Mechanisms of quasi-preconsolidation stress development in clays: A rheological model

Soils and Foundations ]]]];](]):]]]–]]] The Japanese Geotechnical Society

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Mechanisms of quasi-preconsolidation stress development in clays: A rheological model Boning Maa, Balasingam Muhunthanb,n, Xinyu Xiec a

School of Civil and Transportation Engineering, Beijing University of Civil Engineering and Architecture, Beijing, China b Department of Civil and Environmental Engineering, Washington State University, Pullman, WA, USA c Ningbo Institute of Technology, Zhejiang University, Ningbo, Zhejiang, China Received 14 October 2012; received in revised form 16 November 2013; accepted 24 January 2014

Abstract This paper deals with the development of a mechanistic model for the ageing consolidation behavior of clays with the focus on aspects related to the development of quasi-preconsolidation pressure. The initial use of such pressure in design met with criticism, but field and laboratory evidence, which highlights its significance, continues to accumulate. A nonlinear rheological model is used to numerically simulate the consolidation process of clay in laboratory tests and to identify the basic mechanical parameters that contribute to the development of the quasi-preconsolidation phenomenon. Methods to identify the parameters of the model from oedometer tests are described. It is shown that while the variation in soil modulus can be characterized by a linear form in the virgin compression region, it is nonlinear in the recompression region and is best characterized by a hyperbolic function. Changes to the modulus in the recompression region, due to ageing, is shown to be the dominant cause of the development of the quasi-pc phenomenon. Observed results as well as numerical simulations demonstrate that specimens that had aged longer show increased quasi-pc values. While the variation in soil modulus controls the EOP curve of clays, the observed time effects, such as the “vanishing pc” phenomenon, are controlled primarily by changes in soil viscosity. However, this has no bearing on the development of the quasi-pc phenomenon. & 2014 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.

Keywords: Ageing; Soil rheology; Consolidation; Creep; Viscosity; Yield stress; Quasi-preconsolidation pressure

1. Introduction Arthur Casagrande recognized that the reloading compressibility of a soil is much smaller than its virgin compressibility until the current effective overburden stress has exceeded a certain stress level. The 1-D consolidation curves presented in n

Corresponding author. Tel.: þ1 509 335 9578; fax: þ 1 509 335 7632. E-mail address: [email protected] (B. Muhunthan). Peer review under responsibility of The Japanese Geotechnical Society.

his seminal paper (Casagrande, 1932), as well as subsequent data, show that soil compressibility diminishes progressively until an abrupt change or yield occurs and this stress is exceeded. The yield stress is generally defined as the preconsolidation pressure (pc), and the methodology proposed by Casagrande (1936) to determine it, based on laboratory consolidation tests, has been widely used by geotechnical engineers to calculate consolidation settlements in clays. Variations of this methodology have been proposed by a number of investigators (see a summary in Mitchell and Soga, 2005). Significant differences between the observed and the predicted settlements of clays, using the pressure determined from the Casagrande methodology, prompted Gerald A. Leonards of

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Please cite this article as: Ma, B., et al., Mechanisms of quasi-preconsolidation stress development in clays: A rheological model. Soils and Foundations (2014), http://dx.doi.org/10.1016/j.sandf.2014.04.012

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2

Nomenclature cv hc kv mv q t u ua z Hdr

coefficient of consolidation in the consolidation stage depth of the moving boundary coefficient of vertical permeability coefficient of volume compressibility loading pressure time excess pore water pressure average excess pore water pressure depth length of drainage

normal

Purdue University to initiate a systematic program using remolded, artificially sedimented, and field clay samples to identify factors other than the maximum past effective stress state that could affect the value of pc. The results of this program (Raju, 1956; Leonards and Ramiah, 1959; Leonards and Altschaeffl, 1964) showed that the yield stress could exceed pc significantly due to many factors, including thixotropy and volumetric creep. The increased yield stress due to such effects was termed by Leonards as the quasipreconsolidation pressure, pcq. Based on a suite of laboratory results, he used the ratio of pcq/pc E 1.4 to make “class A” predictions of the consolidation settlements of seven buildings in the town of Drammen, Norway (Leonards, 1968). The extraordinary match between the observed and the predicted settlements of these buildings enabled him to consistently use pcq in geotechnical practice with great success (Leonards, 1977, 1980). Bjerrum (1967, 1972) developed hypotheses to explain the quasi-preconsolidation phenomenon, but showed that the pcq would actually vanish at field rates of strain. Since then, his isotache concept has drawn considerable attention and it is widely adopted by researchers to date (Watabe et al., 2012; Tsutsumi and Tanaka, 2012). More recent studies, however, have begun to shed new light onto the generally beneficial effects of the ageing of soils and onto the preconsolidation phenomenon in particular and the factors that influence such effects (Mitchell and Soga, 2005). In the 25th Terzaghi Lecture, Schmertmann (1991) reported experimental results which show that the apparent OCR (ratio of the apparent preconsolidation pressure to the current pressure) of Italian clay equaled 1.5 to 2.0 after creep and, more importantly, even dry Florida quartz sand showed the development of similar effects. Schmertmann termed this phenomenon ageing preconsolidation. The establishment of a functional relationship between the time-effective stress and the apparent overconsolidation rate of artificially sedimented kaolinite clay was the focus of a study by Athanasopoulos (1993). Such a relationship would enable the prediction of the long-term behavior of soils from short-duration tests. The mechanisms of the existence of an apparent preconsolidation pressure and an ageing phenomenon in clays and

