SPLOSNA ANALITIČNA REŠITEV ENODIMENZIONALNE KONSOLIDACIJE ZA NEZASI-ČENE ZEMLJINE PRI RAZLIČNIH OBTEŽNIH POGOJIH Izvleček V članku je predstavljena splošna analitična rešitev za enodimenzionalno konsolidacijo nezasičene zemljine v različnih časovno odvisnih obtežnih pogojih, ki temelji na metodi diferencialne transformacije (DTM). Zlasti so dobljene analitične rešitve za različna razmerja med koeficienti v vodilnih enačbah za konsolidacijo nezasičenih zemljin. Tehnika razširitve Fourierjeve vrste je spremenjena tako, da upošteva tako kontinuirano diferencialno obtežbo, kot tudi periodično posamično obtežbo. Primerjava med rezultati predlagane rešitve in obstoječe teoretične rešitve kaže, da predlagana rešitev daje odlične rezultate, medtem ko je enostavno dobiti analitično rešitev konsolidacije nezasičenih zemljin. Ugotovljeno je bilo tudi, da lahko razlike v koeficientih v vodilnih enačbah znatno vplivajo na disipacijo tako pornega zračnega nadtlaka kot tudi pornega vodnega nadtlaka, čeprav so velikosti njihovih variacij različne. Jiwei Li Taizhou University, College of Civil Engineering and Architecture, Institute of Coastal Environment and Geotechnical Engineering Taizhou, Zhejiang 318000, Kitajska E-posta: iamliji_007@126.com Chinese Academy of Sciences, Institute of Rock and Soil Mechanics, State Key Laboratory of Geomechanics and Geotechnical Eng. Wuhan, Hubei 430071, Kitajska Changfu Wei Chinese Academy of Sciences, Institute of Rock and Soil Mechanics, State Key Laboratory of Geomechanics and Geotechnical Eng. Wuhan, Hubei 430071, Kitajska E-posta: cfwei@whrsm.ac.cn Yanlin Zhao Guangxi University, College of Civil Engineering and Architecture Nanning, Guangxi 530004, Kitajska E-posta: paperyantu@126.com Ključne besede enodimenzionalna konsolidacija; nezasičena zemljina; analitična rešitev; časovno odvisna obremenitev 86. Acta Geotechnica Slovenica, 2018/1 A GENERAL ANALYTICAL SOLUTION TO THE ONE-DIMENSIONAL CONSOLIDATION PROBLEM FOR UNSATU-RATED SOIL UNDER VARIOUS LOADING CONDITIONS Jiwei Li Taizhou University, College of Civil Engineering and Architecture, Institute of Coastal Environment and Geotechnical Engineering Taizhou, Zhejiang 318000, P. R. China E-mail: iamliji_007@126.com Chinese Academy of Sciences, Institute of Rock and Soil Mechanics, State Key Laboratory of Geomechanics and Geotechnical Eng. Wuhan, Hubei 430071, P. R. China Changfu Wei Chinese Academy of Sciences, Institute of Rock and Soil Mechanics, State Key Laboratory of Geomechanics and Geotechnical Eng. Wuhan, Hubei 430071, P. R. China E-mail: cfwei@whrsm.ac.cn Yanlin Zhao Guangxi University, College of Civil Engineering and Architecture Nanning, Guangxi 530004, P. R. China E-mail: paperyantu@126.com Keywords one-dimensional consolidation problem; unsaturated soil; analytical solution; time-dependent loading https://doi.org/10.18690/actageotechslov.15.L87-99.2018 Abstract A general analytical solution is developed for the one-dimensional consolidation problem of unsaturated soil under various time-dependent loading conditions based on a differential transformation method (DTM). In particular, analytical solutions are obtained for different relationships between the coefficients in the governing equations for unsaturated soil consolidation. The Fourier series expansion technique is adopted to account for both the continuous differentiable loading and the periodic piecewise loading. A comparison between the results of the current solution and the existing theoretical solution indicates that the proposed solution yields excellent results, while it is straightforward to obtain the analytical solution of the unsaturated consolidation problems. It was also found that the variations in the coefficients in the governing equations can significantly influence the dissipation of both the excess pore-air pressure and the excess pore-water pressure, though the magnitudes of their variations are different. 