research article nonlinear finite strain consolidation

Research ArticleNonlinear Finite Strain Consolidation Analysis with SecondaryConsolidation Behavior

Jieqing Huang123 Xinyu Xie123 Jifa Zhang4 Jinzhu Li2 and Wenjun Wang2

1 Research Center of Coastal and Urban Geotechnical Engineering Zhejiang University Hangzhou 310058 China2 School of Civil Engineering amp Architecture Ningbo Institute of Technology Zhejiang University Ningbo 315100 China3MOE Key Laboratory of Soft Soils and Geoenvironmental Engineering Zhejiang University Hangzhou 310058 China4 School of Aeronautics and Astronautics Zhejiang University Hangzhou 310027 China

Correspondence should be addressed to Xinyu Xie xiexinyuzjueducn

Received 8 January 2014 Accepted 25 March 2014 Published 17 April 2014

Academic Editor Gianluca Ranzi

Copyright copy 2014 Jieqing Huang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper aims to analyze nonlinear finite strain consolidation with secondary consolidation behavior On the basis of someassumptions about the secondary consolidation behavior the continuity equation of pore water in Gibsonrsquos consolidation theory ismodified Taking the nonlinear compressibility and nonlinear permeability of soils into consideration the governing equation forfinite strain consolidation analysis is derived Based on the experimental data of Hangzhou soft clay samples the new governingequation is solvedwith the finite elementmethod Afterwards the calculation results of this newmethod and other twomethods arecompared It can be found that Gibsonrsquos method may underestimate the excess pore water pressure during primary consolidationThe new method which takes the secondary consolidation behavior the nonlinear compressibility and nonlinear permeability ofsoils into consideration can precisely estimate the settlement rate and the final settlement of Hangzhou soft clay sample

1 Introduction

The secondary consolidation has been studied for decadesIt is considered that the rheological behavior accounts forthe secondary consolidation Nevertheless the controversyregarding whether or not the rheological behavior existsin primary consolidation continues today According to thesummarization of Ladd et al [1] and Jamiolkowski et al[2] there are two hypotheses Hypothesis A and HypothesisB Hypothesis A considers that the rheological behaviorappears after primary consolidation However Hypothesis Bconsiders that the rheological behavior occurs while primaryconsolidation begins Mesri a famous scholar who supportedHypothesis A did long-term researches on the compressionindex 119862

119888and the secondary compression index 119862

120572[3ndash5]

He found that the values of 119862120572119862119888are almost constant for a

majority of inorganic soft clays and the highly organic plasticclays The scholars who support Hypothesis B think that theway of dividing consolidation into primary consolidationand secondary consolidation is highly subjective Bardenthought that the rheological behavior should not be neglected

during primary consolidation [6] Wahls considered that thesettlement of soils includes the time-dependent deformationin primary consolidation [7] Aboshi Leroueil et al andKabbaj et al also documented that the rheological behaviorexists in primary consolidation [8ndash11] Because the rheologi-cal behavior causes the secondary consolidation we call it thesecondary consolidation behavior in the following sections ofthis paper

Gibson et al derived the one-dimensional (1D) finitestrain consolidation equation which was called the Gibsonequation [12] They considered the weight of soils and tookvoid ratio as the control variable in the derivation In theresearch fields of sedimentation and consolidation the Gib-son equation was widely used [13ndash15] Many scholars carriedin-depth researches on finite strain consolidation [16ndash18]However the secondary consolidation behavior is not con-sidered in the majority of existing finite strain consolidationtheories Many experimental and in situ data indicate that thesettlement of soils still increases after primary consolidationthat is under the condition of almost constant effectivestress [19 20] Dan summarized the experimental results

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 979380 8 pageshttpdxdoiorg1011552014979380

2 Mathematical Problems in Engineering

for some Chinese soft clays [19] It is found that Shenzhenclay Tianjin clay Zhengzhou clay Ningbo clay andWenzhouclay all show the secondary consolidation behavior clearlySo it is necessary to consider the secondary consolidationbehavior in the study of finite strain consolidation Ding etal presented a coupling model for 1D finite strain primary-secondary consolidation [21] Nevertheless they did notperform calculation or analysis to verify the rationality of themodel

The objective of this paper is to analyze nonlinear finitestrain consolidation with secondary consolidation behaviorBased on some assumptions about the secondary consol-idation behavior the continuity equation of pore waterin Gibsonrsquos consolidation theory is modified Taking thenonlinear compressibility and nonlinear permeability of soilsinto consideration the new governing equation for numericalanalysis can be derived In order to obtain the calculationparameters for numerical analysis lab tests of Hangzhousoft clay samples are carried out Then the finite elementmethod is chosen to solve the new governing equation andthe calculation results of excess pore water pressure andsettlement are analyzed Afterwards the calculation resultsof this new method and other two methods are comparedin order to find the best method Finally numerical analysisis performed to study the sensitivity of the new secondarycompression index

2 Mathematical Modeling

According to the existing studies [6ndash11] we consider thatthe secondary consolidation behavior should be consideredin the whole process of consolidation As we all know theconventional secondary compression index 119862

120572exists only in

secondary consolidation Thus a new parameter should bedefined to describe the secondary consolidation behaviorWename it the new secondary compression index and use a newsymbol 119862

119904to represent it It should be pointed out that 119862

119904

can be calculated with the classical method of calculating 119862120572

but it exists in both primary consolidation and secondaryconsolidation In order to introduce a nonlinear finite strainconsolidation theory with secondary consolidation behaviorthe following three assumptions are made

(1) The secondary consolidation behavior exists in bothprimary consolidation and secondary consolidation

(2) The new secondary compression indices 119862119904at differ-

ent depths of soils are the same(3) The new secondary compression index 119862

119904remains

constant during consolidation

Gibson et al derived the governing equation of 1D finitestrain consolidation by treating void ratio as the controlvariable [12] The Gibson equation is written as follows

plusmn (119866119904minus 1)