E0 E1 Es Tv U εz γw η0 η1 sz0 p pc pcq

modulus of the spring in the Maxwell body modulus of the spring in the Kelvin body modulus of soil compressibility time factor degree of consolidation vertical strain unit weight of water coefficient of viscosity of the Maxwell body coefficient of viscosity of the Kelvin body vertical effective stress consolidation pressure (total vertical stress) preconsolidation pressure quasi-preconsolidation pressure

sands have been put forward by a number of investigators. Leonards and Deschamps (1995) presented a summary of such mechanisms and listed the following as contributing factors: (a) alteration clay minerals, (b) ions in pore water due to changes in concentration and/or valence, (c) precipitation/ cementation, (d) mineral leaching/internal erosion, and (e) combination of time/volumetric strains at a constant effective stress. While circumstances may dictate some of these factors to be of greater consequence than others, Leonards hypothesized (e) to be the dominant mechanism in the development of the quasi-preconsolidation pressure. Baxter and Mitchell (2004) presented the results of a systematic laboratory testing program to study the influence of above factors (a) to (e) on the presence and magnitude of the ageing effects in sands. Their results suggest that while the influence of most ageing factors of natural deposits can be identified in small-scale laboratory tests, the time-dependency may not be replicated (see also Seng and Tanaka, 2012) Soil particles mainly rearrange during the primary compression process. However, the increased time due to ageing enables the formation of new bonds between particles, this results in an altogether different behavior. These bonds may be mechanical (Schmertmann, 1991), chemical, or biological (Bolton, 2010) depending on the type of soils and stress histories. Such effects on consolidation are well documented and analyzed (Nakai et al., 2011; Deng et al., 2012. The above discussion suggests that despite abundant field and laboratory evidence on the beneficial effects of ageing in soils, their use in practice has been hampered due to the lack of a comprehensive mechanistic model that can explain the different phenomena with clarity. This study uses a rheological model of the consolidation process in laboratory tests to identify the basic mechanical parameters that contribute to the development of the quasi-preconsolidation phenomenon in clays. 2. Features of quasi-preconsolidation pressure in clays While a great deal of recent laboratory and field evidence exists, there is nothing more eloquent in setting the features of quasi-preconsolidation than the set of classical experimental

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data shown in Fig. 1, redrawn from Leonards and Altschaeffl (1964). This figure shows the void ratio-logarithm pressure plot for two specimens which were loaded to 50 kPa and then allowed to remain under this constant pressure for 90 days. It is noted that specimen 6-1 was sedimented in the laboratory with remolded soil, whereas specimen 8-1 was obtained from the field using the best sampler available at that time. The specimens were then unloaded, permitted another 90-day rest period, and finally loaded to pressure levels well in excess of the maximum past pressure experienced. Distinct quasipreconsolidation pressures were obtained for both specimens. It is evident that the nature of the specimens results in the observed differences in behavior of these samples, especially the magnitude of pcq. In the 7th Rankine Lecture of the British Geotechnical Society, Bjerrum (1967) introduced the hypothesis of delayed consolidation illustrated in Fig. 2. Line AB represents “instant” compression during a conventional oedometer test, whereas BC shows the change in void ratio under constant effective stress due to delayed consolidation (secondary compression). If the soil at C is now loaded as in a conventional oedometer test, a new curve, CD, with a distinct pc-effect that represents the instant (primary consolidation only) compression, will be obtained. On the other hand, curve CE depicts the behaviour corresponding to the time-dependent compression; in this case, thirty years. As seen, the preconsolidation effect actually diminishes with the decreased loading rate leading to the

conclusion that the quasi-pc phenomenon would actually vanish in the field. However, as seen from the void ratiologarithmic pressure plot shown in Fig. 1, a much larger quasipreconsolidation pressure was observed than that which could be predicted using the Bjerrum type curves. In addition, the magnitudes of the pcq were different for samples 8-1 and 6-1. These phenomena suggest that the aged-induced contribution to quasi-pc cannot be captured by Bjerrum's hypothesis. This issue has become an ongoing debate among soil science researchers. Recent studies have provided additional insights into this and related issues (Lade et al., 2010; Tatsuoka et al., 2008). Further convincing evidence related to the quasi-pc phenomenon has been presented by Schmertmann (1991) (see also discussions by Crawford, 1992; Ellstein, 1992 and closure by Schmertmann, 1992) and Athanasopoulos (1993). Several phenomenological models, based on plastic yielding as well as the visco-plastic theory (e.g., Yin and Graham, 1999; Kutter and Sathialingam, 1992), have been proposed to account for the time-dependent deformation in clays. The study here proposes a fundamental rheological model whose parameters can be determined from 1-D compression data to quantify the development of pcq.

