaturated soils is accomplished by definition of the average skeleton stress and the void ratio-dependent SWCC.

2.3.Model of mechanical behavior

Natural soils generally form a significant degree of anisotropy due to the sedimentary environment and stress history (Wheeler et al.,2003).Hence,to capture the anisotropic behavior of unsaturated soils,the AMCC model for saturated soils was generalized for unsaturated soils in the framework of Alonso et al.(1990) in this study.The effective stresses in the AMCC model of saturated soils are replaced by the average skeleton stresses.Therefore,the yield function of the extended AMCC model can be given as

where M*is the relative critical state stress ratio with respect to the average skeleton stress;M is the critical state stress ratio;η0is the initial stress ratio pertaining to average skeleton stress; η*is the relative stress ratio pertaining to average skeleton stress; K0denotes the coefficient of static earth pressure with respect to the average skeleton stress; p*0is the initial value of p*; ηijdenotes average skeleton stress ratio tensor and is self-evident,ηij0is the initial value of ηijand ηij0does not change during loading;p*is the average skeleton stress;and p*yis the yield stress at suction s,which is related to the yield stress p′0yof saturated soil as

where λ(0)and λ(s)are the slopes of the normal consolidation line with the suction of 0 and s,respectively; κ is the slope of the swelling line,the value of which is approximately assumed to remain unchanged with suction; and p*nis a reference stress.The λ(s)can be expressed by λ(0)and suction s as(Alonso et al.,1990):

where b and c are the material parameters.

As shown in Fig.2,compared with the yield surface for saturated soil in p*-q stress space(q is the deviatoric shear stress),the yield surface of unsaturated soil is extended into p*-q-s stress space.It is evident that the p*yincreases with increasing s in the p*-s plane,indicating that the suction potentially enhances the strength and the stiffness of unsaturated soils.The two yield surfaces considering suction increase and decrease (SI and SD) are introduced to reproduce the plastic changes of Srduring primary wetting/drying,and SI and SD are denoted as horizontal lines in the p*-s stress plane in Fig.2b.The SI and SD yield functions can be expressed as

where sIand sDare the yield stresses when the suction increases and decreases,respectively.

Since the SWCC employed in this study depends on the void ratio,the SI and SD yield surfaces and LC yield surface are coupled via the void ratio.This means that the sIand sDevolve with the change of void ratio during loading.Following Sun et al.(2008),the sI(sD) can be determined from the initial yield stress at suction increase s0I(s0D),the initial void ratio v0,the current void ratio v,the initial and current degree of saturation (Sr0and Sr),and the initial and current void ratio (e0and e) as follows:

Fig.2.LC yield surface in different stress planes: (a) p*-q plane,and (b) p*-s plane(CSL: critical state line; p*y0 : yield stress of unsaturated soil on K0-line).

Now,the AMCC model of saturated soils has been fully extended to unsaturated conditions and can well simulate the hydrodynamic coupling behavior of anisotropic unsaturated soils.

3.Definitions and assumptions

Fig.3 shows the cavity expansion problem in an infinite crossanisotropic unsaturated soil mass with the far-field stress state.The cavity expands from the initial cavity radius a0to the current cavity radius a with the internal total expansion pressure increasing from σh0to σa(σh0is the initial horizontal total stress,and σais the current cavity pressure).Once the radial stress at the cavity wall is greater than the yield stress of soils,a plastic zone will be formed around the cavity.During the expansion process,a soil particle at the initial position r0is pushed to the current position r.In particular,the soil particle at the elastoplastic boundary (EPB)moves from rp0to rp(rpis the current radius of EPB,and rp0is the initial radial coordinate of rpbefore expansion) with a radial movement Urp(Urp=rp-rp0).

The following assumptions and definitions are proposed for the cavity expansion problem in unsaturated anisotropic soils:

(1) The axial direction of the cavity is perpendicular to the horizontal plane and aligns with the deposition direction of the anisotropic unsaturated soil.Hence,the shear stresses vanish everywhere around the cavity and only three principal stresses and their corresponding strains involve in the derivations;

(2) In accordance with the definition of soil mechanics,the compressive stress and strain are positive;

(3) The plane stress condition means that the vertical total stress σvkeeps unchanged during cavity expansion;

(4) The soil particle is incompressible and the pore-air pressure remains constant;and

(5) The water content keeps constant,which is common in installation of the displacement pile and pressuremeter tests in unsaturated soils.