119889

119889119890(

119896

1 + 119890)120597119890

120597119911+

120597

120597119911[

119896

120574119908(1 + 119890)

1198891205901015840

119889119890

120597119890

120597119911]

+120597119890

120597119905= 0

(1)

where 119911 is the 1D space variable in reduced coordinates 119905 isthe time variable 119866

119904= 120574119904120574119908is the specific weight of soil

particles 120574119904and 120574119908are the unit weights of soil particles and

pore water respectively 119896 is the permeability coefficient 1205901015840is the effective stress 119890 is the void ratio plusmn represents thatthe positive direction of the coordinate is opposite to thedirection of gravity or the same

In their derivation the following four main equationswere applied (1) the equilibrium equation of soils (2)the continuity equation of pore water (3) Darcyrsquos law (4)the effective stress principle Here we modify the conti-nuity equation of pore water We consider that the wholedeformation of soils is equal to the deformation caused bythe primary consolidation behavior plus the deformationcaused by the secondary consolidation behavior Hence thefollowing equation can be obtained

minus119889119881

119889119905=119889119876

119889119905minus119889119881119904

119889119905 (2)

where 119881 is the whole volume of soils 119876 is the volume ofexpelled water 119881

119904is the volume of soils affected by the

secondary consolidation behaviorConsider the following

119889119881 =119881

1 + 119890119889119890 (3)

119889119876 =119889]119889120585

119889119909 119889119910119889120585 119889119905 (4)

119889119881119904=

119881

1 + 119890119889119890119904 (5)

where 119909 119910 and 120585 are the three space variables in Euleriancoordinates the positive direction of the 120585 axis is downward] = ]

119908minus ]119904is the relative velocity of pore water and soil

particles ]119908and ]

119904are the velocities of pore water and soil

particles respectively 119890 is the void ratio 119890119904is the void ratio

affected by the secondary consolidation behavior 119890119904is a

portion of 119890According to (5) we can obtain

minus119889119881119904

119889119905= minus

119881

1 + 119890

119889119890119904

119889119905= minus

119881

1 + 119890

119889119890119904

119889 log 119905119889 log 119905119889119905

(6)

After primary consolidation the deformation of soilsis simply caused by the secondary consolidation behaviorWhen primary consolidation ceases 119890 reduces to 119890

119904

According to the definition of the new secondary com-pression index we can obtain

119862119904= minus

119889119890119904

119889 log 119905 (7)

Substituting (7) into (6) yields

minus119889119881119904

119889119905= 119862119904

119881

1 + 119890

1

119905 ln 10 (8)

Substituting (4) and (8) into (2) yields

minus1

1 + 119890

120597119890

120597119905=120597]120597120585

+119862119904

1 + 119890

1

119905 ln 10 (9)

Mathematical Problems in Engineering 3

The modified Darcy law can be written as [12]

] = ]119908minus ]119904= minus119896119894 = minus

119896

120574119908

120597119906

120597120585 (10)

where 119894 is the hydraulic gradient 119906 is the excess pore waterpressure


minus1

1 + 119890

120597119890

120597119905= minus

120597

120597120585(119896

120574119908

120597119906

120597120585) +

119862119904

1 + 119890

1

119905 ln 10 (11)

The effective stress principle is written as

1205901015840

= 120590 minus 119906 (12)

where 1205901015840 is the effective stress 120590 is the total stressDifferentiating (12) yields

120597119906

120597120585=120597120590

120597120585minus1205971205901015840

120597120585 (13)

As noted the positive direction of the 120585 axis is downwardThe equilibrium equation of soils can be obtained as

120597120590

120597120585=(119866119904minus 1) 120574

119908

1 + 119890 (14)


120597119906

120597120585=(119866119904minus 1) 120574

119908

1 + 119890minus1198891205901015840

119889119890

120597119890

120597120585 (15)

Substituting (15) into (11) we attain the 1D finite strainconsolidation equation with secondary consolidation behav-ior in Eulerian coordinates

(119866119904minus 1)

119889

119889119890(

119896

1 + 119890)120597119890

120597120585minus

120597

120597120585(119896

120574119908

1198891205901015840

119889119890

120597119890

120597120585)

=1

1 + 119890

120597119890

120597119905+

119862119904

1 + 119890

1

119905 ln 10

(16)

Under 1D condition the transformation between Eule-rian coordinates and Lagrangian coordinates and the trans-formation between reduced coordinates and Lagrangiancoordinates are shown as follows

119889120585

119889119886=

1 + 119890

1 + 1198900

119889119911

119889119886=

1

1 + 1198900

(17)

where 119886 is the 1D space variable in Lagrangian coordinates

According to the above two transformations the newconsolidation equations in Lagrangian coordinates andreduced coordinates can be obtained

(119866119904minus 1)

119889

119889119890(

119896

1 + 119890)120597119890

120597119886minus

120597

120597119886(119896

120574119908

1198891205901015840

119889119890

120597119890

120597119886

1 + 1198900

1 + 119890)

=1

1 + 1198900

120597119890

120597119905+

119862119904

1 + 1198900

1

119905 ln 10

(18)

(119866119904minus 1)

119889

119889119890(

119896

1 + 119890)120597119890

120597119911minus

120597

120597119911[

119896

120574119908(1 + 119890)

1198891205901015840

119889119890

120597119890

120597119911]

=120597119890

120597119905+

119862119904

119905 ln 10

(19)

Equation (19) will be used in the next derivation becausea calculation example will be analyzed in the reduced coordi-nates It is worth mentioning that (19) can be converted intothe Gibson equation if 119862

119904is equal to 0

In order to solve (1) Gibson et al made the following twoassumptions [13]

119892 = minus119896

120574119908(1 + 119890)

1198891205901015840

119889119890= constant

120582 = minus119889

119889119890(119889119890

1198891205901015840) = constant

(20)

where 119892 and 120582 are the consolidation coefficient and thegravity coefficient respectively

