Research ArticleAn Elastic-Viscoplastic Model for Time-Dependent Behavior ofSoft Clay and Its Application on Rheological Consolidation

Jinzhu Li12 Wenjun Wang1 Yijun Zhu3 Haofeng Xu2 and Xinyu Xie12

1 Ningbo Institute of Technology Zhejiang University Ningbo 315100 China2 Research Center of Coastal and Urban Geotechnical Engineering Zhejiang University Hangzhou 310058 China3 Zhejiang Provincial Institute of Communications Planning Design amp Research Hangzhou 310006 China

Correspondence should be addressed to Wenjun Wang wwjcumtnitnetcn

Received 10 February 2014 Accepted 12 May 2014 Published 5 June 2014

Academic Editor Gradimir Milovanovic

Copyright copy 2014 Jinzhu Li et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To describe the time-dependent behavior of soft clay this paper extended one-dimensional Nishihara model to three-dimensionalstress state based on the framework of Perzynarsquos overstress theory and modified cam-clay model The yield criterion of modifiedcam-clay model was used to describe the plastic properties of soft clay and the overstress theory was used to describe the strainrate effect Triaxial rheological tests were carried out on Ningbo soft clay and the rheological characteristics were studied Based onlaboratory results the parameters of proposedmodel were determined by curve fitting which show that thismodel is suitable for therheological characteristics of Ningbo soft clay The analysis of parameters shows that the value of parameters changes slightly withdifferent deviatoric stress when the confining pressure was constant but changes notably with the increase of confining pressure Auser material subroutine of the proposed constitutive mode was coded on the platform of the FEM software ABAQUS and verifiedby triaxial compression of soil column A plain strain problem was computed to analyze the rheological consolidation propertiesof soft clay in which the rheological effect and the finite strain effect were considered

1 Introduction

According to the classic consolidation theory presented byTerzaghi when soft clay is consolidated under external loadsthe excessive pore pressure dissipates and the effective stressincreases while the deformation grows until the excessivepore pressure dissipates completely This theory has laid thefoundation of soft soil mechanics and even became the maintheoretical basis of the computation of soft soil foundationHowever in the former works Buisman first discovered thelong term deformation of soils after the excessive pore pres-sure was dissipated in experiments which is also reportedin the works of Zeevaert [1] Leonards and Ramiah [2]and also in lots of engineering practices This long termdeformation which cannot be explained by classic theoriesis usually called rheology which contains creep stress relax-ation long-term strength elastic aftereffect hysteresis effectand so on Because of strict requirement of post-constructionsettlement researchers are paying increasing attention on

the rheology of soft clay To propose a suitable constitutivemodel to describe the rheology of soft clay and to determinethe model parameters accurately by laboratory tests theseare the key points of the topic Since 1960s a numberof constitutive models were proposed for time-dependentbehavior of soft clay Commonly used rheological constitutivemodels can be divided into three categories empirical mod-els component models and microscopic models in whichthe first two categories were most widely used Empiricalmodels such as Singh-Mitchell model [3] Mesri et al model[4 5] are mostly one dimensional models which wereoften used to calculate the rheological deformation in onedimensional problem but were limited in complex stressstate Component models are intuitive in physical meaningwhich can describe numerous rheologicalmodels by differentcombinations of components and can be easily extended tomultidimensional problems to apply in numeral computationfor complex stress state such as the work of Adachi andOka [6] Fodil et al [7] Yin et al [8 9] Hinchberger and

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 587412 14 pageshttpdxdoiorg1011552014587412

2 Mathematical Problems in Engineering

Rowe [10] Hinchberger andQu [11] Based on the frameworkof Perzynarsquos overstress theory [12] and modified cam-claymodel this paper proposed a three-dimensional rheologicalmodelThemodel was verified by laboratory triaxial rheolog-ical tests of Ningbo soft clay and the model parameters weredetermined by curve fitting On the platform of ABAQUSa user material subroutine of the model was coded andverified

2 Model for Soft Clay

As one of commonly used component models Nishiharamodel was connected by Hooke body Kelvin body andBinghambodywhich candescribe elastic deformation creepelastic aftereffect and viscous flow comprehensively such asFigure 1 When a constant external pressure 120590 was given thestress and strain of the model can be written as follows

120590 = 120590119890

= 120590V119890= 120590

V119901 (1a)

120576 = 120576119890

+ 120576V119890+ 120576

V119901 (1b)

In the above equation 120590119890 120590V119890 120590V119901 denote the stress ofHooke body Kelvin body and Bingham body respectively120576 120576119890 120576V119890 120576V119901 denote the total strain the strain of Hooke bodyKelvin body and Bingham body respectively

Substituting the stress-strain relationship of Kelvin bodyand Bingham body into (1a) and (1b) the constitutive equa-tion of one-dimensional Nishihara model can be obtained asfollows

120576 =

120590

1198640

+

120590

1198641

(1 minus 119890minus(11986411205781)119905

) +

⟨120590 minus 120590119904⟩

1205782

119905 (2)

where 1198640denotes the elastic modulus of Hooke body 119864

1

and 1205781denote the elastic modulus and viscosity coefficient

of Kelvin body 1205782and 120590

119904denote the viscosity coefficient

and yield stress of Bingham body ⟨120590 minus 120590119904⟩ denotes a switch

function which can be defined as follows

⟨119891⟩ =

0 (119891 le 0)

119891 (119891 gt 0)

(3)

Equation (2) contains the first two stages of creep whichare often called primary creep stage and steady creep stage Infact for complex stress state (2) cannot be used to calculatethe deformation of soil it is necessary to extend (2) to threedimensional so some hypotheses can be presented as follows

(1) the soil material is isotropic

(2) the volume deformation is just caused by sphericalstress and it is unrelated to shearing stress

(3) the Poissonrsquos ratio of soil is constant and does notchange with the stress or time

120590 120590

120590s

12057821205781

E1

E0

Figure 1 Sketch of Nishihara model

Based on the above hypotheses the stress-strain relation-ship of Hooke body is given as

120576119890

119894119895=

1198681

9119870

120575119894119895+

119904119894119895

21198660

(4)

where 1198681is the first stress invariant and 119870 and 119866

0denote the

bulk modulus and shear modulus of Hooke bodyFor Kelvin body the stress-strain relationship is given as

120576V119890119894119895=

119904119894119895

21198671

+

1198661

1198671

120576V119890119894119895 (5)

where 1198661and119867

1denote the shear modulus and 3D viscosity

coefficient of Kelvin body in which1198671= 2(1+120583)120578

1 120583 is the

Poissonrsquos ratio of soilFor Bingham body the stress-strain relationship is given

as

120576V119901119894119895=

1

1198672

⟨120601 (119891)⟩

120597119876

120597120590119894119895

(6)

In (6) 1198672denotes 3D viscosity coefficient of Bingham

body and ⟨120601(119891)⟩ is a switch function to judge whether plasticyield is occurring and whether the amplitude of the plasticyield occurred 119876 is the plastic potential function when theassociated flow rule is adopted 119876 = 119865 where 119865 is the yieldfunction 120601(119891) can be defined as follows

120601 (119891) = (

119891 minus 1198910

1198910

)

119873

(7)

where 119891 is the current yield function 1198910is the initial yield

function and 119873 is a constant derived from experimentsaccording to the work of Wei [13] the value of 119873 can beassigned to 10 approximately for soft clay

According to the work of Adachi and Oka [6] and Yinet al [8 9] we define a static yield criterion 119891

0 which

represents a reference yield surface for the material suchas Figure 2 Its initial shape depends on the consolidationpressure 119901119904

119888 The expansion of the static yield surface which

describes the hardening of the material is expressed by thevariation of the consolidation pressure due to the inelasticvolumetric strain 120576V119901V as follows

119901119904

119888= 1199010sdot exp(

1 + 1198900

120582 minus 120581

120576V119901V ) (8)

Mathematical Problems in Engineering 3

Critical state line

q

0

1

Static yield surface

Static yield

A

Psc Pd

c PP9984000

M

B Dynamic yieldsurface for B

surface for B

120597Fd120597120590998400ij

Figure 2 Yield criterion of model

In (8) 1199010is the intercept of initial static yield surface in

the 1199011015840 axis for soft clay it can be assigned to the biggestconsolidation stress in history 120582 is the slope of normalconsolidation curve in 119890 sim ln1199011015840 plane 120581 is the slope ofrecompression curve in 119890 sim ln1199011015840 plane

A dynamic yield criterion 119865119889is defined to describe the

current state of stress as follows

119865119889=

1199022

1198722+ 1199011015840

(1199011015840

minus 119901119889

119888) = 0 (9)

From (8) and (9) the 119891 and 1198910in (7) can be written as

119891 = 119901119889

119888= 1199011015840

+

1199022

11987221199011015840

1198910= 119901119904

119888= 1199010sdot exp(

1 + 1198900

120582 minus 120581

120576V)

(10)

So we can write (7) as follows

120601 (119891) =

119901119889

119888

119901119904

119888

minus 1 (11)

Combining (6)sim(11) (6) can be written as follows

120576V119901119894119895=

1

1198672

⟨120601 (119891)⟩(

31199041015840

119894119895

1198722+ (2119901

1015840

minus 119901119889

119888)

120575119894119895

3

) (12)

From (4) (5) and (12) when the stress is constant thetotal strain of the presented model can be written as follows

120576119894119895=

1198681

9119870

120575119894119895+

1199041015840

119894119895

21198660

+

1199041015840

119894119895

21198661

(1 minus 119890minus(11986411205781)119905

)

+

1

1198672

⟨120601 (119891)⟩(

31199041015840

119894119895

1198722+ (2119901

1015840

minus 119901119889

119888)

120575119894119895

3

) 119905

(13)

3 Experiments and Computation

31 Test Program In order to verify the presentedmodel lab-oratory triaxial rheological tests were performed to observethemechanical behavior of soft clay under long-term loading

Table 1 Property of testing material

Property ValueEmbedded depth (m) 35sim45Specific gravity 267Moist unit weight (kNm3) 173Water content () 404Void ratio 108Liquid limit () 425Plastic limit () 249Plasticity index 176Slope of the critical state line 0898Slope of compression curve 0102Slope of the recompression curve 0015

The soil investigated in the experiments is Ningbo soft claywhich is a kind of problematic soil for low strength highcompressibility and time-dependent behavior and depositsin Hangzhou Bay ChinaThe basic properties of the soil weresummarized in Table 1 Bothmoisture content andmoist unitweight of this material are significantly less than those oftypical natural sedimentary deposits

The experimental equipment was refitted on the platformof strain controlled triaxial apparatus the original strainloading method was changed to stress loading method whilethe confining pressure system the back pressure loadingsystem and the measurement system were reserved Thesoil was cut into replicate specimens with a diameter of398mm and a height of 800mm and placed in the triaxialtest apparatus Both soil specimens were consolidated 24hours under a confining pressure of 100 kPa and 200 kParespectively then keep the confining pressure as constantand apply the deviatoric stress increment until the specimenswere damaged or the total strain exceeds 15 In the wholeprocess the free drainage conditions were kept Table 2showed the different loading rates applied in the experimentsIn order to make sure that the creep deformation can be fullydeveloped each load increment lasted no less than 7 daysuntil the deformation of the specimen is lower than 001mmwithin 24 h

32 Experimental Results The experimental results werepresented to observe the rheological behavior of the softclay It is considered that deformation occurs during thewhole process but only the deformation that occurs afterthe dissipation of the excessive pore water pressure can beregarded as creep onlyThe full shearing process stress-strain-time curve of each specimen is illustrated in Figure 3 and themutistage strain-time curves of each specimen are illustratedin Figure 4 The figures show that the creep process undertriaxial loading exhibits attenuation characteristics when thedeviatoric stress is small which contains two stages of creepprimary stage and steady stage while when the deviatoricstress is large itmay exhibit accelerated characteristics whichcontains three stages of creep primary stage steady stage andaccelerated stage such as the last load increment of specimen2

4 Mathematical Problems in Engineering

Table 2 Loading scheme of triaxial rheological tests

Specimen number Confining pressure (kPa) Loading scheme (kPa)1 100 30-60-90-12-150-180-210-2402 200 40-80-120-160-200-240-280-320

20

15

10

5

00

60

120

180

240

300

120576(

)q

(kPa

)

200 400 600 800 1000 1200t (h)

(a) Specimen 1

20

25

15

10

50

80

160

240

320

120576(

)q

(kPa

)

0 200 400 600 800 1000 1200

t (h)

(b) Specimen 2

Figure 3 Full-process press-strain-time curve

0

5

10

15

20

30 kPa60 kPa90 kPa120 kPa

150 kPa180 kPa210 kPa240 kPa

120576(

)

10 1001t (h)

(a) Specimen 1

40 kPa80 kPa120 kPa160 kPa 320 kPa

240 kPa200 kPa

280 kPa

0

5

10

10 100

15

20

25

120576(

)

1t (h)

(b) Specimen 2

Figure 4 Mutistage strain-time curves

33 Computation and Curve Fitting For triaxial rheologicaltests 119868

1= 1205901+ 21205903= 31199011015840 119904101584011= 2(120590

1minus 1205903)3 = 21199023 so (13)

can be written as follows

12057611= 1198751+ 1198752(1 minus 119890

minus1198753119905

) + 1198754119905 (14)

where 1198751= (1199013119870) + (1199023119866

0) 1198752= (1199023119866

1) 1198753= (11986611198671)