1 INTRODUCTION The subsidence induced by the consolidation (or compression) of unsaturated soil under environmental loadings is a subject of great interest in geotechnical engineering practice [1]. Several consolidation models for unsaturated soils have been proposed since 1960. Early contributions include those by Blight [2], Scott [3] and Barden [4]. Perhaps the most popular consolidation model for unsaturated soils is the one proposed by Fredlund and Hasan [5], in which two partial differential equations are employed to describe the dissipation of the pore pressures. Due to its nonlinear nature, it is generally difficult to obtain analytical solutions for the consolidation problem associated with unsaturated soil, and thus most of the existing solutions are numerical [5-7]. Compared with its numerical counterpart, an analytical solution, if available, is much simpler and more robust, providing an exact solution for the verification of computer codes and semi-analytical solutions. Thus far, several efforts have been made to develop analytical solutions to the consolidation problem for unsaturated soil. Using the Laplace transform and the analytical inverse Laplace transform, Qin et al. [8] obtained an analytical solution for the consolidation of a single-layer unsaturated soil subjected to stepwise loading. In this problem, the boundary conditions for the water and air phases are permeable on one side of the surfaces, and impermeable on the other. Adopting the same method, Qin et al. [9] derived another analytical solution in the time domain, in which the external load- DOI Acta Geotechnica Slovenica, 2018/1 87. J. Li et al.: A general analytical solution to the one-dimensional consolidation problem for unsaturated soil under various loading conditions ing exponentially varies with the time. Using the variableseparation method, Shan et al. [10] derived an exact solution to the governing equations for the one-dimensional consolidation of single-layer unsaturated soil. It should be noted that the expressions for the theoretical solutions mentioned above are complicated, so that it is generally difficult to use them. As a result, these analytical solutions have not been extensively applied in practice. Ho et al. [11] obtained an analytical solution using the techniques of eigenfunction expansion and Laplace transformation. In their solutions the temporal change of the total pressure is assumed to be constant. Afterwards, adopting the same method, they derived a series of analytical solutions subjected to different types of external loadings for one-dimensional consolidation [12], 2D plane-strain consolidation [13] and axisym-metric consolidation [14-15] of an unsaturated soil. By considering time-dependent loading under various initial and boundary conditions, Zhou and Zhao [16] obtained an analytical solution for the one-dimensional consolidation of unsaturated soils by introducing two new state variables. However, this solution was developed without considering the correlations among the equation coefficients. In fact, all these coefficients are related to the properties of the soil. It is therefore important to reveal the relationship between the equation coefficients and the solutions. Conte and Troncone [17] developed an analytical solution for the one-dimensional consolidation problem of soils subjected to arbitrarily variable loading. However, this solution does not take into account the effect of the pore-air flow. Despite their own merits, all the above-mentioned solutions suffer from the following two shortcomings: I) their solution procedures are very much involved so that these solutions are generally difficult to apply in practice, and II) the relationships between the equation coefficients and the solutions are not well defined. In addition, although some specific time-dependent loading has been addressed in these solutions, an analytical solution to the problem under a general loading condition, such as cyclic loading and periodic piecewise loading, has yet to be developed. In this paper, on the basis of a differential transform method (DTM) and the Fourier series expansion techniques, a comprehensive analytical solution for Fredlund and Hasan's consolidation model subjected to loading described by an arbitrary function of time is presented within this context. Based on these solutions, the consolidation of the single-layer unsaturated soil subjected to arbitrary external loading is studied. The exact solutions are validated and the consolidation characteristics of the unsaturated soil are discussed by analyzing several examples. 