Based on the above assumptions (1) was transformed into(21) Equation (21) was a linear partial differential equationand then it was solved with the finite difference method [13]

plusmn (120574119904minus 120574119908) 119892120582

120597119890

120597119911minus 119892

1205972

119890

1205971199112+120597119890

120597119905= 0 (21)

Duncan suggested that the variation of consolidationcoefficient should be taken into consideration in the pre-dictions of settlement rates of practical engineering [22]Leroueil suggested that consolidation must be analyzedby considering compressibility and permeability parametersmeasured separately [23]Mesri andRokhsar used the follow-ing two equations to express the relationship between voidratio and effective stress and the relationship between voidratio and permeability coefficient respectively [24]

119890 = 1198900minus 119862119888lg(120590

1015840

1205901015840

0

) (22)

119890 = 1198900+ 119862119896lg( 119896

1198960

) (23)

where 119862119888is the compression index 119862

119896is the permeability

index 1198900is the initial void ratio1205901015840

0is the initial effective stress

1198960is the initial permeability coefficientTavenas et al considered that (23) can depict the variation

of permeability coefficient with void ratio in the majorityof practical engineering [25] Because (22) and (23) aregenerally accepted they are employed to reflect the nonlinear


Table 1 Experimental data of Hangzhou soft clay

119862119888

119862119896

119862119904

1198900

1198960(msdotminuteminus1) 120574

119904(kNsdotmminus3) 119892(m2

sdotminuteminus1) 120582kPaminus1

02418 04191 7735119890(minus3) 118 1419119890(minus8) 2670 3422119890(minus7) 2028119890(minus3)

compressibility and nonlinear permeability for Hangzhousoft clay respectively


(119866119904minus 1) 119896

0

(1 + 119890)2

[ln 10119862119896

(1 + 119890) minus 1] 10(119890minus1198900)119862119896

120597119890

120597119911

+11989601205901015840

0ln 10

120574119908119862119888

120597

120597119911[

1

1 + 11989010(119890minus1198900)119862119896minus(119890minus1198900)119862119888

120597119890

120597119911]

minus120597119890

120597119905minus

119862119904

119905 ln 10= 0

(24)

Equation (24) is the new governing equation for numer-ical analysis in this paper If 119862

119904is equal to 0 (24) can

be converted into (25) which still considers the nonlinearcompressibility and nonlinear permeability of soils [26] Thefinite element software FlexPDE is chosen to solve (24) and(25)

(119866119904minus 1) 119896

0

(1 + 119890)2

[ln 10119862119896

(1 + 119890) minus 1] 10(119890minus1198900)119862119896

120597119890

120597119911

+11989601205901015840

0ln 10

120574119908119862119888

120597

120597119911[

1


120597119890

120597119911]

minus120597119890

120597119905= 0

(25)

If the void ratio at a certain depth and time is attained thecorresponding effective stress and permeability coefficientcan be calculated with (22) and (23) respectively If the voidratio is calculated the settlement at the top surface of soilscan be calculated with the following equation

119878 (119905) = int

119897

0

[119890 (119911 0) minus 119890 (119911 119905)] 119889119911 (26)

3 Calculation Examples and Analysis

31 Problem Description The nonlinear finite strain con-solidation analysis with secondary consolidation behavioris based on the schematic diagram shown in Figure 1 Inthis figure 119897 is the total thickness of the clay layer 119902 is theuniformly distributed load which is instantaneously appliedon the top surface of the clay layer In the numerical analysisthe value of 119902 is 300 kPa The top and bottom surfaces areboth pervious Take the top surface as the datum plane sothe positive direction of the 119911 axis is downward Consideringthe weight of clays the distribution of initial effective stresswith depth is triangular Initial void ratio is correspondingto initial effective stress at different depths of the clay layerInitial excess pore water pressure is constant with depthDistributions of initial effective stress and initial excess porewater pressure with depth are shown in Figure 1

Pervious

Saturated soft clay

Pervious

z

q

lu01205909984000

Figure 1 Schematic diagram of physical modeling

The parameters used in the numerical analysis are basedon the laboratory tests for Hangzhou soft clay and they arelisted in Table 1

In this section the following three methods are used toanalyze the consolidation behavior of Hangzhou soft clay

Method 1 The nonlinear finite strain consolidation equationwith secondary consolidation behavior that is (24) is used

Method 2 The nonlinear finite strain consolidation equationwithout secondary consolidation behavior that is (25) isused

Method 3 The Gibson equation with constant consolidationcoefficient that is (21) is used

In order to find the best analysis method among the threemethods we will compare the excess pore water pressurescalculated with different methods and the time-deformationcurves calculated with different methods

32 Calculation Results of Excess Pore Water Pressure Cal-culation results of excess pore water pressure are shown inFigure 2 It can be seen from Figure 2(a) that the resultscalculated with different methods are almost the same atthe beginning of primary consolidation (119905 = 1 minute)Because the two surfaces of the clay layer are both perviousnear the surfaces the excess pore water pressures dissipaterapidly At the meanwhile in the internal soil the excesspore water pressures do not dissipate With time elapsingconsolidation develops from the pervious surfaces to theinternal soil by degrees At the time 119905 = 50 minutes theexcess pore water pressure calculated with Method 1 is thesmallest while the excess pore water pressure calculated withMethod 3 is the largest The calculation result of Method 2is between the calculation results of the other two methodsAt the centerline of the soil the maximum excess pore water


00

02

04

06

08

10

zl

0 50 100 150 200 250 300 350

u (kPa)Method 1

Method 2

Method 3

(a) t = 1 minute

00

02

04

06

08

10

zl

0 20 40 60 80 100 120

u (kPa)Method 1

Method 2

Method 3

(b) t = 50 minutes

00

02

04

06

08

10

zl

0 10 20 30 40 50

u (kPa)Method 1

Method 2

Method 3

(c) t = 100 minutes

Figure 2 Distributions of excess pore water pressure with normalized depth at three time points

pressures calculated with Methods 1 2 and 3 are 4319 kPa5496 kPa and 7306 kPa respectively At the time 119905 = 100

minutes the excess pore water pressures calculated withMethods 1 and 2 are close to each other and they will reduceto 0 soon The excess pore water pressure calculated withMethod 3 is still the largest and it needs more time todissipate At this moment the maximum excess pore waterpressures calculated with Methods 1 2 and 3 are 273 kPa369 kPa and 1150 kPa respectively