1198754= (1119867

2)⟨120601(119891)⟩((2119902119872

2

) + (13)(21199011015840

minus 119901119889

119888))

Use (14) to fit the experimental data the results wereshowed in Figures 5 and 6 in which the parameters SSEand 1198772 denote the residual sum of squares and squares ofcorrelation coefficient respectively From Figures 5 and 6 wecan see that the model fits testing data of primary creep stageand steady creep stage very well while for the acceleratedcreep stage it needs further study According to the value of

Mathematical Problems in Engineering 5

0023

0024

0025

0026

0027

0 20 40 60 80 100 120 140 160 180

120576 11

SSE = 57349E minus 9R2 = 099069

t (h)

24105E minus 2P114547E minus 3P213321E minus 1P337098E minus 6P4

Test dataFitting curve

(a) 119902 = 30 kPa

0 20 40 60 80 100 120 140 160 180t (h)

0034

0036

0038

0040

0042

0044

120576 11

SSE = 58195E minus 8R2 = 098856

35852E minus 2P150523E minus 3P220259E minus 1P367168E minus 6P4

Test dataFitting curve

(b) 119902 = 60 kPa

SSE = 10771E minus 7R2 = 098854

0 20 40 60 80 100 120 140 160 180

t (h)

0046

0048

0050

0052

0054

0056

0058

120576 11

69284E minus 3P223128E minus 1P310127E minus 5P4

48144E minus 2P1

Test dataFitting curve

(c) 119902 = 90 kPa

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 1712E minus 7R2 = 099035

60822E minus 293268E minus 328662E minus 118749E minus 5

0055

0060

0070

0065

0075120576 1

1

P2P3P4

P1

Test dataFitting curve

(d) 119902 = 120 kPa

SSE = 61207E minus 7R2 = 098043

P1 76261E minus 2P2 12875E minus 3P3 35961E minus 1P4 26325E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0070

0080

0090

0075

0085

0095

120576 11

Test dataFitting curve

(e) 119902 = 150 kPa

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 872E minus 7R2 = 098046

89059E minus 215909E minus 237062E minus 128956E minus 5

0080

0090

0100

0110

0115

0085

0095

0105

120576 11

P2P3P4

P1

Test dataFitting curve

(f) 119902 = 180 kPa

Figure 5 Continued

6 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

0090

0100

0110

0120

0130

0125

0115

0095

0105

120576 11

SSE = 99965E minus 7R2 = 098620

10217E minus 119991E minus 228288E minus 129143E minus 5

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 210 kPa

Figure 5 Fitting curves of specimen 1 (1205903= 100 kPa)

Table 3 Model parameters of specimen 1 (1205903= 100 kPa)

1205901minus 1205903(kPa) 119864

0(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots)

30 249 1856 50160 1607360 251 1069 18993 1795090 249 1169 18198 21297120 247 1158 14544 24253150 236 1049 10497 25049180 236 1018 9891 22046210 235 945 12032 26240

Table 4 Model parameters of specimen 2 (1205903= 200 kPa)

1205901minus 1205903(kPa) 119864

0(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots)

40 1075 1804 13881 1222380 903 1870 25862 11747120 834 1694 29947 12591160 763 1692 30057 12704200 660 1729 26118 11732240 604 1831 27481 12823280 508 1602 28165 11589

1198751 1198752 1198753 and 119875

4 the value of the model parameters 119864

0 1198641

1205781 and 120578

2can be determined easily such as in Tables 3 and 4

34 Discussion Table 3 shows that the model parameters ofspecimen 1 (120590

3= 100 kPa) change slightly with the change

of deviatoric stress except the first stage Table 4 showsthat the model parameters of specimen 2 (120590

3= 200 kPa)

change slightly with the change of deviatoric stress except1198640 Compared with the model parameters of specimen 1 and

specimen 2 it can be found that both 1198640and 119864

1increase

notably with the increase of 1205903 which means that the elastic

strain reduces when 1205903increases and 120578

1increases notably

Table 5 Percentage of strain components of specimen 1 (1205903=

100 kPa 119905 = 7 d)

1205901minus 1205903(kPa) 120576

119890

11() 120576

119907119890

11() 120576

119907119901

11()

30 9206 556 23860 8530 1202 26890 8483 1221 296120 8298 1272 430150 8151 1376 473180 8109 1448 443210 8041 1573 385

Table 6 Percentage of strain components of specimen 2 (1205903=

200 kPa 119905 = 7 d)

1205901minus 1205903(kPa) 120576

119890

11() 120576

119907119890

11() 120576

119907119901

11()

40 7703 1653 64580 6774 1683 1543120 6677 1971 1353160 6713 1981 1306200 6951 1837 1212240 7096 1686 1217280 7397 1740 863

while 1205782is reduced notably with the increase of 120590

3 which

means that the viscoelastic strain rate and viscoplastic strainrate change significantly when 120590

3increases For the lack

of more experiment data the further variation of modelparameters was unable to find out which needed furtherstudy

Based on the parameters in Tables 3 and 4 the com-ponents of elastic strain viscoelastic strain and viscoplasticstrain at any time can be calculated with (13) Tables 5 and 6show 120576

119890

11 120576V11989011 and 120576V119901

11of all loading levels when 119905 = 7 d it can

Mathematical Problems in Engineering 7

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 71424E minus 9R2 = 098984

93026E minus 319959E minus 346778E minus 146368E minus 6

0008

0009

0010

0011

0012

0013120576 1

1

P2P3P4

P1

Test dataFitting curve

(a) 119902 = 40 kPa

0 20 40 60 80 100 120 140 160 180t (h)

SSE = 27165E minus 8R2 = 099567

15502E minus 238505E minus 326029E minus 121015E minus 5

0010

0012

0014

0016

0018

0020

0022

0024

120576 11

P2P3P4

P1

Test dataFitting curve

(b) 119902 = 80 kPa

SSE = 85674E minus 8R2 = 0994

21595E minus 263737E minus 320369E minus 126044E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0016

0018

0020

0022

0024

0026

0028

0030

0032

0034

120576 11

P2P3P4

P1

Test dataFitting curve

(c) 119902 = 120 kPa

SSE = 15199E minus 7R2 = 099385

28845E minus 285110E minus 320264E minus 133395E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0020

0025

0030

0035

0040

0045

120576 11

P2P3P4

P1

Test dataFitting curve

(d) 119902 = 160 kPa

SSE = 26486E minus 7R2 = 099252

39406E minus 210413E minus 223826E minus 140897E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0030

0035

0040

0045

0050

0060

0055

120576 11

P2P3P4

P1

Test dataFitting curve

(e) 119902 = 200 kPa

0040

0045

0050

0060

0065

0075

0070

0055

120576 11

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 20786E minus 7R2 = 099572

49663E minus 211800E minus 223980E minus 150713E minus 5

P2P3P4

P1

Test dataFitting curve

(f) 119902 = 240 kPa

Figure 6 Continued

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 50043E minus 7R2 = 09928

66892E minus 215735E minus 220470E minus 146472E minus 5

00600060

00650065

00700070

00750075

00800080

00850085

00900090

00950095

120576 11

120576 11

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 280 kPa

Figure 6 Fitting curves of specimen 2 (1205903= 200 kPa)

001 10 100

02

04

06

08

10

12

14

16

Visc

ous s

trai

n (

)

Time (h)

VES q = 60kPaVPS q = 60kPaVES q = 120 kPa

VPS q = 120 kPaVES q = 180 kPaVPS q = 180 kPa

(a) Specimen 1

Visc

ous s

trai

n (

)

00

02

04

06

08

10

12

100Time (h)

VES q = 80kPaVPS q = 80kPaVES q = 160 kPa

VPS q = 160 kPaVES q = 240 kPaVPS q = 240 kPa

1 10

(b) Specimen 2

Figure 7 Development of computed viscous strain (VES viscoelastic strain VPS viscoplastic strain)

be found that the elastic strain accounted for the major partof the total strain but the viscoelastic strain and viscoplasticstrain increase with the increase of 120590

3 which takes about

15sim30 for each loading increment of the total and cannotbe neglected

Figure 7 shows several typical curves of the developmentof computed viscoelastic strain and viscoplastic strain ofspecimen 1 and specimen 2 It can be found that theviscoelastic strain increases quickly in the early stage andthen reaches steady in a short time (about 30 hours) whilethe viscoplastic strain increases the whole process whichindicates that the long-term deformation after consolidation

of soft clay foundation is mainly caused by the viscoplasticityrather than viscoelasticity

4 Derivation and Verification of UMAT

41 Derivation of UMAT ABAQUS is a FEM software whichhas powerful computing capabilities of strongly nonlinearproblems A user material subroutine (UMAT) is codedfor the proposed model on the platform of ABAQUS tostudy the elastic-viscoplastic consolidation properties of softclay According to ABAQUS the stiffness coefficient matrix

Mathematical Problems in Engineering 9

(Jacobian matrix) and the equation to update the stress andstrain should be defined in the UAMT The Jacobian matrixD is defined as follows

D =

120597Δ120590

120597Δ120576

(15)

where Δ120590 and Δ120576 denote the stress increment matrix andstrain increment matrix respectively

For the proposedmodel (4)sim(6) can be written as followsin 2D or 3D problems

120576119890

119899=

120583

1198640

A120590119899 (16a)

V119890119899=

1

1205781

A120590119899minus

1198661

1205781

120576V119890119899 (16b)

V119901119899=

1

1205782

⟨120601 (119865)⟩

120597119865

120597120590

(16c)

In (16a) (16b) and (16c) 120590119899 120576119899 120576V119890119899 and 120576V119901

119899denote the

Voigt matrix form of stress and strain component tensors Adenotes the flexibilitymatrix for plane strain problem and 3Dproblems and it can be written as follows

A =

[[[

[

1 minus120583 minus120583 0

minus120583 1 minus120583 0

minus120583 minus120583 1 0

0 0 0 2 (1 + 120583)

]]]

]

(17a)

A =

[[[[[[[

[

1 minus120583 minus120583 0 0 0

minus120583 1 minus120583 0 0 0

minus120583 minus120583 1 0 0 0

0 0 0 2 (1 + 120583) 0 0

0 0 0 0 2 (1 + 120583) 0

0 0 0 0 0 2 (1 + 120583)

]]]]]]]

]

(17b)

Suppose that the time step of the 119899th increment is Δ119905119899 so

the viscoelastic strain increment in the increment step can bewritten as follows

Δ120576V119890119899= Δ119905119899[(1 minus Θ)

V119890119899+ Θ

V119890119899+1] (18)

where Θ is differential coefficient 0 le Θ le 1 Suppose that119861119899= 1(1 + ΘΔ119905

11989911986411205781) C119899= 119861119899ΘΔ119905119899A1205781 (18) can be

written as follows

Δ120576V119890119899= 119861119899V119890119899Δ119905119899+ C119899Δ120590119899 (19)

The viscoplastic strain increment in the increment stepcan be written as follows

Δ120576V119901119899= Δ119905119899[(1 minus Θ)

V119901119899+ Θ

V119901119899+1] (20)

In order to calculate the viscoplastic strain rate at theend of the 119899th increment apply the Taylor series expansionand ignore the second order trace then we can obtain thefollowing equation

V119901119899+1

= V119901119899+

120597V119901119899

120597120590

120597120590

120597119905

Δ119905119899 (21)

Suppose thatH119899= 120597

V119901119899120597120590 (21) can be written as

V119901119899+1

= V119901119899+H119899Δ120590119899 (22)

Substituting (22) into (21) we can obtain the followingequation

Δ120576V119901119899= Δ119905119899V119901119899+ ΘΔ119905

119899H119899Δ120590119899 (23)

From (16c)H119899can be written as

H119899=

1

1205782

[⟨120601 (119865)⟩

120597a119879

120597120590

+

120597 ⟨120601 (119865)⟩

120597119865

aa119879] (24)

where a = 120597119865120597120590 According to (9) a can be written asfollows

a = 120597119865

120597120590

= 1198621a1+ 1198622a2 (25)

where 1198621= 2119902119872

2 1198622= 2119901 minus 119901

0 a1= 120597119902120597120590 a

2= 120597119901120597120590

From (25) the following equation can be derived

120597a119879

120597120590

=

1205971198621

120597119902

a1a1198791+ 1198621

120597a1198791

120597120590

+

1205971198622

120597119901

a2a1198792 (26)

where (120597a1198791120597120590) = (32119902)M

1minus(94119902

3

)M2 a1a1198791= (94119902

2

)M2

a2a1198792= (19)M

3 M1 M2 and M

3are matrices written as

follows

M1=

[[[[[[[

[

23 minus13 minus13 0 0 0

minus13 23 minus13 0 0 0

minus13 minus13 23 0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

]]]]]]]

]

M2=

[[[[[[

[

1198782

111198781111987822

1198781111987833

21198781111987812

21198781111987823

21198781111987813

1198781111987822

1198782

221198782211987833

21198782211987812

21198782211987823

21198782211987813

1198781111987833

1198782211987833

1198782

3321198783311987812

21198783311987823

21198783311987813

21198781111987812

21198782211987812

21198783311987812

41198782

1241198781211987823

41198781211987813

21198781111987823

21198782211987823

21198783311987823

41198781211987823

41198782

2341198782311987813

21198781111987813

21198782211987813

21198783311987813

41198781211987813

41198782311987813

41198782

13

]]]]]]