3. Rheological model The Terzaghi model for the effective stress–volume strain– time relation of the mineral skeleton in clay soils consists of a linear spring, i.e., the volumetric strain is proportional to the effective stress and is time independent. Since this is clearly inadequate to capture the nonlinear and the dependent deformation associated with the consolidation of clays, rheological models of varying complexities have been proposed.

2.7 8-1 in-place 6-1 sampled

2.4 2.1 1.8 e 1.5 1.2 0.9 0.6 1

10

100

3

100

p (kPa)

Fig. 1. Illustration of quasi-pc in 1-D tests (Leonards and Altschaeffl, 1964).

A B pc Void ratio e C D Instant E 3 years 30 300 3000 Consolidation pressure p

Fig. 2. Schematic diagram of Bjerrum's time-line model (Bjerrum, 1967).

Fig. 3. Four-element rheological model (Xie and Liu, 1995).

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4

Taylor and Merchant (1940) were the first to take soil rheology into account to analyze the process of soil consolidation. Gibson and Lo (1961) have used a three-element rheological model to study one-dimensional consolidation. The Gibson–Lo model consisted of a spring connected in a Kelvin body series to represent the mineral skeleton of clays. Zhao (1989) proposed the use of a generalized Kelvin model and used Laplace transformation to solve the governing equations of saturated soft clays. Xie and Liu (1995) extended the Gibson–Lo model with the addition of a dashpot to simulate soil behavior and developed closed-form solutions. This four-unit model is shown in Fig. 3 and consists of a Maxwell body and a Kelvin body in series. Note that E0 and E1 are the spring moduli of the Maxwell and Kelvin body, respectively, and that η0 and η1 represent their viscosities. This four-unit model is adopted here as it has the necessary ingredients to capture the effects of the physical phenomena associated with the consolidation behavior of soils. It is also easily incorporated into the governing equation and is amenable to efficient solution techniques which provide accurate predictions of consolidation behavior, as shown in the following sections. 4. Soil model parameters

log E 0 ¼ m1 log pþ m2

ð1Þ

where m1 and m2 are the slope and the intercept of the line, respectively. However, this is not the case for the recompression region. The variation in the modulus in this region is nonlinear and best characterized by a hyperbolic relation. It is also important to note that the variation is different for samples 6-1 (lab consolidated) and 8-1 (field), and thus reflects their different histories. The simplest form of the hyperbolic relation, relating the logarithmic modulus and the logarithmic pressure, is log E 0 ¼

c1 log p þ c3 log p þ c2

ð2Þ

where c1, c2, and c3 are constants that can be determined by curve-fitting. Table 1 shows the values for the data shown in Fig. 4. Rearranging the above equation results in 1 1 c2 ¼  c1 log E 0  c3 c1 log p

ð3Þ

when consolidation pressure p becomes very small (i.e., p-0), namely, 1 1 log E 0  c3 c1

ð4aÞ

or E0 max ¼ 10ðc1 þ c3 Þ

ð4bÞ

Thus, constants c1 and c2 are related to the maximum elastic soil modulus. 10

4.1. Maxwell body soil elastic moduli E0 The relation between elastic modulus E0 and the consolidation pressure can be established by re-interpreting the consolidation test data in Fig. 1. The data on the e–p plot in Fig. 1 is redrawn in the form of the (dp/dε)-p plot shown in Fig. 4. Since E0 ¼ dp/dε, the E0 variation with p can be obtained from this figure and a functional form develops. The results are

8-1virgin compression 8-1recompression 6-1virgin compression 6-1recompression linear fit hyperbolic fit of 8-1 hyperbolic fit of 6-1

100

E0 (MPa)

In order to use the rheological model, its elastic and viscous parameters must be determined first. Each component in the fourelement model has different effects on the rheological consolidation behavior, as described in Xie and Liu (1995). Specifically, a decrease in the Maxwell body parameters, either in the modulus of the independent spring (E0) or in the coefficient of viscosity η0 of the independent dashpot, decreases the consolidation rate. These two parameters are the major ones that control the consolidation behavior and need to be determined carefully. On the other hand, changes in the parameters, (E1) and η1 in the Kelvin body, were found to have only a minor effect on the consolidation process. Thus, the Kelvin body parameters are kept constant in our analysis. The parameters of the model can be determined from 1-D compression tests, as shown below.

striking in that the variation in the logarithmic elastic modulus with logarithmic stress during the virgin compression, either below pc or beyond pcq, is linear and can be described by

p0

quasi-pc

1000

10

100

1000

p (kPa)

Fig. 4. Soil elastic modulus variation with consolidation pressure.