Compared toMethod 3Method 2 takes thematerial non-linearity of soils into consideration By jointly considering thecalculation results of Methods 2 and 3 in Figures 3(a)ndash3(c) itcan be seen thatMethod 3may underestimate the excess porewater pressure in the intermediate and final stages of primary

consolidation This implies that it is necessary to considerthe nonlinear compressibility and nonlinear permeabilityof soils Compared to Method 2 Method 1 considers thesecondary consolidation behavior By jointly considering thecalculation results of Methods 1 and 2 in Figures 3(a)ndash3(c) it can be observed that Method 2 may underestimatethe excess pore water pressure in the intermediate stageof primary consolidation So the secondary consolidationbehavior should be considered in the study of finite strainconsolidation

33 Calculation Results of Settlement The calculation resultsand the experimental results of settlement are shown inFigure 3 It can be observed that the settlement calculated


with Method 3 is smaller than the observed settlement at thebeginning of consolidation Moreover the final settlement inthe test is about 3mm while the final settlement calculatedwith this method is less than 2mm This shows that Method3 underestimates the final settlement The time-deformationcurve calculated with Method 3 becomes a horizontal lineafter primary consolidation As mentioned Method 3 isGibsonrsquos theory which does not include the secondary con-solidation Therefore the prediction result from this methodis unable to capture secondary consolidation behavior afterthe primary consolidation

It can be found from Figure 3 that Method 2 also under-estimates the settlement at the beginning of consolidationAfter about 10 minutes the calculation result usingMethod 2approaches the experimental result gradually Figure 3 showsthat the final settlement calculated with Method 2 is about26mm So this method slightly underestimates the final set-tlement According to the above calculation and analysis thecalculation result of Method 2 is closer to the experimentalresult than that of Method 3 Perhaps because the nonlinearcompressibility and nonlinear permeability of Hangzhousoft clay are considered in Method 2 this method is morereasonable thanMethod 3 Ichikawa et al performed 1D finitestrain consolidation analysis under change of permeability[27] They considered that the secondary consolidation isinduced by the nonuniform distribution of permeabilityFor Method 2 the distribution of permeability with thedepth of the clay layer is also nonuniform Neverthelessthe secondary consolidation behavior cannot be observed inthe calculated result Consequently we do not agree withIchikawarsquos opinion

It can be observed from Figure 3 that Method 1 canprecisely estimate the settlement rate of Hangzhou soft claysample during consolidation especially in the initial stageof consolidation The analysis methods without consideringthe secondary consolidation behavior such as Methods 2and 3 cannot estimate the settlement rate precisely So it isnecessary to consider the secondary consolidation behaviorin the predictions of settlement rates It can be seen fromFigure 3 that the calculation result of Method 1 exhibits thesecondary consolidation behavior clearly Furthermore thefinal settlement calculated with Method 1 is very close to theactual final settlement

Considering five different values of the new secondaryconsolidation index 119862

119904(0002 0004 0006 0008 and

0010) the time-deformation curves calculated with Method1 are compared in order to study the sensitivity of 119862

119904 All

parameters used in the numerical analysis are the same asthe ones in the aforementioned example except for 119862

119904 The

five time-deformation curves are plotted in Figure 4 It can befound that119862

119904makes a significant influence on the calculation

results of settlement At each time point the larger the119862119904 the

larger the settlementIn the initial stage of primary consolidation 119862

119904has a

great influence on the calculation results For example atthe time 119905 = 01 minute the settlements are approximately021mm 028mm 037mm 054mm and 079mm if 119862

119904

is 0002 0004 0006 0008 and 0010 respectively Withtime elapsing the influence of 119862

119904decreases gradually Near

00

05

10

15

20

25

30

Settl

emen

td

(mm

)

01 1 10 100 1000 10000 100000

Elapsed time t (min)Method 1

Method 2

Method 3

Observed

Figure 3 Time-deformation curves calculated with different meth-ods

00

05

10

15

20

25

30

Settl

emen

td

(mm

)

01 1 10 100 1000 10000 100000

Elapsed time t (min)Cs = 0002

Cs = 0004

Cs = 0006

Cs = 0008

Cs = 0010

Figure 4 Time-deformation curves calculated by consideringdifferent new secondary consolidation indices

the ending time of primary consolidation the influence of119862119904is tiny For instance at the time 119905 = 50 minutes the

settlements are approximately 248mm 252mm 256mm260mm and 265mm if 119862

119904is 0002 0004 0006 0008 and

0010 respectively It is clear that the ending times of primaryconsolidation are almost the same under different conditionsIn the initial stage of secondary consolidation the effect of119862119904is small With increasing time the effect of 119862

119904increases

by degrees For example at the time 119905 = 30000 minutes thesettlements are approximately 267mm 276mm 284mm292mm and 300mm if 119862

119904is 0002 0004 0006 0008

and 0010 respectively Because 119862119904remains constant the

settlements increase linearly in the process of secondaryconsolidation


Overall the influence 119862119904makes on the calculation results

of settlement is evident in the initial stage of primaryconsolidation and the final stage of secondary consolidationHence the value of 119862

119904should be precisely estimated before

performing the numerical analysis

4 Conclusions

Thenonlinear finite strain consolidationwith secondary con-solidation behavior was analyzed in this paperThe followingconclusions can be made

(1) Based on some assumptions about the secondary con-solidation behavior of soils Gibsonrsquos consolidationtheory is modified

(2) Considering the nonlinear compressibility and non-linear permeability of soils the new governing equa-tion for finite strain consolidation analysis is derived