2 GOVERNING EQUATIONS Without loss of generality, it is assumed hereinafter that 1) an isothermal condition and homogeneous soil condition prevail in the spatial domain of concern, 2) the deformation of the soil matrix is linear elastic and infinitesimal, 3) both the solid material and the pore water are incompressible, 4) the pore gas is an ideal gas and continuous, 5) the coefficients of permeability with respect to water and air, and the volume change moduli remain constant during the transient processes, and 6) the effects of air diffusing through the water and the movement of the water vapor are ignored. The governing equations for the consolidation equation of the unsaturated soils was originally proposed by Fredlund and Hasan [5], and later modified for different applications. For a one-dimensional consolidation problem associated with unsaturated soil, which is subjected to infinitely distributed forces on the boundary (Fig. 1), the governing equations can be simplified as duw dt + cw dua dt + CI da dt il cw d 2uw dz2 dua dt + Q duw dt +c: d: dt - Ca d2Ua dz2 (1a) (1b) where C = 1w / w h- w - m2 / mik - ra mik - , W ~ w , C - -— w mw mwk Ywm ww m2 C =- mtk - m2 - (1 - Sr )nrUa,m 1 iUa ) CV = ka RT,„, gUa Ma (K - rf - (1 - Sr )nrUam 1 (u a) ) C=- m1k - m2 - (1 - Sr )nrUa,m 1 Ua uw and ua are the unknown excess pore-water pressure and the excess pore-air pressure, respectively; a is the total vertical stress that is a function of the time; mWk is the coefficient of the water volume change with respect to the change in the net normal stress (a-ua); mW is the coefficient of the water volume change with respect to the change in the matric suction (ua- uw), m'^k is the coefficient of the air volume change with respect to the change of the net normal stress; and ma2 is the coefficient of the air volume change with respect to the change in the matric suction. Subscript k stands for the K0-loading condition (i.e., zero lateral deformation); kw is the water permeability in the unsaturated soil, which is assumed to be constant during the consolidation; yw is the density of the water phase; ka is the air conductivity; u° = u°a + uatm ; R is the universal air constant; Ttem is the absolute a m 2 a m 88. Acta Geotechnica Slovenica, 2018/1 J. Li et al.: A general analytical solution to the one-dimensional consolidation problem for unsaturated soil under various loading conditions temperature; Ma is the average molecular mass of the air phase; u° is the initial excess air pressure; uatm is the atmospheric pressure; Sr is the degree of saturation; and nr is the porosity. The initial conditions and boundary conditions, respectively, are given by (2) ua(z,0) = u0, uw(z,0) = u ua (0, t ) = 0, uw (0, t ) = 0 dua (H, t) _ 0 duw (H, t) _ 0 dz dz (3a) (3b) where, u° and uW are the initial excess air and water a w pressures (at t = 0), respectively. It should be noted that for the situations in which the lower surface of the soil layer is also permeable like the upper surface (i.e., double-drainage condition), the boundary condition dua (H, t) _n duw (H, t) given by dZ " dz " has to be imposed in the middle of the layer (at z = H/2). In other words, the results for the single drainage condition can be adopted to determine the solution for the double drainage condition by interpreting H as the drainage height. 1 1 i 1 H 1 ! ■ -lllllljj / 0 Top surface Unsaturated soil layer Bottom surface \ //////////, '////////// ,Z Figure 1. One-dimensional consolidation in unsaturated soils. Using the same procedure as Conte and Troncone [17], the general loading function a(t) can be expanded into a larger number M of harmonic components using the Fourier series, provided that a(t) is a periodic function satisfying Dirichlet's conditions in the interval (0,T). Namely, we have a M a(t) = "f + ZK cos(wkt) + bk sm(wkt)] (4) 2 k=1 where the amplitudes ak and bk associated with the frequency Wk = 2kn/T (with k = 1, 2, 3 ...) are provided, respectively, by 2 ¡"T T 2 (T ak = TJ0 a(t)cos(wkt)dt (5) 2 r bk = j J0 )smOti)dt (6) T is the period of the a(t), and a0 can be obtained from Eq. (5) by setting w = 0, i.e., 2 rT ao = T Jo a(t)dt 3 ANALYTICAL SOLUTIONS (7) For convenience, Eq. (1) can be written in the following dimensionless forms: du „ du „„ ds d2u —w+c—-+= —w dr dr dx 3Ç du + C u + C° ^ = C dU dt 2 dt " dr 3 dč,2 (8a) (8b) which are subjected to U (£,0) = 1,uwfê,0) = 1 in 0 1 (9) ua (0,t) = 0, vw(0,z) = 0 in z> 0 (10a) (1,t) = 0, (1,T) = 0 (10b) where ua U0' _ uw s _ 0 >A1 0 uu w w , S2 _ - u° z _ H (11a) Cyt H2 , C Cwua C - 0 ' C2 -u w CauW u0 -, C3 = C'y cw (11b) 3.1 Homogeneous cases Under a constant surface traction, i.e., da/dt = 0 , Eq. (8) ends up with the following homogeneous equations: dUw+C u = d2"w dr 1 dr dÇ2 dUa + C dUw _ C ^UL 2 ^£2 (12a) (12b) dr dr d% which are subjected to U (1,0) = 1,MS,0) = 1 in 0 <£< 1 (13) ua(0,t) = 0, Uw(0,t) = 0 in t > 0 (14a) dum = 0, duxhi! = 0 in T>0 (14b) The differential transformation method (DTM) is an analytical method for solving integral equations, ordinary and partial differential equations. The method Acta Geotechnica Slovenica, 2018/1 89. J. Li et al.: A general analytical solution to the one-dimensional consolidation problem for unsaturated soil under various loading conditions provides the solution in terms of a convergent series with readily computable components, i.e., it is an iterative procedure for obtaining the analytical Taylor-series solutions of differential equations. Different applications of the DTM method can be found in [18-29]. Based on the DTM, we have a i! j! d?dvJ Uw CJ)«'r)| (0,0) i! j! d^'drj |(0'0) The corresponding function can be expressed as 1» i» j=0i=0 » 1» ^ = (i, J j=0 i=0 (15a) (15b) (15c) (15d) By substituting Eq. (15) into Eq. (12), we can derive C2(i + 1)(i + 2)Uw(i + 2, j)- C3 (i + 1)(i + 2)Ua(i + 2, j) U (i, j+1) = - (j+1)(CC2 -1) (16a) Uw (i, j + 1) = CC3 (i + 1)(i + 2)Ua(i + 2, j) - (i + 1)(i + 2)Uw (i + 2, j) (j +1)(CC2 -1) (16b) Accordingly, the initial condition, Eq. (13), can be rewritten as I Ua = ± ¿-^ (^^ +1 i=0 n n=0 2n +1 k=0(2k + 1)! 2 (17a) ± aw = - I—¿-^ ((2n+1K)2- i=o n n=0 2n +1 k=o (2k +1)! 2 (17b) which also implies that i-i U. (i,0)^ t ^ ^(^i, ' = 1,3,5. n n-o 2n +1 i! 2 (18a) V. (i,0) = i f ^ i-!^^ ), i = 1,3,5..... n n=o 2n +1 i! 2 (18b) where N is a large positive integer. From the boundary conditions, Eqs.(14), we have i» i» X Ua (0, j)T = 0, X Uw (0, j )T = 0 (19a) j=0 j=0 II (i + 1)U (i, j )T = 0, XI (i + 1)UW (i, j )T = 0 (19b) j=0i=0 j=0 i=0 that is, Ua (0, j) = 0,UW (0, j) = 0 (20) i» i» X (i + 1)Ua (i +1, j) = 0, X (i + 1)Uw (i +1, j) = 0 (21) i=0 i=0 Substituting Eqs. (18) and Eq. (21) into Eq. (16), and using the recursive method, we can derive i—1+2 j U (i i) = 4 f 1 (—1) 2 jj(j) ((2n + 1)gy+2j j nt02n +1 i! j !(C1C2 — 1)j 2 ' i = 1,3,5. (22) i-le2 j U (i j) = 4 f l (-l) 2 (j) ((2n e 1)ny+2j w n fo2ne 1 i! j!(C1C2 -1)j ( 2 , i = 1,3,5...... (23) where aw(j) and aa(j) satisfy the following recursive formula aw (j +1) = CiC3afl (j) - aw (jj (24a) aa (j +1) = C2aw (j) - Ca (j) (24b) where, aa(0) = aw(0) = 1, j = 0, 1, 2... From Eqs. (24), it follows that aa(J) = aw(J)= + 1)(C,C3 -A2 - 1)A/ - (X2 + 1)(C,C3 -Xl - \)XJ (CC3 -x2 - i)\j - (CC3 1)A2J xi-x-2 where A1 and X2 are the eigenvalues of the matrix '-1 C1C3 (25) (26) A = C -C 2 3 Using Eq. (16) and Eqs. (20) and (21), we obtain Ua (i, j) = 0, i = 0,2,4..... Uw (i, j) = 0, i = 0,2,4.... (27) (28) By the differential inverse transform of the two-dimensional DTM, it can be proven that 4 N 1 n n=0 2n + 1 ¡=1,3,5. (-1)2 (2n + \)n , i! 2 E oJj)r-((2n + 1)n)2 tV j! 4(C1C2 -1) J (48) i-1 i-1 90. Acta Geotechnica Slovenica, 2018/1 J. Li et al.: A general analytical solution to the one-dimensional consolidation problem for unsaturated soil under various loading conditions n rxn 4 l 1 L (-1)2 ((2n + 1Ky n n=o 2n +1 t=i35... t! 2 «w (j) r-((2n + 1)n)2 j=0 j! 4(CA -1) T]j (30) Note that the series expression of a sine function is: sin(x) = £ x' i=l,3,5... ' ! Hence Eqs. (29) and (30) can be rewritten as 4 " 1 ^ aa(j) r-((2n +1)n)2 ,(2n +1)n e, n n=o2n +1 j=o j! 4(CjC2 -1) 2 (31) 4 » 1 » aw (j) -((2n +1»2 v- . ,(2n +1> rc n=o 2n +1 pa j! 4(CjC2 -1) (32) Let N ^ x. By using Eqs. (25) and (26) and the series expression of an exponential function, Eqs. (29) and (30) can be written as , ,. 