4.Solution for elastic region

The solution of the elastic region can be obtained by simultaneously combining equilibrium equation,small-strain theory and Hooke’s law(Chen et al.,2020b)for cavity expansion theory under plane strain condition.In their work,the total vertical stress,suction,and degree of saturation in the elastic region remain unchanged during cavity expansion.Therefore,the elastic solution in unsaturated soil under plane strain condition is still applicable to the present case,which gives

where σ*r,σ*θand σ*zare the radial,tangential and vertical average skeleton stresses,respectively;σ*h0and σ*v0are initial horizontal and vertical average soil skeleton stresses,respectively;σ*rpis the σ*rat the EPB;Uris the radial displacement of the soils;v(v=1+e)and v0are the specific value and the corresponding initial value; and G is the shear modulus,which can be expressed as

where μ is the Poisson’s ratio.

5.Formulations of the problem

5.1.Constitutive matrix

A circular plastic region is formed around the cavity after the yielding of soils at the cavity wall,in which strain increment Dεijconsists of elastic component Dεeijand plastic component.The elastic strain increment Dεeijcan be obtained by Hooke’s law as

where E is the elastic modulus.

Applying the associated follow rule for the LC yield function of the newly extended AMCC model for unsaturated anisotropic soils,the plastic strain incrementcan be written as

where Λ is the plastic multiplier,whose specific expression is shown in Appendix A.

As the shear stress and strain disappear during the cavity expansion process,they are not considered in the following derivation.Combining Eqs.(14)and(15)and considering the hydraulic behavior represented by Eqs.(3) and (4),in the plastic region,the elastoplastic constitutive matrix of any soil particle is

The detailed expressions of A,Ar,Aθ,Azand Asare given in Appendix B.Notably,since s and Srkeep constant in the elastic process as reported by Chen et al.(2020b),the hydraulic behavior may be still under elastic state even though the mechanical behavior yields during cavity expansion.Hsis used in the constitutive matrix as its value depends on the state of the soil.Hs=λsr(κsr) represents the hydraulic state moving along the primary wetting/drying (scanning) curve.

The inverse operation of Eq.(16) yields

Fig.3.Schematic diagram of cavity expansion in anisotropic unsaturated soils (σv0:initial vertical total stress; s0: initial suction).

where the detailed expressions of Bij,T and T1are given in Appendix C.

5.2.Constant water content condition

As stated previously,the water content remains unchanged during cavity expansion,thus we have

Accordingly,the incremental form of Eq.(18a)can be expressed as

where w denotes the gravimetric water content,and Gsis the specific gravity.

5.3.Continuity condition

Since the problem considered only involves principal stresses and strains,the continuity condition of the problem under consideration can be expressed as

where εvdenotes the volumetric strain; and εr,εθand εzare the three strain components in radial,tangential and vertical directions,respectively.

To reduce variables and represent the unique location of the state variables,an auxiliary variable ξ (Chen and Abousleiman,2013) is introduced:

The auxiliary variable ξ is used to replace the logarithmic strain relation for large plastic deformation,and the following equation can be obtained:

where “d(·)” denotes the increment of a variable at a specific position r,which is an Eulerian description; and h is the thickness of the soil layer in the vertical direction with h0being its initial value.

From Eqs.(19)and(21),the following equation can be obtained:

Combining Eqs.(21a),(21c) and (22) yields

From Eqs.(21c) and (22),the increments of εr,εθand εzare

It can be seen from Eqs.(24a)-(24c)that the auxiliary variable ξ relates the strain variables to each other,which in fact reduces strain variables and simplifies the problem considered.