(3) The excess pore water pressures calculated with dif-ferent methods are compared It can be seen thatGibsonrsquos method may underestimate the excess porewater pressure in the intermediate and final stages ofprimary consolidation

(4) The time-deformation curves calculated with differ-ent methods are compared it can be found that thenewmethod can precisely estimate the settlement rateand the final settlement of Hangzhou soft clay sample

(5) By considering different new secondary consolidationindices five time-deformation curves are calculatedand compared It can be found that the influenceC119904makes on the calculation results of settlement is

remarkable in the initial stage of primary consolida-tion and the final stage of secondary consolidation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The financial supports from the National Natural ScienceFoundation of China under Grant no 51378469 ZhejiangProvincial Natural Science Foundation of China under Grantno Y1111240 and the Natural Science Foundation of NingboCity China under Grant no 2013A610196 are gratefullyacknowledged

References

[1] C C Ladd R Foott K Ishihara F Schlosser and H G PoulusldquoStress-deformation and strength characteristicsrdquo in Proceed-ings of the 9th International Conference on Soil Mechanics andFoundation Engineering vol 2 pp 421ndash494 Tokyo Japan 1977

[2] M Jamiolkowski C C Ladd J T Germaine and R LancellottaldquoNew developments in field and laboratory testing of soilsrdquoin Proceedings of the 11th International Conference on Soil

Mechanics and Foundation Engineering vol 1 pp 150ndash153 SanFrancisco Calif USA 1985

[3] GMesri and PMGodlewski ldquoTime and stress-compressibilityinterrelationshiprdquo Journal of Geotechnical Engineering vol 103no 5 pp 417ndash430 1977

[4] G Mesri and A Castro ldquo119862120572119862119888concept and Ko during

secondary compressionrdquo Journal of Geotechnical Engineeringvol 113 no 3 pp 230ndash247 1987

[5] G Mesri ldquoPrimary compression and secondary compressionrdquoin Soil Behavior and Soft Ground Construction vol 119 pp 122ndash166 Geotechnical Special Publication ASCE 2003

[6] L Barden ldquoConsolidation of clay with non-linear viscosityrdquoGeotechnique vol 15 no 4 pp 345ndash362 1965

[7] H EWahls ldquoAnalysis of primary and secondary consolidationrdquoJournal of Soil Mechanics and Foundations Division vol 88 no6 pp 207ndash223 1962

[8] H Aboshi ldquoAn experimental investigation on the similitude inthe consolidation of a soft clay including the secondary creepsettlementrdquo in Proceedings of the 8th International Conferenceon Soil Mechanics and Foundation Engineering vol 4 pp 88ndash93 Moscowm Russia 1973

[9] S Leroueil M Kabbaj F Tavenas and R Bouchard ldquoStress-strain-strain rate relation for the compressibility of sensitivenatural claysrdquo Geotechnique vol 35 no 2 pp 159ndash180 1985

[10] S Leroueil M Kabbaj F Tavenas and R Bouchard ldquoReply todiscussion on stress-strain-strain rate relation for the compress-ibility of sensitive natural claysrdquoGeotechnique vol 36 no 2 pp283ndash290 1986

[11] M Kabbaj F Tavenas and S Leroueil ldquoIn situ and laboratorystress-strain relationshipsrdquo Geotechnique vol 38 no 1 pp 83ndash100 1988

[12] R E Gibson G L England andM J L Hussey ldquoThe theory ofone dimensional consolidation of saturated clays I Finite non-linear consolidation of thin homogeneous layersrdquoGeotechniquevol 17 no 3 pp 261ndash273 1967

[13] R E Gibson R L Schiffman and K W Cargill ldquoThe theoryof one-dimensional consolidation of saturated clays II Finitenonlinear consolidation of thick homogeneous layersrdquo Cana-dian Geotechnical Journal vol 18 no 2 pp 280ndash293 1981

[14] X-Y Xie J-F Zhang and G-X Zeng ldquoSimilarity solution ofself-weight consolidation problem for saturated soilrdquo AppliedMathematics and Mechanics vol 26 no 9 pp 1165ndash1171 2005

[15] S Jeeravipoolvarn R J Chalaturnyk and J D ScottldquoSedimentation-consolidation modeling with an interactioncoefficientrdquo Computers and Geotechnics vol 36 no 5 pp751ndash761 2009

[16] R L Schiffman ldquoFinite and infinitesimal strain consolidationrdquoJournal of Geotechnical Engineering vol 106 pp 203ndash207 1980

[17] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering vol 110 no 6 pp 775ndash7951984

[18] S H Toh and M Fahey ldquoNumerical and centrifuge mod-elling of large strain consolidationrdquo in Computer Methods andAdvances in Geomechanics pp 279ndash284 Balkema RotterdamThe Netherlands 1991

[19] H B Dan Time dependent behavior of natural soft clays[PhD dissertation] Zhejiang University Hangzhou China2009 (Chinese)

[20] Y H Liu Study on engineering property and application ofconstitutive model for ningbo soft clay [Ph D dissertation]Zhejiang University Hangzhou China 2008 (Chinese)


[21] Z-X Ding D-J Yuan andH-H Zhu ldquoA novel couplingmodelfor 1D finite-strain primary-secondary consolidationrdquoRock andSoil Mechanics vol 31 no 8 pp 2367ndash2372 2010 (Chinese)

[22] J M Duncan ldquoLimitations of conventional analysis of consoli-dation settlementrdquo Journal of Geotechnical Engineering vol 119no 9 pp 1331ndash1359 1993

[23] S Leroueil ldquoTenth Canadian geotechnical colloquium recentdevelopments in consolidation of natural claysrdquo CanadianGeotechnical Journal vol 25 no 1 pp 85ndash107 1988

[24] G Mesri and A Rokhsar ldquoTheory of consolidation for claysrdquoJournal of Geotechnical and Geoenvironmental Engineering vol100 no 8 pp 889ndash904 1974