]

M3=

[[[[[[[

[

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]]]]]]]

]

(27)

Combining the above equations the value of H119899can

be derived Substituting H119899into (22) the viscoplastic strain

increment Δ120576V119901119899

is obtainedFor current increment Δ120590

119899is calculated by (28) as

follows

Δ120590119899= D119890Δ120576119890

119899 (28)

Substituting (19) and (23) into (28) we can obtain thefollowing equation

Δ120590119899= D (Δ120576

119899minus 119861Δ119905

119899V119890119899minus Δ119905119899V119901119899) (29)

10 Mathematical Problems in Engineering

Table 7 Material parameters of verification model

1198640(MPa) 119864

1(MPa) 120578

1(GPasdots) 120578

2(GPasdots) 119875

0(kPa) 119890

0120583 119872 120582 120581

10 20 1000 1000 50 108 035 0898 0102 0015

12059021205902

1205903

1205903

1205901

1205901

Figure 8 Sketch of verification model

Equation (29) is the relationship between stress incre-ment and strain increment in which matrix D is defined asfollows

D = [(D119890)minus1 + C119899+ ΘΔ119905

119899H119899]

minus1

(30)

Based on the above equations a user material subroutinewas coded for the proposed constitutive model on theplatform of ABAQUS

42 Verification of UMAT In order to verify the validity ofthe UMAT we study a soil column of triaxial compressionsuch as Figure 8 the size of the column is 01m times 01m times

02m Supposing that 1205901= 100 kPa 120590

2= 1205903= 50 kPa

and the model parameters are given in Table 7 The columnwas cut into one element the element type is C3D8 and theconsolidation effect was not considered here Embedding theUMAT coded in this paper on the platform of ABAQUS wecalculated the triaxial compression property of the columnthe results were summarized in Figure 9simFigure 10

From Figure 9(a) we can see the deformation of thecolumn exhibits apparent rheology the rheological deforma-tion takes about 37 of the whole deformation Figure 9(b)shows that the static yield surface gradually expanded withthe time until it reached steady state when it coincided withthe dynamic yield surface From Figures 9(c) and 9(d) wecan see that the viscoelastic strain and the viscoplastic strainboth gradually increased with time until they reached theirsteady state accordingly the viscoelastic strain rate and theviscoplastic strain rate were gradually deduced to nearly zeroThe above results fit the property of Nishihara model andthe hardening law of modified Cambridge model very wellwhich proved the correctness of the UMAT

In the proposed model 1205781 1205782 and 119901

0are the most

important parameters Figure 10 showed the sensitivity ofthese parameters From Figures 10(a) and 10(b) we can seethe final viscous deformation is a constant regardless of thedifferent values of 120578

1 1205782 but the time when it reaches steady

varies greatly so the accuracy of 1205781and 1205782is directly related

to the accuracy of computed displacement in the rheologicalprocess Figure 10(c) shows that the value of 119901

0has a great

influence on the viscoplastic strain the smaller 1199010is the

more rapidly the viscoplastic strain develops and the largerthe final viscoplastic strain occurs and so the accuracy of1199010is not only directly related to the accuracy of computed

displacement in the rheological process but also affected thefinal displacement which needs to be paid more attention

5 Analysis on Rheological Consolidation ofSoft Clay

51 Computed Model Consolidation problems can be classi-fied as small strain theory and finite strain theory accordingto the difference of geometric assumption Small straintheory assumed that the strain is little but the consolida-tion deformation is usually very large for deep soft clayfoundation Take dredger fill for example the deformationmay exceed 80 for which the small strain theory is nolonger suitable [14 15] As the pioneer researchers studiedon finite strain consolidation theory Mikasa [16] and Gib-son et al [17] derived their one-dimensional finite strainconsolidation equations respectively in 1960s and manyscholars developed further research on the topic in thefollowing decades Because of strong nonlinearity most ofexisting researches on finite strain consolidation adoptedsome simplified assumptionswithout considering rheologicalcharacteristics By means of ABAQUS this paper studied therheological consolidation of soft clay considering finite straineffect

The computed model is plane strain problem as shown inFigure 11 The soil foundation is homogeneous anisotropicsaturated and normally consolidated soft clay with a thick-ness of 100m and only the top boundary is permeable Inaddition the semilogarithmic empirical formula proposed byTavenas et al [18] was adopted to consider the variation ofpermeability in the process of consolidation as follows

119896V = 119896V0 sdot 10(119890minus1198900)119862119896 (31)

where 119896V0 is the initial permeability coefficients and 119896V is thepermeability coefficient in the consolidation process 119890

0is the

initial void ratio and 119890 is the void ratio in the consolidationprocess 119862

119896is the permeability index According to the

above work themodel parameters are summarized in Table 8approximately

52 Consolidation Property of Soft Clay From Figure 12 wecan see the differences of the consolidation process betweenfinite strain and small strain Figure 12(a) shows that for anylocation the excess pore pressure of finite strain is less thanthat of small strain at the same time and the difference isincreasing with the increase of time in beginning period

Mathematical Problems in Engineering 11

Table 8 Material parameters of computed model

1198640(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots) 120574sat (kNsdotm

minus3) 1198961199070(msdotsminus1) 119875

0(kPa) 119872 120582 120581 119890

0120583 119862

119896

25 115 150 24000 177 45119864 minus 9 50 0898 0102 0015 108 035 05

100000225

00200

00175

00150

00125

00100

Vert

ical

def

orm

atio

n (m

)

Time (s)10000 100000 1000000 1E7

(a)

40

50

60

70

80

90

100

110

120

1000Time (s)

10000 100000 1000000 1E7

Pcs

(kPa

)(b)

05

00

10

15

20

25

30

35

Viscoelastic strain

Visc

ous s

trai

n (

)

Viscoplastic strain

1000

Time (s)

10000 100000 1000000 1E71E7

(c)

Viscoelastic strain rateViscoplastic strain rate

0

2

4

6

8

10

12

14

16

1000Time (s)

10000 100000 1000000 1E71E7

Visc

ous s

trai

n ra

te (s

minus1)

(d)

Figure 9 Triaxial compression property of soil column

Figure 12(b) shows that the displacement of finite strain isless than that of small strain and the difference increasedsignificantly if the load is increasing For example when 119902 =200 kPa the differencemay take about 20while it is nearly 0when 119902= 50 kPa So it is essential to consider the error causedby small strain if the load is large and the compressibility ofthe soil is large

From Figure 13 we can see the influence of rheologyon the consolidation process Figure 13(a) shows that thedissipation of excess pore pressure is significantly slower ifthe rheological effect is considered for example when 119905 =1000 d the excess pore pressure is nearly 0 if the rheologicaleffect is not considered but it is still larger than 20 kPa ifthe rheological effect is considered Figure 13(a) shows that

the displacement is influenced by rheological effect too forexample in the early stage (119905 le 100 d) the displacement isnearly the same for two cases but the final displacementmay have a difference of more than 35 which indicatesthat the displacement in the early stage is mainly causedby consolidation while the displacement in the late stage ismainly caused by rheology

6 Conclusions

(1) An elastic-viscoplastic model developed to describethe time-dependent behavior of normally consoli-dated soft clay was presented in the paper on the

12 Mathematical Problems in Engineering

00

05

10

15

20

25

30Vi

scoe

lasti

c str

ain

()

100010 100Time (s)

10000 100000 1000000 1E7

1205781 = 10GPamiddots1205781 = 100GPamiddots1205781 = 1000GPamiddots

(a)

00

02

04

06

08

10

12

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

1205782 = 10GPamiddots1205782 = 100GPamiddots1205782 = 1000GPamiddots

(b)

00

04

08

12

16

20

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

P0 = 30kPaP0 = 40kPa

P0 = 50kPaP0 = 60kPa

(c)

Figure 10 Sensitivity of model parameters

O

Permeable

B = 5 mx

z

H = 10m

q = 100 kPa

Figure 11 Calculation model and mesh generation

framework of modified cam-clay elastic plastic modeland Perzynarsquos overstress viscoplastic theory

(2) A series of laboratory triaxial rheological tests ofNingbo soft clay with different confining pressure anddeviatoric stress are performed the results show thatthe presented constitutive model was suitable for therheological characteristic of Ningbo soft clay and it isachievable to determine the parameters of presentedmodel by curve fitting

(3) The analysis on the model parameters shows that thevalue of all parameters is related to confining pressureand deviatoric stress It may need further study on therelationship between parameters and the stress levelso as to use the model conveniently

Mathematical Problems in Engineering 13

0 20 40 60 80 100u (kPa)

0

2

4

6

8

10H

(m)

FS t = 100dSS t = 100dFS t = 500d

SS t = 500dFS t = 1000 dSS t = 1000 d

(a)

12

10

08

06

04

02

00

Def

orm

atio

n (m

)

Time (d)1000101 100 10000 100000

SS q= 100 kPaFS q= 50 kPaSS q= 50 kPa

SS q= 200 kPaFS q= 200 kPa

FS q= 100 kPa

(b)

Figure 12 Results comparison of large strain and small strain (SS small train FS finite strain)

00

2

4

6

8

10

H(m

)

20 40 60 80 100u (kPa)

t = 100d NRt = 100d Rt = 500d NR

t = 500d Rt = 1000 d NRt = 1000 d R

(a)

06

05

04

03

02

01

00

Def

orm

atio

n (m

)

Time (d)

NRR

1000101 100 10000 100000

(b)

Figure 13 Effects of rheology on calculated results (NR rheological effect not considered R rheological effect considered)

(4) In the proposed model the viscoelastic viscositycoefficient 120578

1 the viscoplastic viscosity coefficient 120578

2

and the initial yield surface 1199010are the most important

parameters 1205781and 120578

2have important influence on

the time of viscous strain developed and 1199010have

important influence on both the viscoplastic strainrate and the final viscoplastic strain

(5) The existence of rheology will significantly slow downthe dissipation of the excess pore pressure Consid-ering the rheological effect the total displacement

is larger and the displacement in the early stage ismainly caused by consolidation while in the late stageit is mainly caused by rheology

(6) For consolidation problems of high compressibilitysoft clay foundation great error will be caused bysmall strain assumption so finite strain effect shouldbe considered When finite strain effect is consideredthe consolidation process is developed more rapidlyand the computed consolidation displacement will beless than that of small strain

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Rowe [10] Hinchberger andQu [11] Based on the frameworkof Perzynarsquos overstress theory [12] and modified cam-claymodel this paper proposed a three-dimensional rheologicalmodelThemodel was verified by laboratory triaxial rheolog-ical tests of Ningbo soft clay and the model parameters weredetermined by curve fitting On the platform of ABAQUSa user material subroutine of the model was coded andverified

2 Model for Soft Clay

As one of commonly used component models Nishiharamodel was connected by Hooke body Kelvin body andBinghambodywhich candescribe elastic deformation creepelastic aftereffect and viscous flow comprehensively such asFigure 1 When a constant external pressure 120590 was given thestress and strain of the model can be written as follows

120590 = 120590119890

= 120590V119890= 120590

V119901 (1a)

120576 = 120576119890

+ 120576V119890+ 120576

V119901 (1b)

In the above equation 120590119890 120590V119890 120590V119901 denote the stress ofHooke body Kelvin body and Bingham body respectively120576 120576119890 120576V119890 120576V119901 denote the total strain the strain of Hooke bodyKelvin body and Bingham body respectively

Substituting the stress-strain relationship of Kelvin bodyand Bingham body into (1a) and (1b) the constitutive equa-tion of one-dimensional Nishihara model can be obtained asfollows

120576 =

120590

1198640

+

120590

1198641

(1 minus 119890minus(11986411205781)119905

) +

⟨120590 minus 120590119904⟩

1205782

119905 (2)

where 1198640denotes the elastic modulus of Hooke body 119864

1

and 1205781denote the elastic modulus and viscosity coefficient

of Kelvin body 1205782and 120590

119904denote the viscosity coefficient

and yield stress of Bingham body ⟨120590 minus 120590119904⟩ denotes a switch

function which can be defined as follows

⟨119891⟩ =

0 (119891 le 0)

119891 (119891 gt 0)

(3)

Equation (2) contains the first two stages of creep whichare often called primary creep stage and steady creep stage Infact for complex stress state (2) cannot be used to calculatethe deformation of soil it is necessary to extend (2) to threedimensional so some hypotheses can be presented as follows

(1) the soil material is isotropic

(2) the volume deformation is just caused by sphericalstress and it is unrelated to shearing stress

(3) the Poissonrsquos ratio of soil is constant and does notchange with the stress or time

120590 120590

120590s

12057821205781

E1

E0

Figure 1 Sketch of Nishihara model

Based on the above hypotheses the stress-strain relation-ship of Hooke body is given as

120576119890

119894119895=

1198681

9119870

120575119894119895+

119904119894119895

21198660

(4)

where 1198681is the first stress invariant and 119870 and 119866

0denote the

bulk modulus and shear modulus of Hooke bodyFor Kelvin body the stress-strain relationship is given as