Substituting Eq.(18) and Eqs.(24a)-(24c) into Eqs.(17a)-(17d),the stress increments can be written as follows:

5.4.Plane stress condition

The total vertical stress σzremains unchanged during cavity expansion under plane stress condition:

Further,substituting Eq.(18) (25c) and (25d) into Eq.(26),the following relation can be obtained:

5.5.Equilibrium condition

The equilibrium equation of soil particles during cavity expansion (Chen et al.,2020b) is given by

It should be noted that the equilibrium equation in the vertical direction has already been represented by the plane stress condition given by Eq.(26),and hence is not given herein.

Substituting Eqs.(18a),(18b) and (23) into Eq.(29),the equilibrium equation can be rewritten as

Substituting Eqs.(25a) and (25d) into Eq.(30) and rearranging the equation give

5.6.Governing equations for the problem

Combining Eqs.(28) and (31),the differential equations regarding the specific volume can be given as

where

Now,we have six first-order differential equations,Eqs.(25),(28) and (32),which constitute the governing equations for the cavity expansion problem in unsaturated soils under plane stress condition.Since the radial location of the state variables is represented by the auxiliary variable ξ in the governing equations,it is still necessary to relate the state variables to the radial coordinate r.Recalling Eq.(23),the relationship between the r and ξ can be obtained as

where ξa=1-a0/a.

6.Solving the formulated problem

6.1.Initial conditions

The values of the state variables when the soil just approaches plastic state are taken as the initial condition according to Chen et al.(2020b).Combining Eqs.(5) and (12a)-(12c),the three stress components (σ*r,I,σ*θ,Iand σ*z,I) of the instantaneous state can be determined as (Chen et al.,2019):

where η*pis the value of η*at the EPB and can be determined following Chen et al.(2019).

According to the continuous condition,the sI,vIand Sr,Iat the instant when a soil particle enters the plastic state equal the corresponding values in the elastic region given by Eqs.(12e)-(12f),which gives

Fig.4.Framework of program for the Runge-Kutta algorithm(k is the index of soil partical,m is the total number of soil particals in the solution,and RK[·]is the defined Runge-Kutta function).

From Eq.(36),the plastic mechanical behavior of unsaturated soil takes place before the changes of hydraulic state,which means that the soil reaches the LC yield surface much more quickly than the SI/SD yield surface.From Eq.(20),the value of the auxiliary variable ξIat the instant state when the soil particle enters the plastic region can be given as

where G0is the initial value of G.

6.2.Solving governing equations

As stated previously,the hydraulic characteristics of soil reach the plastic state later than the mechanical state during cavity expansion (Chen et al.,2020b).Therefore,it is necessary to judge and update the state of the hydraulic behavior in every iterative step when solving the governing equations.Considering that the ODE(ordinary differential equation)solver in the MATLAB package cannot judge and update variables,an iterative approach is developed for solving governing differential equations based on the Runge-Kutta algorithm.The key steps of the proposed approach are shown in Fig.4.

7.Results and discussion

Comparisons between the proposed solution and the plane strain solution were conducted so as to reflect the unique expansion response of a near-surface cylindrical cavity.The solution under plane strain condition can be easily obtained by revising the solution of Chen et al.(2020b).The parameters of soils in the parametric analyses were obtained from the cylindrical cavity expansion solution for unsaturated soils by Chen et al.(2020b)under plane strain condition,as shown in Table 1.It should benoted that the Rmis defined as the ratio of the maximum p*yin history to the current p*y0.

Table 1 Soil parameters used in parameter analysis (revised from Chen et al.,2020b).

7.1.Expansion responses at cavity wall

Fig.5 shows the changes of the mechanical and hydraulic state variables at different OCR values under both plane stress and strain conditions.Note that the internal expansion pressure σais the total stress rather than the average skeleton pressure.

Fig.5.Variations of different state variables at the cavity wall with different OCR values: (a) Internal expansion pressure,(b) Suction,and (c) Degree of saturation.

In Fig.5,the expansion responses at the wall surface are similar under plane stress and strain conditions.The pressure on the cavity respectively increases and tends to a constant value under plane stress and strain conditions with the expansion of the cavity radius.It shows that the soil near the cavity wall reaches the critical state faster under plane stress condition than that under the plane strain condition.Compared with plane strain condition,the soil is not strongly constrained in the vertical direction under plane stress condition.As a result,at the same expanded cavity radius,the expansion pressure is far smaller under plane stress condition than the plane strain condition.In addition,since the soils with a larger OCR have higher strength and stiffness,the expansion pressure increases with a higher Rmfor both two solutions.Therefore,the OCR has remarkable impacts on the expansion response of the cavity.