[25] F Tavenas P Jean P Leblond and S Leroueil ldquoThe permeabil-ity of natural soft clays Part II permeability characteristicsrdquoCanadianGeotechnical Journal vol 20 no 4 pp 645ndash660 1983

[26] A-F Hu J-Q Huang X-Y Xie J Wu J-Z Li and K-F LiuldquoStudy on properties of one-dimensional complex nonlinearconsolidation considering self-weight of saturated soilsrdquo Jour-nal of Zhejiang University Engineering Science vol 46 no 3 pp441ndash447 2012 (Chinese)

[27] Y Ichikawa K Kawamura N Theramast and K KitayamaldquoSecondary and tertial consolidation of bentonite clay con-solidation test molecular dynamics simulation and multiscalehomogenization analysisrdquo Mechanics of Materials vol 36 no5-6 pp 487ndash513 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


for some Chinese soft clays [19] It is found that Shenzhenclay Tianjin clay Zhengzhou clay Ningbo clay andWenzhouclay all show the secondary consolidation behavior clearlySo it is necessary to consider the secondary consolidationbehavior in the study of finite strain consolidation Ding etal presented a coupling model for 1D finite strain primary-secondary consolidation [21] Nevertheless they did notperform calculation or analysis to verify the rationality of themodel

The objective of this paper is to analyze nonlinear finitestrain consolidation with secondary consolidation behaviorBased on some assumptions about the secondary consol-idation behavior the continuity equation of pore waterin Gibsonrsquos consolidation theory is modified Taking thenonlinear compressibility and nonlinear permeability of soilsinto consideration the new governing equation for numericalanalysis can be derived In order to obtain the calculationparameters for numerical analysis lab tests of Hangzhousoft clay samples are carried out Then the finite elementmethod is chosen to solve the new governing equation andthe calculation results of excess pore water pressure andsettlement are analyzed Afterwards the calculation resultsof this new method and other two methods are comparedin order to find the best method Finally numerical analysisis performed to study the sensitivity of the new secondarycompression index

2 Mathematical Modeling

According to the existing studies [6ndash11] we consider thatthe secondary consolidation behavior should be consideredin the whole process of consolidation As we all know theconventional secondary compression index 119862

120572exists only in

secondary consolidation Thus a new parameter should bedefined to describe the secondary consolidation behaviorWename it the new secondary compression index and use a newsymbol 119862

119904to represent it It should be pointed out that 119862

119904

can be calculated with the classical method of calculating 119862120572

but it exists in both primary consolidation and secondaryconsolidation In order to introduce a nonlinear finite strainconsolidation theory with secondary consolidation behaviorthe following three assumptions are made

(1) The secondary consolidation behavior exists in bothprimary consolidation and secondary consolidation

(2) The new secondary compression indices 119862119904at differ-

ent depths of soils are the same(3) The new secondary compression index 119862

119904remains

constant during consolidation

Gibson et al derived the governing equation of 1D finitestrain consolidation by treating void ratio as the controlvariable [12] The Gibson equation is written as follows

plusmn (119866119904minus 1)

119889

119889119890(

119896

1 + 119890)120597119890

120597119911+

120597

120597119911[

119896

120574119908(1 + 119890)

1198891205901015840

119889119890

120597119890

120597119911]

+120597119890

120597119905= 0

(1)

where 119911 is the 1D space variable in reduced coordinates 119905 isthe time variable 119866

119904= 120574119904120574119908is the specific weight of soil

particles 120574119904and 120574119908are the unit weights of soil particles and

pore water respectively 119896 is the permeability coefficient 1205901015840is the effective stress 119890 is the void ratio plusmn represents thatthe positive direction of the coordinate is opposite to thedirection of gravity or the same

In their derivation the following four main equationswere applied (1) the equilibrium equation of soils (2)the continuity equation of pore water (3) Darcyrsquos law (4)the effective stress principle Here we modify the conti-nuity equation of pore water We consider that the wholedeformation of soils is equal to the deformation caused bythe primary consolidation behavior plus the deformationcaused by the secondary consolidation behavior Hence thefollowing equation can be obtained

minus119889119881

119889119905=119889119876

119889119905minus119889119881119904

119889119905 (2)

where 119881 is the whole volume of soils 119876 is the volume ofexpelled water 119881

119904is the volume of soils affected by the

secondary consolidation behaviorConsider the following

119889119881 =119881

1 + 119890119889119890 (3)

119889119876 =119889]119889120585

119889119909 119889119910119889120585 119889119905 (4)

119889119881119904=

119881

1 + 119890119889119890119904 (5)

where 119909 119910 and 120585 are the three space variables in Euleriancoordinates the positive direction of the 120585 axis is downward] = ]

119908minus ]119904is the relative velocity of pore water and soil

particles ]119908and ]

119904are the velocities of pore water and soil

particles respectively 119890 is the void ratio 119890119904is the void ratio

affected by the secondary consolidation behavior 119890119904is a

portion of 119890According to (5) we can obtain

minus119889119881119904

119889119905= minus

119881

1 + 119890

119889119890119904

119889119905= minus

119881

1 + 119890

119889119890119904

119889 log 119905119889 log 119905119889119905

(6)

After primary consolidation the deformation of soilsis simply caused by the secondary consolidation behaviorWhen primary consolidation ceases 119890 reduces to 119890

119904

According to the definition of the new secondary com-pression index we can obtain

119862119904= minus

119889119890119904

119889 log 119905 (7)


minus119889119881119904

119889119905= 119862119904

119881

1 + 119890

1

119905 ln 10 (8)


minus1

1 + 119890

120597119890

120597119905=120597]120597120585

+119862119904

1 + 119890

1

119905 ln 10 (9)




119896

120574119908

120597119906

120597120585 (10)



minus1

1 + 119890

120597119890

120597119905= minus

120597

120597120585(119896

120574119908

120597119906

120597120585) +

119862119904

1 + 119890

1

119905 ln 10 (11)


1205901015840

= 120590 minus 119906 (12)