120576V119890119894119895=

119904119894119895

21198671

+

1198661

1198671

120576V119890119894119895 (5)

where 1198661and119867

1denote the shear modulus and 3D viscosity

coefficient of Kelvin body in which1198671= 2(1+120583)120578

1 120583 is the

Poissonrsquos ratio of soilFor Bingham body the stress-strain relationship is given

as

120576V119901119894119895=

1

1198672

⟨120601 (119891)⟩

120597119876

120597120590119894119895

(6)

In (6) 1198672denotes 3D viscosity coefficient of Bingham

body and ⟨120601(119891)⟩ is a switch function to judge whether plasticyield is occurring and whether the amplitude of the plasticyield occurred 119876 is the plastic potential function when theassociated flow rule is adopted 119876 = 119865 where 119865 is the yieldfunction 120601(119891) can be defined as follows

120601 (119891) = (

119891 minus 1198910

1198910

)

119873

(7)

where 119891 is the current yield function 1198910is the initial yield

function and 119873 is a constant derived from experimentsaccording to the work of Wei [13] the value of 119873 can beassigned to 10 approximately for soft clay

According to the work of Adachi and Oka [6] and Yinet al [8 9] we define a static yield criterion 119891

0 which

represents a reference yield surface for the material suchas Figure 2 Its initial shape depends on the consolidationpressure 119901119904

119888 The expansion of the static yield surface which

describes the hardening of the material is expressed by thevariation of the consolidation pressure due to the inelasticvolumetric strain 120576V119901V as follows

119901119904

119888= 1199010sdot exp(

1 + 1198900

120582 minus 120581

120576V119901V ) (8)

Mathematical Problems in Engineering 3

Critical state line

q

0

1

Static yield surface

Static yield

A

Psc Pd

c PP9984000

M

B Dynamic yieldsurface for B

surface for B

120597Fd120597120590998400ij

Figure 2 Yield criterion of model

In (8) 1199010is the intercept of initial static yield surface in

the 1199011015840 axis for soft clay it can be assigned to the biggestconsolidation stress in history 120582 is the slope of normalconsolidation curve in 119890 sim ln1199011015840 plane 120581 is the slope ofrecompression curve in 119890 sim ln1199011015840 plane

A dynamic yield criterion 119865119889is defined to describe the

current state of stress as follows

119865119889=

1199022

1198722+ 1199011015840

(1199011015840

minus 119901119889

119888) = 0 (9)

From (8) and (9) the 119891 and 1198910in (7) can be written as

119891 = 119901119889

119888= 1199011015840

+

1199022

11987221199011015840

1198910= 119901119904

119888= 1199010sdot exp(

1 + 1198900

120582 minus 120581

120576V)

(10)

So we can write (7) as follows

120601 (119891) =

119901119889

119888

119901119904

119888

minus 1 (11)

Combining (6)sim(11) (6) can be written as follows

120576V119901119894119895=

1

1198672

⟨120601 (119891)⟩(

31199041015840

119894119895

1198722+ (2119901

1015840

minus 119901119889

119888)

120575119894119895

3

) (12)

From (4) (5) and (12) when the stress is constant thetotal strain of the presented model can be written as follows

120576119894119895=

1198681

9119870

120575119894119895+

1199041015840

119894119895

21198660

+

1199041015840

119894119895

21198661

(1 minus 119890minus(11986411205781)119905

)

+

1

1198672

⟨120601 (119891)⟩(

31199041015840

119894119895

1198722+ (2119901

1015840

minus 119901119889

119888)

120575119894119895

3

) 119905

(13)

3 Experiments and Computation

31 Test Program In order to verify the presentedmodel lab-oratory triaxial rheological tests were performed to observethemechanical behavior of soft clay under long-term loading

Table 1 Property of testing material

Property ValueEmbedded depth (m) 35sim45Specific gravity 267Moist unit weight (kNm3) 173Water content () 404Void ratio 108Liquid limit () 425Plastic limit () 249Plasticity index 176Slope of the critical state line 0898Slope of compression curve 0102Slope of the recompression curve 0015

The soil investigated in the experiments is Ningbo soft claywhich is a kind of problematic soil for low strength highcompressibility and time-dependent behavior and depositsin Hangzhou Bay ChinaThe basic properties of the soil weresummarized in Table 1 Bothmoisture content andmoist unitweight of this material are significantly less than those oftypical natural sedimentary deposits

The experimental equipment was refitted on the platformof strain controlled triaxial apparatus the original strainloading method was changed to stress loading method whilethe confining pressure system the back pressure loadingsystem and the measurement system were reserved Thesoil was cut into replicate specimens with a diameter of398mm and a height of 800mm and placed in the triaxialtest apparatus Both soil specimens were consolidated 24hours under a confining pressure of 100 kPa and 200 kParespectively then keep the confining pressure as constantand apply the deviatoric stress increment until the specimenswere damaged or the total strain exceeds 15 In the wholeprocess the free drainage conditions were kept Table 2showed the different loading rates applied in the experimentsIn order to make sure that the creep deformation can be fullydeveloped each load increment lasted no less than 7 daysuntil the deformation of the specimen is lower than 001mmwithin 24 h

32 Experimental Results The experimental results werepresented to observe the rheological behavior of the softclay It is considered that deformation occurs during thewhole process but only the deformation that occurs afterthe dissipation of the excessive pore water pressure can beregarded as creep onlyThe full shearing process stress-strain-time curve of each specimen is illustrated in Figure 3 and themutistage strain-time curves of each specimen are illustratedin Figure 4 The figures show that the creep process undertriaxial loading exhibits attenuation characteristics when thedeviatoric stress is small which contains two stages of creepprimary stage and steady stage while when the deviatoricstress is large itmay exhibit accelerated characteristics whichcontains three stages of creep primary stage steady stage andaccelerated stage such as the last load increment of specimen2

4 Mathematical Problems in Engineering

Table 2 Loading scheme of triaxial rheological tests

Specimen number Confining pressure (kPa) Loading scheme (kPa)1 100 30-60-90-12-150-180-210-2402 200 40-80-120-160-200-240-280-320

20

15

10

5

00

60

120

180

240

300

120576(

)q

(kPa

)

200 400 600 800 1000 1200t (h)

(a) Specimen 1

20

25

15

10

50

80

160

240

320

120576(

)q

(kPa

)

0 200 400 600 800 1000 1200

t (h)

(b) Specimen 2

Figure 3 Full-process press-strain-time curve

0

5

10

15

20

30 kPa60 kPa90 kPa120 kPa

150 kPa180 kPa210 kPa240 kPa

120576(

)

10 1001t (h)

(a) Specimen 1

40 kPa80 kPa120 kPa160 kPa 320 kPa

240 kPa200 kPa

280 kPa

0

5

10

10 100

15

20

25

120576(

)

1t (h)

(b) Specimen 2

Figure 4 Mutistage strain-time curves

33 Computation and Curve Fitting For triaxial rheologicaltests 119868

1= 1205901+ 21205903= 31199011015840 119904101584011= 2(120590

1minus 1205903)3 = 21199023 so (13)

can be written as follows

12057611= 1198751+ 1198752(1 minus 119890

minus1198753119905

) + 1198754119905 (14)

where 1198751= (1199013119870) + (1199023119866

0) 1198752= (1199023119866

1) 1198753= (11986611198671)

1198754= (1119867

2)⟨120601(119891)⟩((2119902119872

2

) + (13)(21199011015840

minus 119901119889

119888))

Use (14) to fit the experimental data the results wereshowed in Figures 5 and 6 in which the parameters SSEand 1198772 denote the residual sum of squares and squares ofcorrelation coefficient respectively From Figures 5 and 6 wecan see that the model fits testing data of primary creep stageand steady creep stage very well while for the acceleratedcreep stage it needs further study According to the value of

Mathematical Problems in Engineering 5

0023

0024

0025

0026

0027

0 20 40 60 80 100 120 140 160 180

120576 11

SSE = 57349E minus 9R2 = 099069

t (h)

24105E minus 2P114547E minus 3P213321E minus 1P337098E minus 6P4

Test dataFitting curve

(a) 119902 = 30 kPa

0 20 40 60 80 100 120 140 160 180t (h)

0034

0036

0038

0040

0042

0044

120576 11

SSE = 58195E minus 8R2 = 098856

35852E minus 2P150523E minus 3P220259E minus 1P367168E minus 6P4

Test dataFitting curve

(b) 119902 = 60 kPa

SSE = 10771E minus 7R2 = 098854

0 20 40 60 80 100 120 140 160 180

t (h)

0046

0048

0050

0052

0054

0056

0058

120576 11

69284E minus 3P223128E minus 1P310127E minus 5P4

48144E minus 2P1

Test dataFitting curve

(c) 119902 = 90 kPa

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 1712E minus 7R2 = 099035

60822E minus 293268E minus 328662E minus 118749E minus 5

0055

0060

0070

0065

0075120576 1

1

P2P3P4

P1

Test dataFitting curve

(d) 119902 = 120 kPa

SSE = 61207E minus 7R2 = 098043

P1 76261E minus 2P2 12875E minus 3P3 35961E minus 1P4 26325E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0070

0080

0090

0075

0085

0095

120576 11

Test dataFitting curve

(e) 119902 = 150 kPa

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 872E minus 7R2 = 098046

89059E minus 215909E minus 237062E minus 128956E minus 5

0080

0090

0100

0110

0115

0085

0095

0105

120576 11

P2P3P4

P1

Test dataFitting curve

(f) 119902 = 180 kPa

Figure 5 Continued

6 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

0090

0100

0110

0120

0130

0125

0115

0095

0105

120576 11

SSE = 99965E minus 7R2 = 098620

10217E minus 119991E minus 228288E minus 129143E minus 5

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 210 kPa

Figure 5 Fitting curves of specimen 1 (1205903= 100 kPa)

Table 3 Model parameters of specimen 1 (1205903= 100 kPa)

1205901minus 1205903(kPa) 119864

0(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots)

30 249 1856 50160 1607360 251 1069 18993 1795090 249 1169 18198 21297120 247 1158 14544 24253150 236 1049 10497 25049180 236 1018 9891 22046210 235 945 12032 26240

Table 4 Model parameters of specimen 2 (1205903= 200 kPa)

1205901minus 1205903(kPa) 119864

0(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots)

40 1075 1804 13881 1222380 903 1870 25862 11747120 834 1694 29947 12591160 763 1692 30057 12704200 660 1729 26118 11732240 604 1831 27481 12823280 508 1602 28165 11589

1198751 1198752 1198753 and 119875

4 the value of the model parameters 119864

0 1198641

1205781 and 120578

2can be determined easily such as in Tables 3 and 4

34 Discussion Table 3 shows that the model parameters ofspecimen 1 (120590

3= 100 kPa) change slightly with the change

of deviatoric stress except the first stage Table 4 showsthat the model parameters of specimen 2 (120590

3= 200 kPa)

change slightly with the change of deviatoric stress except1198640 Compared with the model parameters of specimen 1 and

specimen 2 it can be found that both 1198640and 119864

1increase

notably with the increase of 1205903 which means that the elastic

strain reduces when 1205903increases and 120578

1increases notably

Table 5 Percentage of strain components of specimen 1 (1205903=

100 kPa 119905 = 7 d)

1205901minus 1205903(kPa) 120576

119890

11() 120576

119907119890

11() 120576

119907119901

11()

30 9206 556 23860 8530 1202 26890 8483 1221 296120 8298 1272 430150 8151 1376 473180 8109 1448 443210 8041 1573 385

Table 6 Percentage of strain components of specimen 2 (1205903=

200 kPa 119905 = 7 d)

1205901minus 1205903(kPa) 120576

119890

11() 120576

119907119890

11() 120576

119907119901

11()

40 7703 1653 64580 6774 1683 1543120 6677 1971 1353160 6713 1981 1306200 6951 1837 1212240 7096 1686 1217280 7397 1740 863

while 1205782is reduced notably with the increase of 120590

3 which

means that the viscoelastic strain rate and viscoplastic strainrate change significantly when 120590

3increases For the lack

of more experiment data the further variation of modelparameters was unable to find out which needed furtherstudy

Based on the parameters in Tables 3 and 4 the com-ponents of elastic strain viscoelastic strain and viscoplasticstrain at any time can be calculated with (13) Tables 5 and 6show 120576

119890

11 120576V11989011 and 120576V119901

11of all loading levels when 119905 = 7 d it can

Mathematical Problems in Engineering 7

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 71424E minus 9R2 = 098984

93026E minus 319959E minus 346778E minus 146368E minus 6

0008

0009

0010

0011

0012

0013120576 1

1

P2P3P4

P1

Test dataFitting curve

(a) 119902 = 40 kPa

0 20 40 60 80 100 120 140 160 180t (h)

SSE = 27165E minus 8R2 = 099567

15502E minus 238505E minus 326029E minus 121015E minus 5

0010

0012

0014

0016

0018

0020

0022

0024

120576 11

P2P3P4

P1

Test dataFitting curve

(b) 119902 = 80 kPa

SSE = 85674E minus 8R2 = 0994

21595E minus 263737E minus 320369E minus 126044E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0016