Table 1 Parameters of soil modulus. Virgin compression

8-1 Recompression

6-1 Recompression

Soil viscosity

m1

m2

c1

c2

c3

c1

c2

c3

n1

n2

1.067

0.275

0.153

 2.074

4.006

0.289

2.311

3.797

3.984  1012

6.305  1012

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4.2. Maxwell body soil viscosity η0

η0

Crawford (1964) presented the results of incremental oedometer tests on undisturbed “Leda” clay samples from the St. Lawrence and Ottawa River valleys that have been used in several other past investigations (see a summary in Mitchell and Soga, 2005). These results are reproduced in Fig. 5. It can be seen that the preconsolidation pressure is about 18.5 kPa for the one-day curve and 13.3 kPa for the one-week curve. This observation was instrumental in the “vanishing preconsolidation pressure” phenomenon referred to by a number of investigators. Bjerrum (1967) showed that using vertical deformation ε (vertical distance between the time lines) and the time elapsed t (time difference between lines), the soil viscosity under certain effective stress can be estimated by η0 ¼ p  t/ε. This technique was applied to the data in Fig. 5, and the corresponding viscosity parameters at each stress level are shown in Fig. 6. It can be seen that the viscosity corresponding to effective stress levels below pc are fairly constant albeit with some scatter, whereas the values corresponding to stress levels beyond pc have a definite linear relationship with effective stress. Notice also that the viscosity in the recompression region is an order of magnitude higher than those in the virgin compression indicating that the creep effect is minimal in this region. The proposed variation in soil viscosity is shown in Fig. 7. Note that the effects of soil creep at stresses beyond pc have been found to attenuate with time in the 1-D tests (Vyalov and Sapunov, 1986). The 0 pc=26.5 kPa

pc=13.3 kPa 10 ε (%)

pc=18.5 kPa

20 EOP 1d 7d

30 10

100 p (kPa)

Fig. 5. Time duration and apparent preconsolidation pressure (after Crawford, 1964).

12

η0(1012Pa·s)

5

8

log t

p pc

Fig. 7. Proposed viscosity model.

non-dimensional form for the variation in η0, after the effective stress reaches pc, can be described by

 p t η0 ¼ n1 þ n2 log ð5Þ pc t0 where t0 is the time when the excess pore water pressure dissipates completely, i.e., the EOP (end of primary consolidation) moment. 4.3. Kelvin body parameters E1 and η1 Unlike Maxwell parameters E0 and η0, the determination of Kelvin parameters E1 and η1 is exhaustive and makes use of the curve-fit matching of laboratory excess pore water pressure with time results to the analytical solution of the governing equation by Xie and Pan (1995) described later and provided in Appendix A. Details of this process are provided in the dissertation along with the computer program by Li (2005). Briefly, the program module contains Xie and Pan's analytical solution as curve-fit and uses a non-linear least squares method to find the best fit to the laboratory curve. With Maxwell parameters E0 and η0 set, the initial values for Kevin body parameters E1 and η1 are input first. The program will iterate until an exact match is found, and then outputs in a window both the original and the fit curves, the times for the iteration, and the final Kelvin body model parameters. The choice of sound initial input parameters is key to the efficient matching of the curves. In this study, an E1 of 5000 kPa and an η1 of 4  108 kPa s were found to be good initial values that resulted in the final values of E1 ¼ 8000 kPa and η1 ¼ 3.7  108 kPa s for an exact match. Fortunately, past numerical simulations have shown that values affect only the rate of consolidation towards the tail end of the consolidation process (Xie and Liu, 1995). Thus, we have used E1 ¼ 8000 kPa and η1 ¼ 3.7  108 kPa s and kept them the same in all of the analyses here without the loss of generality.

4

5. Governing equation 0 10

20

30

40

50

60

70

p (kPa)

Fig. 6. Viscosity variation in clays.

80

90

The schematic diagram of a double drained thin layer of clay specimen of thickness H, in an oedometer undergoing consolidation, is shown in Fig. 8. The soil is subjected to an instantaneously applied load q on the surface that induces

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6

z¼H

q

u ¼0

ðpermeable surfaceÞ

ð11bÞ

and the initial condition is given by ð12Þ

t ¼ 0 u ¼ qð0Þ ¼ qu

Note that when parameter η0-1, the model decreases to the Gibson–Lo model, and the governing equation becomes   Z t 0 sz  ðE1 =ηi Þðt  τÞ kv ∂2 u ∂ s0z  ¼ þ e dτ ð13Þ ∂t E0 γ w ∂z2 0 ηi

H

Soillayer

Furthermore, when η1-1 or E1-1, there is no creep involved, and Eq. (13) falls to the linear elastic onedimensional consolidation equation, identical to that of Terzaghi's.

z

∂s0 kv ∂2 u ∂u  2 ¼ z ¼ ∂t γ w mv ∂z ∂t

Fig. 8. Schematic of consolidation test.

constant total stress throughout the layer during consolidation. The addition of the surcharge will result in an instantaneous excess pore pressure equal to q at the very beginning of the consolidation process. With time, it will be begin to decrease and eventually become zero. According to the Terzaghi's principle, the added effective stress of the soil at depth z, sz0 , is given by 0

sz ¼ q u

ð6Þ

where u is the excess pore water pressure. The equation governing the consolidation of the clay layer is given by Gibson and Lo (1961), Xie et al. (2008) k v ∂2 u ∂εz  ¼ γ w ∂z2 ∂t