120597119906

120597120585=120597120590

120597120585minus1205971205901015840

120597120585 (13)


120597120590

120597120585=(119866119904minus 1) 120574

119908

1 + 119890 (14)


120597119906

120597120585=(119866119904minus 1) 120574

119908

1 + 119890minus1198891205901015840

119889119890

120597119890

120597120585 (15)


(119866119904minus 1)

119889

119889119890(

119896

1 + 119890)120597119890

120597120585minus

120597

120597120585(119896

120574119908

1198891205901015840

119889119890

120597119890

120597120585)

=1

1 + 119890

120597119890

120597119905+

119862119904

1 + 119890

1

119905 ln 10

(16)


119889120585

119889119886=

1 + 119890

1 + 1198900

119889119911

119889119886=

1

1 + 1198900

(17)



(119866119904minus 1)

119889

119889119890(

119896

1 + 119890)120597119890

120597119886minus

120597

120597119886(119896

120574119908

1198891205901015840

119889119890

120597119890

120597119886

1 + 1198900

1 + 119890)

=1

1 + 1198900

120597119890

120597119905+

119862119904

1 + 1198900

1

119905 ln 10

(18)

(119866119904minus 1)

119889

119889119890(

119896

1 + 119890)120597119890

120597119911minus

120597

120597119911[

119896

120574119908(1 + 119890)

1198891205901015840

119889119890

120597119890

120597119911]

=120597119890

120597119905+

119862119904

119905 ln 10

(19)


119904is equal to 0


119892 = minus119896

120574119908(1 + 119890)

1198891205901015840

119889119890= constant

120582 = minus119889

119889119890(119889119890

1198891205901015840) = constant

(20)



plusmn (120574119904minus 120574119908) 119892120582

120597119890

120597119911minus 119892

1205972

119890

1205971199112+120597119890

120597119905= 0 (21)


119890 = 1198900minus 119862119888lg(120590

1015840

1205901015840

0

) (22)

119890 = 1198900+ 119862119896lg( 119896

1198960

) (23)









119862119888

119862119896

119862119904

1198900







(119866119904minus 1) 119896

0

(1 + 119890)2

[ln 10119862119896

(1 + 119890) minus 1] 10(119890minus1198900)119862119896

120597119890

120597119911

+11989601205901015840

0ln 10

120574119908119862119888

120597

120597119911[

1


120597119890

120597119911]

minus120597119890

120597119905minus

119862119904

119905 ln 10= 0

(24)




(119866119904minus 1) 119896

0

(1 + 119890)2

[ln 10119862119896

(1 + 119890) minus 1] 10(119890minus1198900)119862119896

120597119890

120597119911

+11989601205901015840

0ln 10

120574119908119862119888

120597

120597119911[

1


120597119890

120597119911]

minus120597119890

120597119905= 0

(25)


119878 (119905) = int

119897

0

[119890 (119911 0) minus 119890 (119911 119905)] 119889119911 (26)



Pervious

Saturated soft clay

Pervious

z

q

lu01205909984000










00

02

04

06

08

10

zl

0 50 100 150 200 250 300 350

u (kPa)Method 1

Method 2

Method 3

(a) t = 1 minute

00

02

04

06

08

10

zl

0 20 40 60 80 100 120

u (kPa)Method 1

Method 2

Method 3

(b) t = 50 minutes

00

02

04

06

08

10

zl

0 10 20 30 40 50

u (kPa)Method 1

Method 2

Method 3

(c) t = 100 minutes












119904(0002 0004 0006 0008 and


119904 All


119904 The





119904has a


119904



00

05

10

15

20

25

30

Settl

emen

td

(mm

)

01 1 10 100 1000 10000 100000


Method 2

Method 3

Observed


00

05

10

15

20

25

30

Settl

emen

td

(mm

)

01 1 10 100 1000 10000 100000


Cs = 0004

Cs = 0006

Cs = 0008

Cs = 0010




119904is 0002 0004 0006 0008 and


119904increases


119904is 0002 0004 0006 0008








4 Conclusions










Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119896

120574119908

120597119906

120597120585 (10)



minus1

1 + 119890

120597119890

120597119905= minus

120597

120597120585(119896

120574119908

120597119906

120597120585) +

119862119904

1 + 119890

1

119905 ln 10 (11)


1205901015840

= 120590 minus 119906 (12)


120597119906

120597120585=120597120590

120597120585minus1205971205901015840

120597120585 (13)


120597120590

120597120585=(119866119904minus 1) 120574

119908

1 + 119890 (14)


120597119906

120597120585=(119866119904minus 1) 120574

119908

1 + 119890minus1198891205901015840

119889119890

120597119890

120597120585 (15)


(119866119904minus 1)

119889

119889119890(

119896

1 + 119890)120597119890

120597120585minus

120597

120597120585(119896

120574119908

1198891205901015840

119889119890

120597119890

120597120585)

=1

1 + 119890

120597119890

120597119905+

119862119904

1 + 119890

1

119905 ln 10

(16)


119889120585

119889119886=

1 + 119890

1 + 1198900

119889119911

119889119886=

1

1 + 1198900

(17)



(119866119904minus 1)

119889

119889119890(

119896

1 + 119890)120597119890

120597119886minus

120597

120597119886(119896

120574119908

1198891205901015840

119889119890

120597119890

120597119886

1 + 1198900

1 + 119890)

=1

1 + 1198900

120597119890

120597119905+

119862119904

1 + 1198900

1

119905 ln 10

(18)

(119866119904minus 1)

119889

119889119890(

119896

1 + 119890)120597119890

120597119911minus

120597

120597119911[

119896

120574119908(1 + 119890)

1198891205901015840

119889119890

120597119890

120597119911]

=120597119890

120597119905+

119862119904

119905 ln 10

(19)


119904is equal to 0


119892 = minus119896

120574119908(1 + 119890)