4 » Be + B2eA . (2n +1)* ,. Ua = -f-sin^ (33) * „=o 2n +1 2 =± (34) ^ n=o 2n +1 where A = -((2» + 1)n)2 4(CC -1) B = b2 = 4(cic2 -1) (A+ 1)(CCC3 -a2 -1) -(A2 + IXC1C3 -1) C1C3 -l2 — 1 A = D =- — A2 ■(QC3 -^-1) Äi — Ä2 Finally, we obtain the analytical solutions for the excess pore-air and water pressures in unsaturated soils as ^ (z> f) = 4«: - Bil^ s^^JL) (35) n n=0 2n +1 2 H where t (= C™t/H2) is the characteristic time of the excess pore-water pressure dissipation. 3.2 Non-homogeneous cases According to the exact solutions for the above homogenous governing equations and the principle of superposition [30], the exact solutions for the non-homogenous governing equations, Eqs. (8)-(10), assume the following forms: ® 2n +1 Ua (Z, t) = Z (tUZ) n=0 2H ® 2n +1 uw (zf) = Z % (t nz) n=0 2H (37) (38) Substituting Eqs. (37) and (38) into Eqs. (8)-(10), and . 2« +1 multiplying the resultants by sm(_2H~ nz), and integrating the resultants of Eq. (8) from 0 to H with respect to z, we obtain a family of ordinary differential equations as Y„ = MY + N n n n which is subjected to Yn (0)=2 iH sin(2n+1 nz)dz n MJ0 If/ where 2H Y„ (t) = (fa (t ),108s). 0.200.150,100.050: \ fa) \ > X " Sfc el al(N] \ \ -W'.i \ \ —yfcjf1 * \ \ -----iyl^lG V~ -"Ar»« \ ' \ \ V '-. \ * \ \ 10 10 10 10 t (s) 10s 10 0.4 0.30.20.1 " X ™ s v X. * \ \ (b) - , viA ■ Qin et al[«] \ tçAr0'1 \ -- W4 \ V^-i» \ — - KAv=ioi) \ 10 10' 10 10 t{ S) 10 10 Figure 2. Variations in the excess pore-air (a) and excess pore-water (b) pressures with dimensionless 0.25- 0.20- 0.15- 0.10- 0.05- — — VAM .....KAv-'° - - V^-ioo —/ " / / / / ! / 1 / 1 / / ! / / ! / / / / / / / .. - ^ y jS I - 10 10 10 î (s) 10 10 10 Figure 3. Compression of a single-layer unsaturated soil varia- time for different values of ka/kw. tion with time for different values of ka/kw. Acta Geotechnica Slovenica, 2018/1 93. J. Li et al.: A general analytical solution to the one-dimensional consolidation problem for unsaturated soil under various loading conditions Variations of the normalized settlement S/S0 (S0= mSkq0H0) caused by the constant surface traction are illustrated in Fig. 3.It is clear that at the beginning of the consolidation the settlement generally proceeds faster as ka/kw increases, thus characterizing the inverse sigmoid shapes. However, at the later stage of consolidation, for all the ka/ kw values, the settlement curves increase unanimously after 108s. Now, consider the situation that the coefficients Ca , Cw and Ca satisfy Eq.(50). In this case, the solutions of the problem are given by Eqs. (53) and (54). In the calculations, it is assumed that C1 = 1.0, H = 10m, u0 = 20 kPa, and uW = 40 kPa. The calculated results are illustrated in Fig.4, which shows the variations of uw and ua with time under different C3 (i.e., C3 = Ca/CW) for the single drainage condition. The top surface is permeable to water and air, whereas the bottom surface is impermeable to water and air. Clearly, both the excess pore-water pressure and excess pore-air pressure gradually decrease with time, and finally approach a stable value. In contrast to Fig. 2, however, both uw and ua dissipate almost simultaneously to a stable value. In addition, it is clear that the excess pore-air pressure decreases with an increase of C3 = C^/Cy, whereas the excess pore-water pressure variation shows a different tendency. For the situation that the coefficients Ca , Cw and Ca satisfy Eq.(55), the solutions are given by Eq. (58) and (59). In the calculations, the parameters are selected as: H = 10m, u0a = 20 kPa, and u0w = 40 kPa. The calculated results are given in Fig.5 for the different values of C1 at C2 = -0.2 . It is clear that, in practice, a variation of C1 has no effect on the dissipation of the pore pressure. To explore the influence of C2, Fig. 6 shows the results of Eqs. (58) and (59) for different C2 at C1 = -0.07 . It is clear that with the change of time, the excess pore-air pressure decreases and it finally attains a stable value. 0.200.15" ii — 0.10% 0.05- oi 0.4- 0.3- u — 0>| 0.1-