ð7Þ

where εz, u, and sz0 are the vertical strain, the excess pore water pressure, and the vertical effective stress, respectively, kv is the coefficient of vertical permeability of the soil, γw is the unit weight of water, and t and z are time and depth, respectively. For the four-unit rheological model (Fig. 3), the vertical strain of the soil mass, εz, can be expressed in an integral form as Z t 0 Z t 0 s0z sz sz  ðE1 =η1 Þðt  τÞ εz ¼ þ dt þ e dτ ð8Þ E0 η 0 0 0 η1 where t is the time after the application of the load and τ represents a dummy variable. The substitution of Eq. (8) into Eq. (7) results in   Z t 0 Z t 0 E sz sz  η 1 ðt  τÞ k v ∂2 u ∂ s0z  ¼ þ dt þ e 1 dτ ð9Þ ∂t E 0 γ w ∂z2 0 η0 0 η1 Using the effective stress relationship (Eq. (6)) in the above equation leads to   Z t Z t k v ∂2 u ∂ u u u  ðE1 =η1 Þðt  τÞ  2 ¼ þ dt þ e dτ ð10Þ ∂t E 0 γ w ∂z 0 η0 0 η1 The boundary conditions of the system are given by (Fig. 1) z¼0 u¼0

ðpermeable surfaceÞ

ð11aÞ

ð14Þ

where mv is the coefficient of volume compressibility, mv ¼ 1/Es. In addition to the modulus and viscous parameters, the solution of Eq. (10) requires input on soil permeability. 5.1. Model of soil permeability Permeability decreases nonlinearly during compression and a suitable expression is needed to capture its variation. Various relationships between permeability and consolidation parameters, such as the void ratio or pressure, have been proposed in the literature. Based on its simplicity and widespread usage, the following equation relating permeability to void ratio is adopted: For the recompression region, e e0 ¼ Cke logðkv =kv0 Þ

ð15aÞ

and for the virgin compression region, e ec ¼ C k logðk v =k vc Þ

ð15bÞ

where Cke and Ck are the slopes of the recompression stage and the virgin consolidation stage, respectively, ec is the void ratio of the soil when the effective stress is pc, kvc is the corresponding permeability, and kv0 is the initial parameter of permeability determined in the laboratory. The above relationships, shown in Fig. 9, have been extensively verified for many clayey soils e e0 ec

C ke

Ck

k vc

k v0

log k v

Fig. 9. Proposed non-linear permeability model.

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(Hvorslev, 1960; Raymond, 1966; Mesri and Rokhsar, 1974; Leroueil et al., 1992).

7

for mvei and mvi discretized as mvik 

1 e  ðE1i =ηi Þtk  1 k  1 þ ∑ ½_uai ðt j ÞðeðE1i =ηi Þtj  ðE1i =ηi Þtj  1 Þ E 0i E1i u_ ai ðt k  1 Þ j ¼ 1

6. Solution of governing equation

ð19Þ

Due to the nonlinearities involved, the solution of the governing equation makes use of a hybrid combination of analytical as well as numerical methods. This is done by first transforming the governing equation into a linearized form for which an analytical solution can be obtained. The discretized forms of these solutions, for individual sublayers, are then summed up numerically to obtain a solution for the entire layer. 6.1. Discretization of time and space The finite medium is uniformly divided into sublayers, marked by subscript “i”, and the time is divided into short intervals, marked by subscript “k”. Accordingly, the governing equation for each sublayer becomes   Z Z E0i k vi ∂2 ui ∂ui E 0i t E0i t 1þ  2 ¼ RðτÞdτ þ RðτÞe  ðE1i =η1i Þðt  τÞ dτ γw ∂z ∂t η0i 0 η1i 0

 ∂ui  i where Ri ðτÞ ¼ ∂u = ∂τ ∂τ  

τ¼t

ð16Þ  . Note that the above equations

satisfy the continuity conditions, namely, ui ¼ ui þ 1 ;

z ¼ zi

kvi

∂ui ∂z

¼ kvði þ 1Þ

∂ui þ 1 ; ∂z

¼ 1; 2; …; n  1

i ð17aÞ

where u_ ai ðt j Þ ¼ ðuai ðt j Þ uai ðt j  1 Þ=Δt j Þ is the rate of change in the of average pore pressure. Note that for the initial step, k¼ 1, mvi1 ¼ 1=E 0i , and cvi1 ¼ k vi E0i =γ w . It is important to note that while the boundary and the initial conditions remain the same for each sublayer (Eqs. (17a) and (17b)), the initial condition for each time interval is different and is given by u0i ¼ uai ðt k  1 Þ

where uai(tk  1) is the average excess pore water pressure of the ith layer at the end of the previous time interval, namely, Z 1 zi uai ¼ ui ðz; t k  1 Þdz ð21Þ hi z i  1 where hi is the thickness of the ith sublayer. Note that for the first time interval,uai ¼ ui ðz; 0Þ ¼ qð0Þ. 6.2. Excess pore water pressure distribution For linear elastic layered systems, Xie and Pan (1995) have developed analytical solutions to the governing set of equations in the same form as in Eq. (18a). A summary of their solutions (originally in Chinese) is provided in Appendix A. Using their solution (Eq. (A6)), the pore water pressure at the bottom of each sublayer at a selected time interval tk is given by 1

ui ðz; t k Þ ¼ ∑ C m gmi ðzÞe  βm Δtk

u ¼ qð0Þ ¼ qu

ð17bÞ

Using a set of modified variables, the governing equation (Eq. (16)) for each sublayer can be cast in a familiar form as cvi