In Fig.5b,at Rm=1,the suction at cavity wall does not change drastically when the vertical boundary condition changes from plane stress to plane strain.However,a significant difference between plane stress and strain solutions can be seen for overconsolidated soils,especially for the heavily overconsolidated soils with Rm=7.Interestingly,the final suction at the cavity wall is greater than the initial suction under the plane stress condition,but less than the initial suction under plane strain condition when Rm=7.This indicates that the heavily overconsolidated soil might experience dilatancy and contraction under the plane stress and strain conditions,respectively.In particular,a horizontal straight segment appears on the curves for Rm=3,5 and 7 at the initial stage of cavity expansion,which represents the elastic expansion of the cavity wall.After that,the suction decreases with the cavity radius for Rm=3,while increases and then decreases with the cavity radius for Rm=5 and 7 under plane stress condition.This is because the normally and slightly overconsolidated soils(Rm=1 and 3) undergo shear contraction,while the heavily overconsolidated soils(Rm=5 and 7) experience shear dilatancy and then contraction during cavity expansion under plane stress condition.

As shown in Fig.5c,the variation of Srat the cavity wall is also significantly different under two plane conditions.In general,Srunder plane stress is less than that under plane strain.When the OCR is small (Rm=1 and 3),Srincreases with increase of the cavity radius.If the OCR is large(Rm=5 and 7),Srdecreases with increase of the cavity radius.

Fig.6 shows the changes of the mechanical and hydraulic state variables of a cavity wall with different initial suctions under both plane stress and strain conditions.As anticipated,the suction shows remarkable influences on the expansion responses under the two conditions.Generally,the increase of suction in soils improves the strength of soils.As a result,the expansion pressure and suction of soils increase,and Srdecreases with increase of the initial suction.It is interesting that an inflection point appears on the curve for s0=70 kPa,which indicates the yield of soil on the SI or SD yield curves.As a result,the evolution of suction at s0=70 kPa coincides with the trajectory of suction at s0=40 kPa,which moves along the main wetting curve.

Fig.7 shows the changes of the mechanical and hydraulic state variables at cavity wall with different Sr0values under both plane stress and strain conditions.Again,the expansion responses of the cavity wall are significantly different under the two plane conditions.The internal expansion pressure and Srare smaller,and the suction is larger under the plane stress condition than that under the plane strain condition.As indicated by Eq.(1),the average skeleton stress increases with increase of the suction.Hence,increase of Srstrengthens the soils.Therefore,the expansion pressure increases and the suction decreases with increase of the initial saturation.In Fig.7b,the hydraulic characteristics change during cavity expansion for each case,which is considered as an inflection point on the suction curve.

Figs.6 and 7 show that the hydraulic parameters have a great influence on the expansion response at the cavity wall because the hydraulic and mechanical properties are coupled and interactively affected during the cavity expansion.

Fig.6.Variations of different state variables at the cavity wall of cylindrical cavity with different initial suctions:(a)Internal expansion pressure,(b)Suction,and(c)Degree of saturation.

7.2.Distributions of state variables

Fig.8 shows the state variables surrounding an expanded cavity at a/a0=2 for two OCR values(Rm=1 and 5)under both plane stress and strain conditions.From Fig.8a,the vertical average skeleton stress around the cavity barely changes under the plane stress condition.This is because the total vertical stress is constant,and Srand s only fluctuate slightly near cavity under the plane stress condition.Oppositely,the vertical skeleton stress near the cavity wall increases significantly due to the restriction of vertical deformation under the plane strain condition.Moreover,the radial average skeleton stress around the cavity is larger under plane strain condition than that under plane stress condition.This is because the restriction of deformation in the vertical direction increases p*value of the soil.In addition,the vertical deformation of soils is not fully constrained under plane stress condition,thus the change of specific volume of unsaturated soils is smaller under plane stress condition than that under plane strain condition,as shown in Fig.8b.Interestingly,when Rm=5,soils first exhibit obvious dilation and then marked shrinkage near the cavity wall under plane stress condition.In contrast,the soils only show shear contraction under plane strain condition.This demonstrates that the vertical boundary condition has great impacts on the expansion responses.From Fig.8b,negative vertical strain develops primarily around the cavity wall under plane stress condition,which means that the soils around the cavity wall are pushed outwards and upwards simultaneously.