0018

0020

0022

0024

0026

0028

0030

0032

0034

120576 11

P2P3P4

P1

Test dataFitting curve

(c) 119902 = 120 kPa

SSE = 15199E minus 7R2 = 099385

28845E minus 285110E minus 320264E minus 133395E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0020

0025

0030

0035

0040

0045

120576 11

P2P3P4

P1

Test dataFitting curve

(d) 119902 = 160 kPa

SSE = 26486E minus 7R2 = 099252

39406E minus 210413E minus 223826E minus 140897E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0030

0035

0040

0045

0050

0060

0055

120576 11

P2P3P4

P1

Test dataFitting curve

(e) 119902 = 200 kPa

0040

0045

0050

0060

0065

0075

0070

0055

120576 11

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 20786E minus 7R2 = 099572

49663E minus 211800E minus 223980E minus 150713E minus 5

P2P3P4

P1

Test dataFitting curve

(f) 119902 = 240 kPa

Figure 6 Continued

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 50043E minus 7R2 = 09928

66892E minus 215735E minus 220470E minus 146472E minus 5

00600060

00650065

00700070

00750075

00800080

00850085

00900090

00950095

120576 11

120576 11

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 280 kPa

Figure 6 Fitting curves of specimen 2 (1205903= 200 kPa)

001 10 100

02

04

06

08

10

12

14

16

Visc

ous s

trai

n (

)

Time (h)

VES q = 60kPaVPS q = 60kPaVES q = 120 kPa

VPS q = 120 kPaVES q = 180 kPaVPS q = 180 kPa

(a) Specimen 1

Visc

ous s

trai

n (

)

00

02

04

06

08

10

12

100Time (h)

VES q = 80kPaVPS q = 80kPaVES q = 160 kPa

VPS q = 160 kPaVES q = 240 kPaVPS q = 240 kPa

1 10

(b) Specimen 2

Figure 7 Development of computed viscous strain (VES viscoelastic strain VPS viscoplastic strain)

be found that the elastic strain accounted for the major partof the total strain but the viscoelastic strain and viscoplasticstrain increase with the increase of 120590

3 which takes about

15sim30 for each loading increment of the total and cannotbe neglected

Figure 7 shows several typical curves of the developmentof computed viscoelastic strain and viscoplastic strain ofspecimen 1 and specimen 2 It can be found that theviscoelastic strain increases quickly in the early stage andthen reaches steady in a short time (about 30 hours) whilethe viscoplastic strain increases the whole process whichindicates that the long-term deformation after consolidation

of soft clay foundation is mainly caused by the viscoplasticityrather than viscoelasticity

4 Derivation and Verification of UMAT

41 Derivation of UMAT ABAQUS is a FEM software whichhas powerful computing capabilities of strongly nonlinearproblems A user material subroutine (UMAT) is codedfor the proposed model on the platform of ABAQUS tostudy the elastic-viscoplastic consolidation properties of softclay According to ABAQUS the stiffness coefficient matrix

Mathematical Problems in Engineering 9

(Jacobian matrix) and the equation to update the stress andstrain should be defined in the UAMT The Jacobian matrixD is defined as follows

D =

120597Δ120590

120597Δ120576

(15)

where Δ120590 and Δ120576 denote the stress increment matrix andstrain increment matrix respectively

For the proposedmodel (4)sim(6) can be written as followsin 2D or 3D problems

120576119890

119899=

120583

1198640

A120590119899 (16a)

V119890119899=

1

1205781

A120590119899minus

1198661

1205781

120576V119890119899 (16b)

V119901119899=

1

1205782

⟨120601 (119865)⟩

120597119865

120597120590

(16c)

In (16a) (16b) and (16c) 120590119899 120576119899 120576V119890119899 and 120576V119901

119899denote the

Voigt matrix form of stress and strain component tensors Adenotes the flexibilitymatrix for plane strain problem and 3Dproblems and it can be written as follows

A =

[[[

[

1 minus120583 minus120583 0

minus120583 1 minus120583 0

minus120583 minus120583 1 0

0 0 0 2 (1 + 120583)

]]]

]

(17a)

A =

[[[[[[[

[

1 minus120583 minus120583 0 0 0

minus120583 1 minus120583 0 0 0

minus120583 minus120583 1 0 0 0

0 0 0 2 (1 + 120583) 0 0

0 0 0 0 2 (1 + 120583) 0

0 0 0 0 0 2 (1 + 120583)

]]]]]]]

]

(17b)

Suppose that the time step of the 119899th increment is Δ119905119899 so

the viscoelastic strain increment in the increment step can bewritten as follows

Δ120576V119890119899= Δ119905119899[(1 minus Θ)

V119890119899+ Θ

V119890119899+1] (18)

where Θ is differential coefficient 0 le Θ le 1 Suppose that119861119899= 1(1 + ΘΔ119905

11989911986411205781) C119899= 119861119899ΘΔ119905119899A1205781 (18) can be

written as follows

Δ120576V119890119899= 119861119899V119890119899Δ119905119899+ C119899Δ120590119899 (19)

The viscoplastic strain increment in the increment stepcan be written as follows

Δ120576V119901119899= Δ119905119899[(1 minus Θ)

V119901119899+ Θ

V119901119899+1] (20)

In order to calculate the viscoplastic strain rate at theend of the 119899th increment apply the Taylor series expansionand ignore the second order trace then we can obtain thefollowing equation

V119901119899+1

= V119901119899+

120597V119901119899

120597120590

120597120590

120597119905

Δ119905119899 (21)

Suppose thatH119899= 120597

V119901119899120597120590 (21) can be written as

V119901119899+1

= V119901119899+H119899Δ120590119899 (22)

Substituting (22) into (21) we can obtain the followingequation

Δ120576V119901119899= Δ119905119899V119901119899+ ΘΔ119905

119899H119899Δ120590119899 (23)

From (16c)H119899can be written as

H119899=

1

1205782

[⟨120601 (119865)⟩

120597a119879

120597120590

+

120597 ⟨120601 (119865)⟩

120597119865

aa119879] (24)

where a = 120597119865120597120590 According to (9) a can be written asfollows

a = 120597119865

120597120590

= 1198621a1+ 1198622a2 (25)

where 1198621= 2119902119872

2 1198622= 2119901 minus 119901

0 a1= 120597119902120597120590 a

2= 120597119901120597120590

From (25) the following equation can be derived

120597a119879

120597120590

=

1205971198621

120597119902

a1a1198791+ 1198621

120597a1198791

120597120590

+

1205971198622

120597119901

a2a1198792 (26)

where (120597a1198791120597120590) = (32119902)M

1minus(94119902

3

)M2 a1a1198791= (94119902

2

)M2

a2a1198792= (19)M

3 M1 M2 and M

3are matrices written as

follows

M1=

[[[[[[[

[

23 minus13 minus13 0 0 0

minus13 23 minus13 0 0 0

minus13 minus13 23 0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

]]]]]]]

]

M2=

[[[[[[

[

1198782

111198781111987822

1198781111987833

21198781111987812

21198781111987823

21198781111987813

1198781111987822

1198782

221198782211987833

21198782211987812

21198782211987823

21198782211987813

1198781111987833

1198782211987833

1198782

3321198783311987812

21198783311987823

21198783311987813

21198781111987812

21198782211987812

21198783311987812

41198782

1241198781211987823

41198781211987813

21198781111987823

21198782211987823

21198783311987823

41198781211987823

41198782

2341198782311987813

21198781111987813

21198782211987813

21198783311987813

41198781211987813

41198782311987813

41198782

13

]]]]]]

]

M3=

[[[[[[[

[

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]]]]]]]

]

(27)

Combining the above equations the value of H119899can

be derived Substituting H119899into (22) the viscoplastic strain

increment Δ120576V119901119899

is obtainedFor current increment Δ120590

119899is calculated by (28) as

follows

Δ120590119899= D119890Δ120576119890

119899 (28)

Substituting (19) and (23) into (28) we can obtain thefollowing equation

Δ120590119899= D (Δ120576

119899minus 119861Δ119905

119899V119890119899minus Δ119905119899V119901119899) (29)

10 Mathematical Problems in Engineering

Table 7 Material parameters of verification model

1198640(MPa) 119864

1(MPa) 120578

1(GPasdots) 120578

2(GPasdots) 119875

0(kPa) 119890

0120583 119872 120582 120581

10 20 1000 1000 50 108 035 0898 0102 0015

12059021205902

1205903

1205903

1205901

1205901

Figure 8 Sketch of verification model

Equation (29) is the relationship between stress incre-ment and strain increment in which matrix D is defined asfollows

D = [(D119890)minus1 + C119899+ ΘΔ119905

119899H119899]

minus1

(30)

Based on the above equations a user material subroutinewas coded for the proposed constitutive model on theplatform of ABAQUS

42 Verification of UMAT In order to verify the validity ofthe UMAT we study a soil column of triaxial compressionsuch as Figure 8 the size of the column is 01m times 01m times

02m Supposing that 1205901= 100 kPa 120590

2= 1205903= 50 kPa

and the model parameters are given in Table 7 The columnwas cut into one element the element type is C3D8 and theconsolidation effect was not considered here Embedding theUMAT coded in this paper on the platform of ABAQUS wecalculated the triaxial compression property of the columnthe results were summarized in Figure 9simFigure 10

From Figure 9(a) we can see the deformation of thecolumn exhibits apparent rheology the rheological deforma-tion takes about 37 of the whole deformation Figure 9(b)shows that the static yield surface gradually expanded withthe time until it reached steady state when it coincided withthe dynamic yield surface From Figures 9(c) and 9(d) wecan see that the viscoelastic strain and the viscoplastic strainboth gradually increased with time until they reached theirsteady state accordingly the viscoelastic strain rate and theviscoplastic strain rate were gradually deduced to nearly zeroThe above results fit the property of Nishihara model andthe hardening law of modified Cambridge model very wellwhich proved the correctness of the UMAT

In the proposed model 1205781 1205782 and 119901

0are the most

important parameters Figure 10 showed the sensitivity ofthese parameters From Figures 10(a) and 10(b) we can seethe final viscous deformation is a constant regardless of thedifferent values of 120578

1 1205782 but the time when it reaches steady

varies greatly so the accuracy of 1205781and 1205782is directly related

to the accuracy of computed displacement in the rheologicalprocess Figure 10(c) shows that the value of 119901

0has a great

influence on the viscoplastic strain the smaller 1199010is the

more rapidly the viscoplastic strain develops and the largerthe final viscoplastic strain occurs and so the accuracy of1199010is not only directly related to the accuracy of computed

displacement in the rheological process but also affected thefinal displacement which needs to be paid more attention

5 Analysis on Rheological Consolidation ofSoft Clay

51 Computed Model Consolidation problems can be classi-fied as small strain theory and finite strain theory accordingto the difference of geometric assumption Small straintheory assumed that the strain is little but the consolida-tion deformation is usually very large for deep soft clayfoundation Take dredger fill for example the deformationmay exceed 80 for which the small strain theory is nolonger suitable [14 15] As the pioneer researchers studiedon finite strain consolidation theory Mikasa [16] and Gib-son et al [17] derived their one-dimensional finite strainconsolidation equations respectively in 1960s and manyscholars developed further research on the topic in thefollowing decades Because of strong nonlinearity most ofexisting researches on finite strain consolidation adoptedsome simplified assumptionswithout considering rheologicalcharacteristics By means of ABAQUS this paper studied therheological consolidation of soft clay considering finite straineffect

The computed model is plane strain problem as shown inFigure 11 The soil foundation is homogeneous anisotropicsaturated and normally consolidated soft clay with a thick-ness of 100m and only the top boundary is permeable Inaddition the semilogarithmic empirical formula proposed byTavenas et al [18] was adopted to consider the variation ofpermeability in the process of consolidation as follows

119896V = 119896V0 sdot 10(119890minus1198900)119862119896 (31)

where 119896V0 is the initial permeability coefficients and 119896V is thepermeability coefficient in the consolidation process 119890

0is the

initial void ratio and 119890 is the void ratio in the consolidationprocess 119862

119896is the permeability index According to the

above work themodel parameters are summarized in Table 8approximately

52 Consolidation Property of Soft Clay From Figure 12 wecan see the differences of the consolidation process betweenfinite strain and small strain Figure 12(a) shows that for anylocation the excess pore pressure of finite strain is less thanthat of small strain at the same time and the difference isincreasing with the increase of time in beginning period

Mathematical Problems in Engineering 11

Table 8 Material parameters of computed model

1198640(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots) 120574sat (kNsdotm

minus3) 1198961199070(msdotsminus1) 119875

0(kPa) 119872 120582 120581 119890

0120583 119862

119896

25 115 150 24000 177 45119864 minus 9 50 0898 0102 0015 108 035 05

100000225

00200

00175

00150

00125

00100

Vert

ical

def

orm

atio

n (m

)

Time (s)10000 100000 1000000 1E7

(a)

40

50

60

70

80

90

100

110

120

1000Time (s)

10000 100000 1000000 1E7

Pcs

(kPa

)(b)

05

00

10

15

20

25

30

35

Viscoelastic strain

Visc

ous s

trai

n (

)

Viscoplastic strain

1000

Time (s)

10000 100000 1000000 1E71E7

(c)

Viscoelastic strain rateViscoplastic strain rate

0

2

4

6

8

10

12

14

16

1000Time (s)