∂ 2 ui ∂ui ¼ ∂z2 ∂t

i ¼ 1; 2; …; n

k vi γ w mvi

hc r zr H

C m e  βm Δtk  1 ½Ami ðC i  Di þ 1 Þ þ Bmi ðBi þ 1  Ai Þ μi ρi λm i¼1 n

uai ðt k  1 Þ ¼ ∑

ð23Þ

i ¼ k; k þ 1; …; m

ð18bÞ

Note that the definition of variables, such as A and B in Eqs. (22) and (23), are the same as those found in Xie and Pan's solution (Appendix A). 6.3. Deformation of clay layer

and mvi ¼

Substituting Eq. (22) into Eq. (21), the average excess pore pressure of each sublayer uai(tk  1) can be obtained as

ð18aÞ

where cvi ¼

ð22Þ

m¼1

and the initial condition is given by t¼0

ð20Þ

1 1 þ E0i ηi

Z

t

RðτÞe



E 1i ηi ðt  τÞ

dτ

ð18cÞ

0

where cvei, cvi, kvi, mvei, mvi, E0ei, E0i, E1ei, E1i, ηei, and ηi are the modified set of consolidation parameters of the rheological model for the ith sublayer, respectively. It is assumed that the variation in soil parameters within a sublayer is negligible for short-time increment Δtk. Thus, the soil parameters within a sublayer at time tk are assumed to be the same as those at tk  1, where tk ¼ tk  1 þ Δtk. Using this assumption, the integrals of Eqs. (18b) and (18c) can be evaluated, and the resulting forms

The deformation of a sublayer is given by δi ðt k Þ ¼ εzi ðt k Þ  hi k Z 1 where 0

s ðt k Þ εzi ðt k Þ ¼ zi þ E0i

Z

tk 0

0

szi ðτÞ dt þ η1i

ð24Þ Z 0

tk

0

szi ðτÞ  ðE1i =η1i Þðtk  τÞ e dτ η1i ð25Þ

tc is the time at which the effective stress of the ith layer reaches the preconsolidation pressure. Thus, the total deformation of the

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8

0.0

entire clay layer at a given time t is n

St ðt k Þ ¼ ∑ δi ðt k Þ t¼1

ð26Þ

p=40 kPa, Tv=0.01 p=70 kPa, Tv=0.01 p=500 kPa, Tv=0.01

0.2

0.4 h/H

6.4. Average degree of consolidation

0.6

The average degree of consolidation is defined as the ratio of the average effective stress at any time to the final effective stress of the clay layer. Thus, the average degree of consolidation of a sublayer is given by U i ðt k Þ ¼

q0  uai ðt k  1 Þ q0

0.8

1.0 0.0

ð27Þ

U p ¼ ∑ ρi U i i¼1

1.0

500 kPa 70 kPa 40 kPa

0.2

0.4 U 0.6

The numerical solution of the governing equation with the rheological model parameters enables the simulation of many of the features associated with the ageing consolidation of clays including the development of the quasi-preconsolidation pressure. The thickness of the clay layer is assumed to be 20 mm, and the E0 and η0 parameters are those shown in Figs. 4 and 6, respectively. As mentioned previously, constant values are assumed for E1 and η1 and are set to 1.65 MPa and 1.43  1010 kPa s, respectively.

0.8

The samples used in the tests reported by Leonards and Altchaeffl (1964) were consolidated at pc=50 kPa, but the observed pcq was about 80 kPa (Fig. 1). In order to highlight the difference in excess pore pressure dissipation rates, analyses were conducted with external pressure values below pc, between pc and pcq, and above pcq. The excess pore water pressure variations with time within the sample corresponding to q¼ 40 kPa, 70 kPa, and 500 kPa are as shown in Fig. 10. Note that time factor T v ¼ cv  t=H dr 2 , which is non-dimensional, is adopted in the analysis as it captures the soil consolidation behavior more distinctly than real time t. It can be observed that excess pore water pressure dissipates much faster when subjected to 40 kPa and comparatively slower when subjected to 70 kPa, but very much slower under 500 kPa. A similar pattern is observed for the degree of consolidation versus the time factor shown in Fig. 11. For example, the excess pore pressure is completely dissipated (EOP) in only about 15 min under 40 kPa, in 0.92 h under 70 kPa, and in 9.1 h under 500 kPa. This indicates that the consolidation rate is controlled by changes in the soil modulus. It is our belief that mechanical ageing is the dominant factor that causes an

0.8

0.0

7. Simulations

7.1. Consolidation rates

0.6

Fig. 10. Excess pore water pressure dissipation.

ð28Þ

where ρi ¼ Hhi . Note that ∑ni ¼ 1 ρi ¼ 1.

0.4 u/p

Consequently, the average degree of consolidation for the entire clay layer can be obtained from Eq. (27) by summation as n

0.2

1.0 1E-4

1E-3

0.01

0.1

1

10

Tv

Fig. 11. Comparison of consolidation rates of soils subjected to differing stresses.

increase in the soil modulus, as reported by Schmertmann (1991). Although soil particles mainly rearrange during the primary compression process, the long elapse of time during the ageing process enables the formation of new bonds between soil particles resulting in an increased soil modulus. In turn, this increase leads to differences in the observed consolidation behavior, i.e., ageing-induced soil stiffness contributes to faster consolidation.