Fig.7.Effect of initial saturation on state variables at the cavity wall: (a) Internal expansion pressure,(b) Suction,and (c) Degree of saturation.

As shown in Fig.8c,the suction distribution curves of the two solutions share similar pattern conditions at Rm=1.The suction around the cavity increases with r and an obvious inflection point occurs on the curve for Rm=1 under plane stress/strain condition.In contrast,the suction distribution curves for the case Rm=5 show totally different patterns under the two plane conditions.The suction increases and then decreases with r near the cavity for Rm=5 under the plane stress condition due to shear dilatancy,while the suction increases with r under plane strain condition.In Eq.(18b),Sris inversely proportional to the specific volume of the unsaturated soils under the condition of constant water content,and thus its distribution curve is opposite to that of the specific volume around the cavity.

Fig.8.Influence of OCR on different state variables along logarithmic coordinates surrounding the cavity at instant a/a0=2: (a) Stress components,(b) Strain,and (c) Suction.

In order to explore the influence of OCR and state variables on the expansion responses under both plane stress and strain conditions.Fig.9 gives the distribution of state variables around the cavity with four OCR values under both plane stress and strain conditions.

From Fig.9,the radial average skeleton stress under plane strain condition is larger than those under plane stress condition,due to the enhanced strength of soils under plane strain condition.For tangential average skeleton stress,it is larger in the vicinity of the cavity under plane stress condition than that under plane strain condition.Considering the plane stress condition and slight changes of suction and Srnear the cavity wall (Fig.9e and f),the vertical average skeleton stress near the cavity changes slightly under plane stress condition.In contrast,the vertical average skeleton stress increases significantly under the plane strain condition.It is noted that the overconsolidated soils with Rm=7 exhibit obvious dilatancy under two plane conditions(see Fig.9d).As a result,apparent increase of suction and decrease of Srappear near the cavity,as illustrated in Fig.9e and f.All of above observations manifest that the overconsolidation and the vertical boundary condition have appreciable impacts during cavity expansion.

7.3.Stress trajectories of typical elements at cavity wall

Fig.10 shows the projection of stress trajectory and LC yield surface under different hydraulic conditions.Note that only the stress path(SP)under plane stress conditions is included in Fig.10 for clarity and the pnet0and q are set to be constant in the cases investigated.

As shown in Fig.10,for all cases,the SP in the p*-q plane deviates from the initial stress point on the K0-consolidation lines.With the expansion of the cavity,the SP moves upwards and rightwards and finally approaches the critical state line(CSL).The location of the final stress state is higher for soils with larger Sr0and s0,indicating that both Sr0and s0enhance the strength of unsaturated soils.It is also interesting that the gradient of LC yield surface decreases after cavity expansion in Fig.10a.This is because the pnet0and q are kept the same,and the initial p*increases with s0.On the other hand,the SP begins from the initial LC curve and then moves rightward and downward in the p*-s stress space in Fig.10b.In particular,inflection points occur on the SP when s0=70 kPa,Sr0=0.6 and 0.8.This represents the yielding of the hydraulic behavior of unsaturated soils.As shown in Fig.10c,the SP starts from the σ*zaxis and moves downward and rightward in the π plane.The SP finally hits the intersection of the CSL and the final LC yield locus near the σ*raxis.Furthermore,the final LC yield surface expands with increase of Sr0and s0,which indicates the increases of Sr0and s0potentially enhance the strength of unsaturated soils.Again,the above results show that the hydraulic variables have a significant effect on cavity expansion responses.

Fig.9.Distributions of different state variables along logarithmic coordinates around the cavity wall with different OCR values during the expansion process of cylindrical cavity:(a)Radial average skeleton stress,(b) Tangential average skeleton stress,(c) Vertical average skeleton stress,(d) Specific volume,(e) Suction,and (f) Degree of saturation.