10000 100000 1000000 1E71E7

Visc

ous s

trai

n ra

te (s

minus1)

(d)

Figure 9 Triaxial compression property of soil column

Figure 12(b) shows that the displacement of finite strain isless than that of small strain and the difference increasedsignificantly if the load is increasing For example when 119902 =200 kPa the differencemay take about 20while it is nearly 0when 119902= 50 kPa So it is essential to consider the error causedby small strain if the load is large and the compressibility ofthe soil is large

From Figure 13 we can see the influence of rheologyon the consolidation process Figure 13(a) shows that thedissipation of excess pore pressure is significantly slower ifthe rheological effect is considered for example when 119905 =1000 d the excess pore pressure is nearly 0 if the rheologicaleffect is not considered but it is still larger than 20 kPa ifthe rheological effect is considered Figure 13(a) shows that

the displacement is influenced by rheological effect too forexample in the early stage (119905 le 100 d) the displacement isnearly the same for two cases but the final displacementmay have a difference of more than 35 which indicatesthat the displacement in the early stage is mainly causedby consolidation while the displacement in the late stage ismainly caused by rheology

6 Conclusions

(1) An elastic-viscoplastic model developed to describethe time-dependent behavior of normally consoli-dated soft clay was presented in the paper on the

12 Mathematical Problems in Engineering

00

05

10

15

20

25

30Vi

scoe

lasti

c str

ain

()

100010 100Time (s)

10000 100000 1000000 1E7

1205781 = 10GPamiddots1205781 = 100GPamiddots1205781 = 1000GPamiddots

(a)

00

02

04

06

08

10

12

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

1205782 = 10GPamiddots1205782 = 100GPamiddots1205782 = 1000GPamiddots

(b)

00

04

08

12

16

20

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

P0 = 30kPaP0 = 40kPa

P0 = 50kPaP0 = 60kPa

(c)

Figure 10 Sensitivity of model parameters

O

Permeable

B = 5 mx

z

H = 10m

q = 100 kPa

Figure 11 Calculation model and mesh generation

framework of modified cam-clay elastic plastic modeland Perzynarsquos overstress viscoplastic theory

(2) A series of laboratory triaxial rheological tests ofNingbo soft clay with different confining pressure anddeviatoric stress are performed the results show thatthe presented constitutive model was suitable for therheological characteristic of Ningbo soft clay and it isachievable to determine the parameters of presentedmodel by curve fitting

(3) The analysis on the model parameters shows that thevalue of all parameters is related to confining pressureand deviatoric stress It may need further study on therelationship between parameters and the stress levelso as to use the model conveniently

Mathematical Problems in Engineering 13

0 20 40 60 80 100u (kPa)

0

2

4

6

8

10H

(m)

FS t = 100dSS t = 100dFS t = 500d

SS t = 500dFS t = 1000 dSS t = 1000 d

(a)

12

10

08

06

04

02

00

Def

orm

atio

n (m

)

Time (d)1000101 100 10000 100000

SS q= 100 kPaFS q= 50 kPaSS q= 50 kPa

SS q= 200 kPaFS q= 200 kPa

FS q= 100 kPa

(b)

Figure 12 Results comparison of large strain and small strain (SS small train FS finite strain)

00

2

4

6

8

10

H(m

)

20 40 60 80 100u (kPa)

t = 100d NRt = 100d Rt = 500d NR

t = 500d Rt = 1000 d NRt = 1000 d R

(a)

06

05

04

03

02

01

00

Def

orm

atio

n (m

)

Time (d)

NRR

1000101 100 10000 100000

(b)

Figure 13 Effects of rheology on calculated results (NR rheological effect not considered R rheological effect considered)

(4) In the proposed model the viscoelastic viscositycoefficient 120578

1 the viscoplastic viscosity coefficient 120578

2

and the initial yield surface 1199010are the most important

parameters 1205781and 120578

2have important influence on

the time of viscous strain developed and 1199010have

important influence on both the viscoplastic strainrate and the final viscoplastic strain

(5) The existence of rheology will significantly slow downthe dissipation of the excess pore pressure Consid-ering the rheological effect the total displacement

is larger and the displacement in the early stage ismainly caused by consolidation while in the late stageit is mainly caused by rheology

(6) For consolidation problems of high compressibilitysoft clay foundation great error will be caused bysmall strain assumption so finite strain effect shouldbe considered When finite strain effect is consideredthe consolidation process is developed more rapidlyand the computed consolidation displacement will beless than that of small strain

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Critical state line

q

0

1

Static yield surface

Static yield

A

Psc Pd

c PP9984000

M

B Dynamic yieldsurface for B

surface for B

120597Fd120597120590998400ij

Figure 2 Yield criterion of model

In (8) 1199010is the intercept of initial static yield surface in

the 1199011015840 axis for soft clay it can be assigned to the biggestconsolidation stress in history 120582 is the slope of normalconsolidation curve in 119890 sim ln1199011015840 plane 120581 is the slope ofrecompression curve in 119890 sim ln1199011015840 plane

A dynamic yield criterion 119865119889is defined to describe the

current state of stress as follows

119865119889=

1199022

1198722+ 1199011015840

(1199011015840

minus 119901119889

119888) = 0 (9)

From (8) and (9) the 119891 and 1198910in (7) can be written as

119891 = 119901119889

119888= 1199011015840

+

1199022

11987221199011015840

1198910= 119901119904

119888= 1199010sdot exp(

1 + 1198900

120582 minus 120581

120576V)

(10)

So we can write (7) as follows

120601 (119891) =

119901119889

119888

119901119904

119888

minus 1 (11)

Combining (6)sim(11) (6) can be written as follows

120576V119901119894119895=

1

1198672

⟨120601 (119891)⟩(

31199041015840

119894119895

1198722+ (2119901

1015840

minus 119901119889

119888)

120575119894119895

3

) (12)

From (4) (5) and (12) when the stress is constant thetotal strain of the presented model can be written as follows

120576119894119895=

1198681

9119870

120575119894119895+

1199041015840

119894119895

21198660

+

1199041015840

119894119895

21198661

(1 minus 119890minus(11986411205781)119905

)

+

1

1198672

⟨120601 (119891)⟩(

31199041015840

119894119895

1198722+ (2119901

1015840

minus 119901119889

119888)

120575119894119895

3

) 119905

(13)

3 Experiments and Computation

31 Test Program In order to verify the presentedmodel lab-oratory triaxial rheological tests were performed to observethemechanical behavior of soft clay under long-term loading

Table 1 Property of testing material

Property ValueEmbedded depth (m) 35sim45Specific gravity 267Moist unit weight (kNm3) 173Water content () 404Void ratio 108Liquid limit () 425Plastic limit () 249Plasticity index 176Slope of the critical state line 0898Slope of compression curve 0102Slope of the recompression curve 0015

The soil investigated in the experiments is Ningbo soft claywhich is a kind of problematic soil for low strength highcompressibility and time-dependent behavior and depositsin Hangzhou Bay ChinaThe basic properties of the soil weresummarized in Table 1 Bothmoisture content andmoist unitweight of this material are significantly less than those oftypical natural sedimentary deposits

The experimental equipment was refitted on the platformof strain controlled triaxial apparatus the original strainloading method was changed to stress loading method whilethe confining pressure system the back pressure loadingsystem and the measurement system were reserved Thesoil was cut into replicate specimens with a diameter of398mm and a height of 800mm and placed in the triaxialtest apparatus Both soil specimens were consolidated 24hours under a confining pressure of 100 kPa and 200 kParespectively then keep the confining pressure as constantand apply the deviatoric stress increment until the specimenswere damaged or the total strain exceeds 15 In the wholeprocess the free drainage conditions were kept Table 2showed the different loading rates applied in the experimentsIn order to make sure that the creep deformation can be fullydeveloped each load increment lasted no less than 7 daysuntil the deformation of the specimen is lower than 001mmwithin 24 h

32 Experimental Results The experimental results werepresented to observe the rheological behavior of the softclay It is considered that deformation occurs during thewhole process but only the deformation that occurs afterthe dissipation of the excessive pore water pressure can beregarded as creep onlyThe full shearing process stress-strain-time curve of each specimen is illustrated in Figure 3 and themutistage strain-time curves of each specimen are illustratedin Figure 4 The figures show that the creep process undertriaxial loading exhibits attenuation characteristics when thedeviatoric stress is small which contains two stages of creepprimary stage and steady stage while when the deviatoricstress is large itmay exhibit accelerated characteristics whichcontains three stages of creep primary stage steady stage andaccelerated stage such as the last load increment of specimen2

4 Mathematical Problems in Engineering

Table 2 Loading scheme of triaxial rheological tests

Specimen number Confining pressure (kPa) Loading scheme (kPa)1 100 30-60-90-12-150-180-210-2402 200 40-80-120-160-200-240-280-320

20

15

10

5

00

60

120

180

240

300

120576(

)q

(kPa

)

200 400 600 800 1000 1200t (h)

(a) Specimen 1

20

25

15

10

50

80

160

240

320

120576(

)q

(kPa

)

0 200 400 600 800 1000 1200

t (h)

(b) Specimen 2

Figure 3 Full-process press-strain-time curve

0

5

10

15

20

30 kPa60 kPa90 kPa120 kPa

150 kPa180 kPa210 kPa240 kPa

120576(

)

10 1001t (h)

(a) Specimen 1

40 kPa80 kPa120 kPa160 kPa 320 kPa

240 kPa200 kPa

280 kPa

0

5

10

10 100

15

20

25

120576(

)

1t (h)

(b) Specimen 2

Figure 4 Mutistage strain-time curves

33 Computation and Curve Fitting For triaxial rheologicaltests 119868

1= 1205901+ 21205903= 31199011015840 119904101584011= 2(120590

1minus 1205903)3 = 21199023 so (13)

can be written as follows

12057611= 1198751+ 1198752(1 minus 119890

minus1198753119905

) + 1198754119905 (14)

where 1198751= (1199013119870) + (1199023119866

0) 1198752= (1199023119866

1) 1198753= (11986611198671)

1198754= (1119867

2)⟨120601(119891)⟩((2119902119872

2

) + (13)(21199011015840

minus 119901119889

119888))

Use (14) to fit the experimental data the results wereshowed in Figures 5 and 6 in which the parameters SSEand 1198772 denote the residual sum of squares and squares ofcorrelation coefficient respectively From Figures 5 and 6 wecan see that the model fits testing data of primary creep stageand steady creep stage very well while for the acceleratedcreep stage it needs further study According to the value of

Mathematical Problems in Engineering 5

0023

0024

0025

0026

0027

0 20 40 60 80 100 120 140 160 180

120576 11

SSE = 57349E minus 9R2 = 099069

t (h)

24105E minus 2P114547E minus 3P213321E minus 1P337098E minus 6P4

Test dataFitting curve

(a) 119902 = 30 kPa

0 20 40 60 80 100 120 140 160 180t (h)

0034

0036

0038

0040

0042

0044

120576 11

SSE = 58195E minus 8R2 = 098856

35852E minus 2P150523E minus 3P220259E minus 1P367168E minus 6P4

Test dataFitting curve

(b) 119902 = 60 kPa

SSE = 10771E minus 7R2 = 098854

0 20 40 60 80 100 120 140 160 180

t (h)

0046

0048

0050

0052

0054

0056

0058

120576 11

69284E minus 3P223128E minus 1P310127E minus 5P4

48144E minus 2P1

Test dataFitting curve

(c) 119902 = 90 kPa

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 1712E minus 7R2 = 099035

60822E minus 293268E minus 328662E minus 118749E minus 5

0055

0060

0070

0065

0075120576 1

1

P2P3P4

P1

Test dataFitting curve

(d) 119902 = 120 kPa

SSE = 61207E minus 7R2 = 098043

P1 76261E minus 2P2 12875E minus 3P3 35961E minus 1P4 26325E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0070

0080

0090

0075

0085

0095

120576 11

Test dataFitting curve

(e) 119902 = 150 kPa

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 872E minus 7R2 = 098046

89059E minus 215909E minus 237062E minus 128956E minus 5

0080

0090

0100

0110

0115

0085

0095

0105

120576 11

P2P3P4

P1

Test dataFitting curve

(f) 119902 = 180 kPa

Figure 5 Continued

6 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

0090

0100

0110

0120

0130

0125

0115

0095

0105

120576 11

SSE = 99965E minus 7R2 = 098620

10217E minus 119991E minus 228288E minus 129143E minus 5

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 210 kPa

Figure 5 Fitting curves of specimen 1 (1205903= 100 kPa)

Table 3 Model parameters of specimen 1 (1205903= 100 kPa)

1205901minus 1205903(kPa) 119864

0(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots)

30 249 1856 50160 1607360 251 1069 18993 1795090 249 1169 18198 21297120 247 1158 14544 24253150 236 1049 10497 25049180 236 1018 9891 22046210 235 945 12032 26240

Table 4 Model parameters of specimen 2 (1205903= 200 kPa)

1205901minus 1205903(kPa) 119864

0(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots)

40 1075 1804 13881 1222380 903 1870 25862 11747120 834 1694 29947 12591160 763 1692 30057 12704200 660 1729 26118 11732240 604 1831 27481 12823280 508 1602 28165 11589