7.2. Recompression and quasi-pc The numerical solution was also used to predict the variation in void ratio-logarithm pressure corresponding to the consolidation history of laboratory specimen 8-1, as shown in Fig. 12. It can be seen that the results capture the development of the quasi-pc phenomenon elegantly. pcq falls beyond the original compression line with the ratio pcq/pc E1.6. This ratio is a reflection of the nonlinear compressibility within stress level pc opopcq (Fig. 4). It is important to note, however, that the recompression line eventually merges with the original compression line after pcq (Fig. 12). The variation in void ratio-logarithmic pressure, corresponding to the consolidation history of specimens 8-1 and 6-1, are compared in Fig. 13. After the same period of creep, they both

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B. Ma et al. / Soils and Foundations ] (]]]]) ]]]–]]] 2.4

9

2.0 1d compression line 7d compression line 30d compression line

1.6

1.8 e

e 1.2

virgin compression recompression creep and rebound

1.2

0.8 0.6

10

100

100

p (kPa)

Fig. 12. Variation in void ratio with pressure and quasi-pc.

0.4 10

100 p (kPa)

100

Fig. 14. Simulation of the vanishing pc. 1.7 virgin compression 8-1 recompression 6-1 recompression

1.6

parallel, which follows the delayed compression hypothesis (Bjerrum, 1967) highlighted in Fig. 2. However, since the variation in elastic modulus in the recompression region is kept the same, the abrupt change must necessarily occur at decreasing pressures to enable the transition into the appropriate virgin compression curve. This is what is reflected in the vanishing of the preconsolidation pressure.

e 1.5

1.4

1.3 45

55

65

75

85

p (kPa)

Fig. 13. Quasi pc development with ageing.

show the development of quasi-pc, but with a slightly larger value for field sample 8-1 as it had been aged much longer. 7.3. Vanishing pc The decrease in preconsolidation pressure with the increase in the duration of the load application in the laboratory tests, first highlighted by Crawford (1964), and the subsequent usage of the concept of vanishing pc phenomenon (Bjerrum, 1972; Leroueil et al., 1985) posed significant difficulties to the rationale for the quasi-pc phenomenon. We believe that the quasi-pc phenomenon is the consequence of the increase in the age-induced modulus in the solid skeleton, as documented by Schmertmann (1991, 1993) and Athanasopoulos (1993) and numerically simulated here, whereas the rate-dependent value of the preconsolidation pressures observed in the laboratory is the result of the changes in viscosity of the solid skeleton due to secondary compression. This phenomenon is difficult to reproduce in the laboratory, except possibly through the use of smaller load increments around the preconsolidation stress on specimens, as advocated by Leonards (1977). In order to examine this issue, we performed an additional series of numerical simulations by keeping the variation in hyperbolic modulus in the recompression region constant while changing the viscosity. The variation in void ratio-logarithm pressure corresponding to the specimens subjected to load durations of 1 day, 1 week, and 1 month are as shown in Fig. 14. It can be seen that in the normally consolidated range, the compression curves are approximately

8. Conclusions Significant ageing effects occurring in virtually all types of soils, including dry sands, have now been well documented in geotechnical practice. Soil ageing over periods of time can generally cause significant improvements in key soil properties. Several theories for the mechanisms contributing to such improvements have been proposed in the past, but few fundamental mechanistic models have been developed to quantify such behavior. This study has been concerned with the development of a mechanistic model for the ageing consolidation behavior of clays with the focus on the aspects related to the development of the quasi-preconsolidation pressure. Benefits from the use of increased pressure in the design are significant, but it has met with skepticism in the past due to difficulties in reconciling various interpretations of the consolidation tests. This study uses a four-unit rheological model incorporated into the governing equation of consolidation to simulate many of the features associated with the consolidation behavior of ageing clays. The parameters of the model are determined from oedometer tests. The governing equation of the consolidation process with the nonlinear rheological model is solved with appropriate boundary and initial conditions. The solution makes use of a hybrid combination of analytical and numerical methods in that the set of governing equations are first transformed into equivalent linearized forms for which analytical solutions are obtained. The discretized forms of these solutions for individual sublayers are then summed up numerically to obtain the solution for the entire layer. The numerical solution enables the simulation of many of the features associated with the ageing and the consolidation of clays, including the development of the quasi-preconsolidation pressure.

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B. Ma et al. / Soils and Foundations ] (]]]]) ]]]–]]]

Based on the simulations, the following conclusions are drawn: (i) An increase in the elastic soil modulus, due to the ageing of clays, is the dominant cause of the development of the quasi-pc phenomenon. The observed experimental results confirmed with the numerical simulations here show that specimens which have been aged longer have increased pcq values. The soil modulus is linear in the virgin compression region, but is nonlinear in the recompression region and is best characterized by a hyperbolic function. In order to capture this nonlinear variation and its effect on quasi-pc accurately, smaller load increments around the preconsolidation stress region are needed (Leonards, 1977; Schmertmann, 1991, 1993). The recompression curve eventually merges with the virgin compression line after pcq, resulting in the form of the curves of the type shown in Fig. 1. (ii) Changes in the modulus result in the consolidation rate in clays being very much dependent on the magnitude of the applied load with respect to the yield or the preconsolidation stress. Excess pore pressure dissipates much faster (correspondingly, clay consolidates quicker) when the stress level is below the preconsolidation value, but considerably slower in the virgin compression region. (iii) While the variation in soil elastic modulus controls the EOP curve of clays, the effects of the observed time, such as the “vanishing pc” phenomenon, are controlled primarily by the changes in soil viscosity. It is found that the coefficient of viscosity is higher, but relatively constant (Fig. 6), before p reaches pc. However, when soil enters the virgin compression region, i.e., the effective stress surpasses pc, the viscosity is relatively smaller and increases linearly with pressure and logarithm time. However, this has no bearing on the development of the quasi-pc phenomenon.