1198891205901015840

119889119890= constant

120582 = minus119889

119889119890(119889119890

1198891205901015840) = constant

(20)



plusmn (120574119904minus 120574119908) 119892120582

120597119890

120597119911minus 119892

1205972

119890

1205971199112+120597119890

120597119905= 0 (21)


119890 = 1198900minus 119862119888lg(120590

1015840

1205901015840

0

) (22)

119890 = 1198900+ 119862119896lg( 119896

1198960

) (23)









119862119888

119862119896

119862119904

1198900







(119866119904minus 1) 119896

0

(1 + 119890)2

[ln 10119862119896

(1 + 119890) minus 1] 10(119890minus1198900)119862119896

120597119890

120597119911

+11989601205901015840

0ln 10

120574119908119862119888

120597

120597119911[

1


120597119890

120597119911]

minus120597119890

120597119905minus

119862119904

119905 ln 10= 0

(24)




(119866119904minus 1) 119896

0

(1 + 119890)2

[ln 10119862119896

(1 + 119890) minus 1] 10(119890minus1198900)119862119896

120597119890

120597119911

+11989601205901015840

0ln 10

120574119908119862119888

120597

120597119911[

1


120597119890

120597119911]

minus120597119890

120597119905= 0

(25)


119878 (119905) = int

119897

0

[119890 (119911 0) minus 119890 (119911 119905)] 119889119911 (26)



Pervious

Saturated soft clay

Pervious

z

q

lu01205909984000










00

02

04

06

08

10

zl

0 50 100 150 200 250 300 350

u (kPa)Method 1

Method 2

Method 3

(a) t = 1 minute

00

02

04

06

08

10

zl

0 20 40 60 80 100 120

u (kPa)Method 1

Method 2

Method 3

(b) t = 50 minutes

00

02

04

06

08

10

zl

0 10 20 30 40 50

u (kPa)Method 1

Method 2

Method 3

(c) t = 100 minutes












119904(0002 0004 0006 0008 and


119904 All


119904 The





119904has a


119904



00

05

10

15

20

25

30

Settl

emen

td

(mm

)

01 1 10 100 1000 10000 100000


Method 2

Method 3

Observed


00

05

10

15

20

25

30

Settl

emen

td

(mm

)

01 1 10 100 1000 10000 100000


Cs = 0004

Cs = 0006

Cs = 0008

Cs = 0010




119904is 0002 0004 0006 0008 and


119904increases


119904is 0002 0004 0006 0008








4 Conclusions










Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119862119888

119862119896

119862119904

1198900







(119866119904minus 1) 119896

0

(1 + 119890)2

[ln 10119862119896

(1 + 119890) minus 1] 10(119890minus1198900)119862119896

120597119890

120597119911

+11989601205901015840

0ln 10

120574119908119862119888

120597

120597119911[

1


120597119890

120597119911]

minus120597119890

120597119905minus

119862119904

119905 ln 10= 0

(24)




(119866119904minus 1) 119896

0

(1 + 119890)2

[ln 10119862119896

(1 + 119890) minus 1] 10(119890minus1198900)119862119896

120597119890

120597119911

+11989601205901015840

0ln 10

120574119908119862119888

120597

120597119911[

1


120597119890

120597119911]

minus120597119890

120597119905= 0

(25)


119878 (119905) = int

119897

0

[119890 (119911 0) minus 119890 (119911 119905)] 119889119911 (26)



Pervious

Saturated soft clay

Pervious

z

q

lu01205909984000










00

02

04

06

08

10

zl

0 50 100 150 200 250 300 350

u (kPa)Method 1

Method 2

Method 3

(a) t = 1 minute

00

02

04

06

08

10

zl

0 20 40 60 80 100 120

u (kPa)Method 1

Method 2

Method 3

(b) t = 50 minutes

00

02

04

06

08

10

zl

0 10 20 30 40 50

u (kPa)Method 1

Method 2

Method 3

(c) t = 100 minutes












119904(0002 0004 0006 0008 and


119904 All


119904 The





119904has a


119904



00

05

10

15

20

25

30

Settl

emen

td

(mm

)

01 1 10 100 1000 10000 100000


Method 2

Method 3

Observed


00

05

10

15

20

25

30

Settl

emen

td

(mm

)

01 1 10 100 1000 10000 100000


Cs = 0004

Cs = 0006

Cs = 0008

Cs = 0010




119904is 0002 0004 0006 0008 and


119904increases


119904is 0002 0004 0006 0008








4 Conclusions










Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

02

04

06

08

10

zl

0 50 100 150 200 250 300 350

u (kPa)Method 1

Method 2

Method 3

(a) t = 1 minute

00

02

04

06

08

10

zl

0 20 40 60 80 100 120

u (kPa)Method 1

Method 2

Method 3

(b) t = 50 minutes

00

02

04

06

08

10

zl

0 10 20 30 40 50

u (kPa)Method 1

Method 2

Method 3

(c) t = 100 minutes












119904(0002 0004 0006 0008 and


119904 All


119904 The





119904has a


119904



00

05

10

15

20

25

30

Settl

emen

td

(mm

)

01 1 10 100 1000 10000 100000


Method 2

Method 3

Observed


00

05

10

15

20

25

30

Settl

emen

td

(mm

)

01 1 10 100 1000 10000 100000


Cs = 0004

Cs = 0006

Cs = 0008

Cs = 0010




119904is 0002 0004 0006 0008 and


119904increases


119904is 0002 0004 0006 0008








4 Conclusions










Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119904(0002 0004 0006 0008 and


119904 All


119904 The





119904has a


119904



00

05

10

15

20

25

30

Settl

emen

td

(mm

)

01 1 10 100 1000 10000 100000


Method 2

Method 3

Observed


00

05

10

15

20

25

30

Settl

emen

td

(mm

)

01 1 10 100 1000 10000 100000


Cs = 0004

Cs = 0006

Cs = 0008

Cs = 0010




119904is 0002 0004 0006 0008 and


119904increases


119904is 0002 0004 0006 0008








4 Conclusions










Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














4 Conclusions










Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article nonlinear finite strain consolidation

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