1198751 1198752 1198753 and 119875

4 the value of the model parameters 119864

0 1198641

1205781 and 120578

2can be determined easily such as in Tables 3 and 4

34 Discussion Table 3 shows that the model parameters ofspecimen 1 (120590

3= 100 kPa) change slightly with the change

of deviatoric stress except the first stage Table 4 showsthat the model parameters of specimen 2 (120590

3= 200 kPa)

change slightly with the change of deviatoric stress except1198640 Compared with the model parameters of specimen 1 and

specimen 2 it can be found that both 1198640and 119864

1increase

notably with the increase of 1205903 which means that the elastic

strain reduces when 1205903increases and 120578

1increases notably

Table 5 Percentage of strain components of specimen 1 (1205903=

100 kPa 119905 = 7 d)

1205901minus 1205903(kPa) 120576

119890

11() 120576

119907119890

11() 120576

119907119901

11()

30 9206 556 23860 8530 1202 26890 8483 1221 296120 8298 1272 430150 8151 1376 473180 8109 1448 443210 8041 1573 385

Table 6 Percentage of strain components of specimen 2 (1205903=

200 kPa 119905 = 7 d)

1205901minus 1205903(kPa) 120576

119890

11() 120576

119907119890

11() 120576

119907119901

11()

40 7703 1653 64580 6774 1683 1543120 6677 1971 1353160 6713 1981 1306200 6951 1837 1212240 7096 1686 1217280 7397 1740 863

while 1205782is reduced notably with the increase of 120590

3 which

means that the viscoelastic strain rate and viscoplastic strainrate change significantly when 120590

3increases For the lack

of more experiment data the further variation of modelparameters was unable to find out which needed furtherstudy

Based on the parameters in Tables 3 and 4 the com-ponents of elastic strain viscoelastic strain and viscoplasticstrain at any time can be calculated with (13) Tables 5 and 6show 120576

119890

11 120576V11989011 and 120576V119901

11of all loading levels when 119905 = 7 d it can

Mathematical Problems in Engineering 7

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 71424E minus 9R2 = 098984

93026E minus 319959E minus 346778E minus 146368E minus 6

0008

0009

0010

0011

0012

0013120576 1

1

P2P3P4

P1

Test dataFitting curve

(a) 119902 = 40 kPa

0 20 40 60 80 100 120 140 160 180t (h)

SSE = 27165E minus 8R2 = 099567

15502E minus 238505E minus 326029E minus 121015E minus 5

0010

0012

0014

0016

0018

0020

0022

0024

120576 11

P2P3P4

P1

Test dataFitting curve

(b) 119902 = 80 kPa

SSE = 85674E minus 8R2 = 0994

21595E minus 263737E minus 320369E minus 126044E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0016

0018

0020

0022

0024

0026

0028

0030

0032

0034

120576 11

P2P3P4

P1

Test dataFitting curve

(c) 119902 = 120 kPa

SSE = 15199E minus 7R2 = 099385

28845E minus 285110E minus 320264E minus 133395E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0020

0025

0030

0035

0040

0045

120576 11

P2P3P4

P1

Test dataFitting curve

(d) 119902 = 160 kPa

SSE = 26486E minus 7R2 = 099252

39406E minus 210413E minus 223826E minus 140897E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0030

0035

0040

0045

0050

0060

0055

120576 11

P2P3P4

P1

Test dataFitting curve

(e) 119902 = 200 kPa

0040

0045

0050

0060

0065

0075

0070

0055

120576 11

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 20786E minus 7R2 = 099572

49663E minus 211800E minus 223980E minus 150713E minus 5

P2P3P4

P1

Test dataFitting curve

(f) 119902 = 240 kPa

Figure 6 Continued

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 50043E minus 7R2 = 09928

66892E minus 215735E minus 220470E minus 146472E minus 5

00600060

00650065

00700070

00750075

00800080

00850085

00900090

00950095

120576 11

120576 11

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 280 kPa

Figure 6 Fitting curves of specimen 2 (1205903= 200 kPa)

001 10 100

02

04

06

08

10

12

14

16

Visc

ous s

trai

n (

)

Time (h)

VES q = 60kPaVPS q = 60kPaVES q = 120 kPa

VPS q = 120 kPaVES q = 180 kPaVPS q = 180 kPa

(a) Specimen 1

Visc

ous s

trai

n (

)

00

02

04

06

08

10

12

100Time (h)

VES q = 80kPaVPS q = 80kPaVES q = 160 kPa

VPS q = 160 kPaVES q = 240 kPaVPS q = 240 kPa

1 10

(b) Specimen 2

Figure 7 Development of computed viscous strain (VES viscoelastic strain VPS viscoplastic strain)

be found that the elastic strain accounted for the major partof the total strain but the viscoelastic strain and viscoplasticstrain increase with the increase of 120590

3 which takes about

15sim30 for each loading increment of the total and cannotbe neglected

Figure 7 shows several typical curves of the developmentof computed viscoelastic strain and viscoplastic strain ofspecimen 1 and specimen 2 It can be found that theviscoelastic strain increases quickly in the early stage andthen reaches steady in a short time (about 30 hours) whilethe viscoplastic strain increases the whole process whichindicates that the long-term deformation after consolidation

of soft clay foundation is mainly caused by the viscoplasticityrather than viscoelasticity

4 Derivation and Verification of UMAT

41 Derivation of UMAT ABAQUS is a FEM software whichhas powerful computing capabilities of strongly nonlinearproblems A user material subroutine (UMAT) is codedfor the proposed model on the platform of ABAQUS tostudy the elastic-viscoplastic consolidation properties of softclay According to ABAQUS the stiffness coefficient matrix

Mathematical Problems in Engineering 9

(Jacobian matrix) and the equation to update the stress andstrain should be defined in the UAMT The Jacobian matrixD is defined as follows

D =

120597Δ120590

120597Δ120576

(15)

where Δ120590 and Δ120576 denote the stress increment matrix andstrain increment matrix respectively

For the proposedmodel (4)sim(6) can be written as followsin 2D or 3D problems

120576119890

119899=

120583

1198640

A120590119899 (16a)

V119890119899=

1

1205781

A120590119899minus

1198661

1205781

120576V119890119899 (16b)

V119901119899=

1

1205782

⟨120601 (119865)⟩

120597119865

120597120590

(16c)

In (16a) (16b) and (16c) 120590119899 120576119899 120576V119890119899 and 120576V119901

119899denote the

Voigt matrix form of stress and strain component tensors Adenotes the flexibilitymatrix for plane strain problem and 3Dproblems and it can be written as follows

A =

[[[

[

1 minus120583 minus120583 0

minus120583 1 minus120583 0

minus120583 minus120583 1 0

0 0 0 2 (1 + 120583)

]]]

]

(17a)

A =

[[[[[[[

[

1 minus120583 minus120583 0 0 0

minus120583 1 minus120583 0 0 0

minus120583 minus120583 1 0 0 0

0 0 0 2 (1 + 120583) 0 0

0 0 0 0 2 (1 + 120583) 0

0 0 0 0 0 2 (1 + 120583)

]]]]]]]

]

(17b)

Suppose that the time step of the 119899th increment is Δ119905119899 so

the viscoelastic strain increment in the increment step can bewritten as follows

Δ120576V119890119899= Δ119905119899[(1 minus Θ)

V119890119899+ Θ

V119890119899+1] (18)

where Θ is differential coefficient 0 le Θ le 1 Suppose that119861119899= 1(1 + ΘΔ119905

11989911986411205781) C119899= 119861119899ΘΔ119905119899A1205781 (18) can be

written as follows

Δ120576V119890119899= 119861119899V119890119899Δ119905119899+ C119899Δ120590119899 (19)

The viscoplastic strain increment in the increment stepcan be written as follows

Δ120576V119901119899= Δ119905119899[(1 minus Θ)

V119901119899+ Θ

V119901119899+1] (20)

In order to calculate the viscoplastic strain rate at theend of the 119899th increment apply the Taylor series expansionand ignore the second order trace then we can obtain thefollowing equation

V119901119899+1

= V119901119899+

120597V119901119899

120597120590

120597120590

120597119905

Δ119905119899 (21)

Suppose thatH119899= 120597

V119901119899120597120590 (21) can be written as

V119901119899+1

= V119901119899+H119899Δ120590119899 (22)

Substituting (22) into (21) we can obtain the followingequation

Δ120576V119901119899= Δ119905119899V119901119899+ ΘΔ119905

119899H119899Δ120590119899 (23)

From (16c)H119899can be written as

H119899=

1

1205782

[⟨120601 (119865)⟩

120597a119879

120597120590

+

120597 ⟨120601 (119865)⟩

120597119865

aa119879] (24)

where a = 120597119865120597120590 According to (9) a can be written asfollows

a = 120597119865

120597120590

= 1198621a1+ 1198622a2 (25)

where 1198621= 2119902119872

2 1198622= 2119901 minus 119901

0 a1= 120597119902120597120590 a

2= 120597119901120597120590

From (25) the following equation can be derived

120597a119879

120597120590

=

1205971198621

120597119902

a1a1198791+ 1198621

120597a1198791

120597120590

+

1205971198622

120597119901

a2a1198792 (26)

where (120597a1198791120597120590) = (32119902)M

1minus(94119902

3

)M2 a1a1198791= (94119902

2

)M2

a2a1198792= (19)M

3 M1 M2 and M

3are matrices written as

follows

M1=

[[[[[[[

[

23 minus13 minus13 0 0 0

minus13 23 minus13 0 0 0

minus13 minus13 23 0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

]]]]]]]

]

M2=

[[[[[[

[

1198782

111198781111987822

1198781111987833

21198781111987812

21198781111987823

21198781111987813

1198781111987822

1198782

221198782211987833

21198782211987812

21198782211987823

21198782211987813

1198781111987833

1198782211987833

1198782

3321198783311987812

21198783311987823

21198783311987813

21198781111987812

21198782211987812

21198783311987812

41198782

1241198781211987823

41198781211987813

21198781111987823

21198782211987823

21198783311987823

41198781211987823

41198782

2341198782311987813

21198781111987813

21198782211987813

21198783311987813

41198781211987813

41198782311987813

41198782

13

]]]]]]

]

M3=

[[[[[[[

[

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]]]]]]]

]

(27)

Combining the above equations the value of H119899can

be derived Substituting H119899into (22) the viscoplastic strain

increment Δ120576V119901119899

is obtainedFor current increment Δ120590

119899is calculated by (28) as

follows

Δ120590119899= D119890Δ120576119890

119899 (28)

Substituting (19) and (23) into (28) we can obtain thefollowing equation

Δ120590119899= D (Δ120576

119899minus 119861Δ119905

119899V119890119899minus Δ119905119899V119901119899) (29)

10 Mathematical Problems in Engineering

Table 7 Material parameters of verification model

1198640(MPa) 119864

1(MPa) 120578

1(GPasdots) 120578

2(GPasdots) 119875

0(kPa) 119890

0120583 119872 120582 120581

10 20 1000 1000 50 108 035 0898 0102 0015

12059021205902

1205903

1205903

1205901

1205901

Figure 8 Sketch of verification model

Equation (29) is the relationship between stress incre-ment and strain increment in which matrix D is defined asfollows

D = [(D119890)minus1 + C119899+ ΘΔ119905

119899H119899]

minus1

(30)

Based on the above equations a user material subroutinewas coded for the proposed constitutive model on theplatform of ABAQUS

42 Verification of UMAT In order to verify the validity ofthe UMAT we study a soil column of triaxial compressionsuch as Figure 8 the size of the column is 01m times 01m times

02m Supposing that 1205901= 100 kPa 120590

2= 1205903= 50 kPa

and the model parameters are given in Table 7 The columnwas cut into one element the element type is C3D8 and theconsolidation effect was not considered here Embedding theUMAT coded in this paper on the platform of ABAQUS wecalculated the triaxial compression property of the columnthe results were summarized in Figure 9simFigure 10

From Figure 9(a) we can see the deformation of thecolumn exhibits apparent rheology the rheological deforma-tion takes about 37 of the whole deformation Figure 9(b)shows that the static yield surface gradually expanded withthe time until it reached steady state when it coincided withthe dynamic yield surface From Figures 9(c) and 9(d) wecan see that the viscoelastic strain and the viscoplastic strainboth gradually increased with time until they reached theirsteady state accordingly the viscoelastic strain rate and theviscoplastic strain rate were gradually deduced to nearly zeroThe above results fit the property of Nishihara model andthe hardening law of modified Cambridge model very wellwhich proved the correctness of the UMAT

In the proposed model 1205781 1205782 and 119901

0are the most

important parameters Figure 10 showed the sensitivity ofthese parameters From Figures 10(a) and 10(b) we can seethe final viscous deformation is a constant regardless of thedifferent values of 120578

1 1205782 but the time when it reaches steady

varies greatly so the accuracy of 1205781and 1205782is directly related

to the accuracy of computed displacement in the rheologicalprocess Figure 10(c) shows that the value of 119901

0has a great

influence on the viscoplastic strain the smaller 1199010is the

more rapidly the viscoplastic strain develops and the largerthe final viscoplastic strain occurs and so the accuracy of1199010is not only directly related to the accuracy of computed

displacement in the rheological process but also affected thefinal displacement which needs to be paid more attention