the volume compressibility are hi, kvi, and mvi (i¼ 1, 2, …, n), respectively. The governing equation of each soil layer has the following form cvi

∂ 2 ui ∂ui ¼ 2 ∂z ∂t

ðA1Þ

The boundary conditions are z ¼ 0 : u1 ¼ 0

ðA2aÞ

z ¼ H : un ¼ 0

ðA2bÞ

The continuity conditions are z ¼ zi : ui ¼ ui þ 1 ; kvi

∂ui ∂ui þ 1 ¼ k vði þ 1Þ i ¼ 1; 2; …; n 1 ∂z ∂z ðA3Þ

The initial condition is t ¼ 0 : u1 ¼ qð0Þ ¼ qu

ðA4Þ

The following non-dimensional parameters are defined for brevity. ai ¼

kvi kv1

ðA5aÞ

bi ¼

mvi E s1 ¼ mv1 E si

ðA5bÞ

hi H rffiffiffiffiffiffi rffiffiffiffi cv1 bi μi ¼ ¼ cvi ai

ðA5cÞ

ρi ¼

ðA5dÞ

The solution of each layer is in unified form as 1

ui ¼ ∑ C m gmi ðzÞe  βm t m¼1

i ¼ 1; 2; …; n

ðA6Þ

where Acknowledgments The intriguing quasi-pc phenomenon in clays was first introduced to the second author (Muhunthan) by Professor Gerald A. Leonards while he was a student at Purdue University in the late 1980s. GAL exhorted him to use the fundamental mechanistic model to simulate the quasi-pc phenomenon one day and this paper is the result of this endeavor nearly twenty-five years later. This paper is dedicated to Professor Leonards's memory. The research was supported by the National Nature Science Foundation of China (No. 51079126) and the Natural Science Foundation of Zhejiang Province (No. Y1090971). Their support is gratefully acknowledged. Appendix A. The analytical solution of the consolidation of layered elastic systems (Xie and Pan, 1995) The sketch of a layered soil system is shown in Fig. A1. The total thickness of the clay layer is H. The thickness of individual layers, the coefficient of vertical permeability, and

βm ¼ λm 2 cv1 =H 2 gmi ðzÞ ¼ Ami

ðA7Þ

  z z sin μi λm þ Bmi cos μi λm H H

ðA8Þ

Ami and Bmi are given by ½Am1

Bm1 T ¼ ½1

½Ami

Bmi T ¼ Si ½Amði  1Þ

ðA9aÞ

0T Bmði  1Þ T

i ¼ 1; 2; U U U; n

ðA9bÞ

q Drainage 1st

kv1 mv1

ith

kvi mvi

nth

kvn mvn

z0 z1

h1

zi-1 zi

hi

H

zn-1 zn

z

hn Drainage

Fig. A1. Layered consolidation system.

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B. Ma et al. / Soils and Foundations ] (]]]]) ]]]–]]]

where " Si ¼

Ai Bi þ di C i Di

Ai Di  di C i Bi

C i Bi  d i Ai Di

Ci Di þ di Ai Bi

# i ¼ 1; 2; U U U; n ðA10Þ



Ai ¼ sin μi λm

zi  1 H



ðA11aÞ

 zi  1  Bi ¼ sin μi  1 λm H  zi  1  C i ¼ cos μi λm H  zi  1  Di ¼ cos μi  1 λm H rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kvði  1Þ cvi ai  1 bi  1 ¼ di ¼ k vi cvði  1Þ ai bi

ðA11bÞ ðA11cÞ ðA11dÞ ðA11eÞ

λm is the positive root of the following transcendental equation: Sn þ 1  Sn  Sn  1 ⋯S2  S1 ¼ 0

ðA12Þ

where S1 ¼ ½1

ðA13aÞ

0T

Sn þ 1 ¼ ½Bn þ 1

Dn þ 1 

ðA13bÞ

C m is given by n

2q0 ∑ Cm ¼

i¼1

11

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pffiffiffiffiffiffiffiffi ai bi ½Ami ðCi  Di þ 1 Þþ Bmi ðBi þ 1  Ai Þ

pffiffiffiffiffiffiffiffi ∑ ai bi ½μi ρi λm ðAmi 2 þ Bmi 2 Þ þ ðBmi 2  Ami 2 ÞðDi þ 1 Bi þ 1  Ci Ai Þþ 2Ami Bmi ðC i 2  Di þ 1 2 Þ n

ðA14Þ

i¼1

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