Fig.10.Projections of stress trajectories and LC yield surfaces under different stress planes: (a) p*-q plane,(b) p*-s plane,and (c) π plane (Orange and blue points denote the initial and final state of soil at cavity wall,respectively).

7.4.Hydraulic responses

Fig.11 shows the AMCC of soils at cavity wall under different hydraulic conditions during cavity expansion.Only the cases under plane stress condition are investigated in this section.As shown in Fig.11a,the trajectory of the hydraulic state during cavity expansion is significantly affected by the initial suction value.When the initial suction is smaller than sD(s0=10 kPa and 40 kPa),the hydraulic state moves along the main wetting curve during the cavity expansion.When the initial suction is greater than sDand less than sI(s0=70 kPa),the hydraulic state moves first along the scanning line and then along the main wetting line,which indicates the hydraulic behavior initially responds elastically and then yields with further expansion of the cavity.Moreover,the hydraulic state always moves along the scanning line when s0=100 kPa and 130 kPa.

From Fig.11b,the soils with a higher Rmwould experience smaller changes of Srand s during expansion.Due to the coupled motion of the SD and the LC yield surface,the hydraulic state first moves along the scanning curve and then along the main wetting curve at s0=40 kPa.Accordingly,the hydraulic state moves along the scanning curve for s0=70 kPa.Due to the shear dilatancy of overconsolidated soils at the beginning of expansion,the suction increases and exceeds the yield stress sIfor s0=100 kPa and 130 kPa.Hence,the hydraulic state first moves along the main drying curve and then along the scanning curve,which indicates the overconsolidated soils would experience dilation and then contraction during cavity expansion.

As shown in Fig.11c,the hydraulic state always moves along the scanning curve for the case Sr0=0.8.The hydraulic state moves along the scanning curve and then the main wetting curve for other cases (Sr0=0.4,0.5,0.6 and 0.7).This indicates that the hydraulic behavior is only elastic for Sr0=0.8,while a typical elastoplastic process occurs for (Sr0=0.4,0.5,0.6 and 0.7).When the initial suction is constant,the soils with a higher Sr0value are expected to experience a larger change of Srwith the same change of the suction,which means that there is an obvious interaction between mechanical and hydraulic responses.

Fig.11.The relationship curve between logarithmic suction and degree of saturation of a soil element at the cavity wall during cavity expansion for different hydraulic conditions: (a) Different initial suctions with Rm=1,(b) Different initial suctions with Rm=5,and (c) Different initial degrees of saturation.

8.Conclusions

A rigorous analytical solution of cylindrical cavity expansion in unsaturated soils under the plane stress condition was developed in this paper.The AMCC model was firstly extended for unsaturated soils by introducing suction into the yield function and incorporating the SWCC model.Based on the plane stress condition and the extended AMCC model,the coupled hydraulic and mechanical behaviors of unsaturated soils were considered and the cavity expansion problem in unsaturated anisotropic soils under plane stress condition was formed.The Lagrangian description was used to transform the problem into a system of first-order differential equations.Runge-Kutta algorithm was used to solve the problem.Then extensive parameter studies were carried out by analyzing the influence of OCR and initial suction and saturation on the response of cavity expansion for the unsaturated soils under plane stress.To highlight the importance of the vertical boundary condition,the solution under plane strain condition was also compared with the present solution.

The results show that the hydromechanical coupling response of unsaturated soils during cavity expansion process can be well described.Compared with the plane strain solution,the expansion pressure,degree of saturation,and compression of specific volume are smaller,and the suction is larger at the cavity wall under the plane stress condition.Due to free restriction in the vertical direction,the heavily overconsolidated unsaturated soils near the cavity show apparent shear dilatancy during cavity expansion.The OCR,initial suction and degree of saturation have significant influences on the expansion response of unsaturated soils during cavity expansion under both plane stress and strain conditions.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We acknowledge the funding support from the National Natural Science Foundation of China (Grant No.U1934213),the National Key Research and Development Program of China (Grant Nos.2021YFB2600600 and 2021YFB2600601).

Appendix A.Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jrmge.2022.03.015.