5 Analysis on Rheological Consolidation ofSoft Clay

51 Computed Model Consolidation problems can be classi-fied as small strain theory and finite strain theory accordingto the difference of geometric assumption Small straintheory assumed that the strain is little but the consolida-tion deformation is usually very large for deep soft clayfoundation Take dredger fill for example the deformationmay exceed 80 for which the small strain theory is nolonger suitable [14 15] As the pioneer researchers studiedon finite strain consolidation theory Mikasa [16] and Gib-son et al [17] derived their one-dimensional finite strainconsolidation equations respectively in 1960s and manyscholars developed further research on the topic in thefollowing decades Because of strong nonlinearity most ofexisting researches on finite strain consolidation adoptedsome simplified assumptionswithout considering rheologicalcharacteristics By means of ABAQUS this paper studied therheological consolidation of soft clay considering finite straineffect

The computed model is plane strain problem as shown inFigure 11 The soil foundation is homogeneous anisotropicsaturated and normally consolidated soft clay with a thick-ness of 100m and only the top boundary is permeable Inaddition the semilogarithmic empirical formula proposed byTavenas et al [18] was adopted to consider the variation ofpermeability in the process of consolidation as follows

119896V = 119896V0 sdot 10(119890minus1198900)119862119896 (31)

where 119896V0 is the initial permeability coefficients and 119896V is thepermeability coefficient in the consolidation process 119890

0is the

initial void ratio and 119890 is the void ratio in the consolidationprocess 119862

119896is the permeability index According to the

above work themodel parameters are summarized in Table 8approximately

52 Consolidation Property of Soft Clay From Figure 12 wecan see the differences of the consolidation process betweenfinite strain and small strain Figure 12(a) shows that for anylocation the excess pore pressure of finite strain is less thanthat of small strain at the same time and the difference isincreasing with the increase of time in beginning period

Mathematical Problems in Engineering 11

Table 8 Material parameters of computed model

1198640(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots) 120574sat (kNsdotm

minus3) 1198961199070(msdotsminus1) 119875

0(kPa) 119872 120582 120581 119890

0120583 119862

119896

25 115 150 24000 177 45119864 minus 9 50 0898 0102 0015 108 035 05

100000225

00200

00175

00150

00125

00100

Vert

ical

def

orm

atio

n (m

)

Time (s)10000 100000 1000000 1E7

(a)

40

50

60

70

80

90

100

110

120

1000Time (s)

10000 100000 1000000 1E7

Pcs

(kPa

)(b)

05

00

10

15

20

25

30

35

Viscoelastic strain

Visc

ous s

trai

n (

)

Viscoplastic strain

1000

Time (s)

10000 100000 1000000 1E71E7

(c)

Viscoelastic strain rateViscoplastic strain rate

0

2

4

6

8

10

12

14

16

1000Time (s)

10000 100000 1000000 1E71E7

Visc

ous s

trai

n ra

te (s

minus1)

(d)

Figure 9 Triaxial compression property of soil column

Figure 12(b) shows that the displacement of finite strain isless than that of small strain and the difference increasedsignificantly if the load is increasing For example when 119902 =200 kPa the differencemay take about 20while it is nearly 0when 119902= 50 kPa So it is essential to consider the error causedby small strain if the load is large and the compressibility ofthe soil is large

From Figure 13 we can see the influence of rheologyon the consolidation process Figure 13(a) shows that thedissipation of excess pore pressure is significantly slower ifthe rheological effect is considered for example when 119905 =1000 d the excess pore pressure is nearly 0 if the rheologicaleffect is not considered but it is still larger than 20 kPa ifthe rheological effect is considered Figure 13(a) shows that

the displacement is influenced by rheological effect too forexample in the early stage (119905 le 100 d) the displacement isnearly the same for two cases but the final displacementmay have a difference of more than 35 which indicatesthat the displacement in the early stage is mainly causedby consolidation while the displacement in the late stage ismainly caused by rheology

6 Conclusions

(1) An elastic-viscoplastic model developed to describethe time-dependent behavior of normally consoli-dated soft clay was presented in the paper on the

12 Mathematical Problems in Engineering

00

05

10

15

20

25

30Vi

scoe

lasti

c str

ain

()

100010 100Time (s)

10000 100000 1000000 1E7

1205781 = 10GPamiddots1205781 = 100GPamiddots1205781 = 1000GPamiddots

(a)

00

02

04

06

08

10

12

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

1205782 = 10GPamiddots1205782 = 100GPamiddots1205782 = 1000GPamiddots

(b)

00

04

08

12

16

20

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

P0 = 30kPaP0 = 40kPa

P0 = 50kPaP0 = 60kPa

(c)

Figure 10 Sensitivity of model parameters

O

Permeable

B = 5 mx

z

H = 10m

q = 100 kPa

Figure 11 Calculation model and mesh generation

framework of modified cam-clay elastic plastic modeland Perzynarsquos overstress viscoplastic theory

(2) A series of laboratory triaxial rheological tests ofNingbo soft clay with different confining pressure anddeviatoric stress are performed the results show thatthe presented constitutive model was suitable for therheological characteristic of Ningbo soft clay and it isachievable to determine the parameters of presentedmodel by curve fitting

(3) The analysis on the model parameters shows that thevalue of all parameters is related to confining pressureand deviatoric stress It may need further study on therelationship between parameters and the stress levelso as to use the model conveniently

Mathematical Problems in Engineering 13

0 20 40 60 80 100u (kPa)

0

2

4

6

8

10H

(m)

FS t = 100dSS t = 100dFS t = 500d

SS t = 500dFS t = 1000 dSS t = 1000 d

(a)

12

10

08

06

04

02

00

Def

orm

atio

n (m

)

Time (d)1000101 100 10000 100000

SS q= 100 kPaFS q= 50 kPaSS q= 50 kPa

SS q= 200 kPaFS q= 200 kPa

FS q= 100 kPa

(b)

Figure 12 Results comparison of large strain and small strain (SS small train FS finite strain)

00

2

4

6

8

10

H(m

)

20 40 60 80 100u (kPa)

t = 100d NRt = 100d Rt = 500d NR

t = 500d Rt = 1000 d NRt = 1000 d R

(a)

06

05

04

03

02

01

00

Def

orm

atio

n (m

)

Time (d)

NRR

1000101 100 10000 100000

(b)

Figure 13 Effects of rheology on calculated results (NR rheological effect not considered R rheological effect considered)

(4) In the proposed model the viscoelastic viscositycoefficient 120578

1 the viscoplastic viscosity coefficient 120578

2

and the initial yield surface 1199010are the most important

parameters 1205781and 120578

2have important influence on

the time of viscous strain developed and 1199010have

important influence on both the viscoplastic strainrate and the final viscoplastic strain

(5) The existence of rheology will significantly slow downthe dissipation of the excess pore pressure Consid-ering the rheological effect the total displacement

is larger and the displacement in the early stage ismainly caused by consolidation while in the late stageit is mainly caused by rheology

(6) For consolidation problems of high compressibilitysoft clay foundation great error will be caused bysmall strain assumption so finite strain effect shouldbe considered When finite strain effect is consideredthe consolidation process is developed more rapidlyand the computed consolidation displacement will beless than that of small strain

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Table 2 Loading scheme of triaxial rheological tests

Specimen number Confining pressure (kPa) Loading scheme (kPa)1 100 30-60-90-12-150-180-210-2402 200 40-80-120-160-200-240-280-320

20

15

10

5

00

60

120

180

240

300

120576(

)q

(kPa

)

200 400 600 800 1000 1200t (h)

(a) Specimen 1

20

25

15

10

50

80

160

240

320

120576(

)q

(kPa

)

0 200 400 600 800 1000 1200

t (h)

(b) Specimen 2

Figure 3 Full-process press-strain-time curve

0

5

10

15

20

30 kPa60 kPa90 kPa120 kPa

150 kPa180 kPa210 kPa240 kPa

120576(

)

10 1001t (h)

(a) Specimen 1

40 kPa80 kPa120 kPa160 kPa 320 kPa

240 kPa200 kPa

280 kPa

0

5

10

10 100

15

20

25

120576(

)

1t (h)

(b) Specimen 2

Figure 4 Mutistage strain-time curves

33 Computation and Curve Fitting For triaxial rheologicaltests 119868

1= 1205901+ 21205903= 31199011015840 119904101584011= 2(120590

1minus 1205903)3 = 21199023 so (13)

can be written as follows

12057611= 1198751+ 1198752(1 minus 119890

minus1198753119905

) + 1198754119905 (14)

where 1198751= (1199013119870) + (1199023119866

0) 1198752= (1199023119866

1) 1198753= (11986611198671)

1198754= (1119867

2)⟨120601(119891)⟩((2119902119872

2

) + (13)(21199011015840

minus 119901119889

119888))

Use (14) to fit the experimental data the results wereshowed in Figures 5 and 6 in which the parameters SSEand 1198772 denote the residual sum of squares and squares ofcorrelation coefficient respectively From Figures 5 and 6 wecan see that the model fits testing data of primary creep stageand steady creep stage very well while for the acceleratedcreep stage it needs further study According to the value of

Mathematical Problems in Engineering 5

0023

0024

0025

0026

0027

0 20 40 60 80 100 120 140 160 180

120576 11

SSE = 57349E minus 9R2 = 099069

t (h)

24105E minus 2P114547E minus 3P213321E minus 1P337098E minus 6P4

Test dataFitting curve

(a) 119902 = 30 kPa

0 20 40 60 80 100 120 140 160 180t (h)

0034

0036

0038

0040

0042

0044

120576 11

SSE = 58195E minus 8R2 = 098856

35852E minus 2P150523E minus 3P220259E minus 1P367168E minus 6P4

Test dataFitting curve

(b) 119902 = 60 kPa

SSE = 10771E minus 7R2 = 098854

0 20 40 60 80 100 120 140 160 180

t (h)

0046

0048

0050

0052

0054

0056

0058

120576 11

69284E minus 3P223128E minus 1P310127E minus 5P4

48144E minus 2P1

Test dataFitting curve

(c) 119902 = 90 kPa

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 1712E minus 7R2 = 099035

60822E minus 293268E minus 328662E minus 118749E minus 5

0055

0060

0070

0065

0075120576 1

1

P2P3P4

P1

Test dataFitting curve

(d) 119902 = 120 kPa

SSE = 61207E minus 7R2 = 098043

P1 76261E minus 2P2 12875E minus 3P3 35961E minus 1P4 26325E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0070

0080

0090

0075

0085

0095

120576 11

Test dataFitting curve

(e) 119902 = 150 kPa

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 872E minus 7R2 = 098046

89059E minus 215909E minus 237062E minus 128956E minus 5

0080

0090

0100

0110

0115

0085

0095

0105

120576 11

P2P3P4

P1

Test dataFitting curve

(f) 119902 = 180 kPa

Figure 5 Continued

6 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

0090

0100

0110

0120

0130

0125

0115

0095

0105

120576 11

SSE = 99965E minus 7R2 = 098620

10217E minus 119991E minus 228288E minus 129143E minus 5

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 210 kPa

Figure 5 Fitting curves of specimen 1 (1205903= 100 kPa)

Table 3 Model parameters of specimen 1 (1205903= 100 kPa)

1205901minus 1205903(kPa) 119864

0(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots)

30 249 1856 50160 1607360 251 1069 18993 1795090 249 1169 18198 21297120 247 1158 14544 24253150 236 1049 10497 25049180 236 1018 9891 22046210 235 945 12032 26240

Table 4 Model parameters of specimen 2 (1205903= 200 kPa)

1205901minus 1205903(kPa) 119864

0(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots)

40 1075 1804 13881 1222380 903 1870 25862 11747120 834 1694 29947 12591160 763 1692 30057 12704200 660 1729 26118 11732240 604 1831 27481 12823280 508 1602 28165 11589

1198751 1198752 1198753 and 119875

4 the value of the model parameters 119864

0 1198641

1205781 and 120578

2can be determined easily such as in Tables 3 and 4

34 Discussion Table 3 shows that the model parameters ofspecimen 1 (120590

3= 100 kPa) change slightly with the change

of deviatoric stress except the first stage Table 4 showsthat the model parameters of specimen 2 (120590

3= 200 kPa)

change slightly with the change of deviatoric stress except1198640 Compared with the model parameters of specimen 1 and

specimen 2 it can be found that both 1198640and 119864

1increase

notably with the increase of 1205903 which means that the elastic

strain reduces when 1205903increases and 120578

1increases notably

Table 5 Percentage of strain components of specimen 1 (1205903=

100 kPa 119905 = 7 d)

1205901minus 1205903(kPa) 120576

119890

11() 120576

119907119890

11() 120576

119907119901

11()

30 9206 556 23860 8530 1202 26890 8483 1221 296120 8298 1272 430150 8151 1376 473180 8109 1448 443210 8041 1573 385

Table 6 Percentage of strain components of specimen 2 (1205903=

200 kPa 119905 = 7 d)

1205901minus 1205903(kPa) 120576

119890

11() 120576

119907119890

11() 120576

119907119901

11()

40 7703 1653 64580 6774 1683 1543120 6677 1971 1353160 6713 1981 1306200 6951 1837 1212240 7096 1686 1217280 7397 1740 863

while 1205782is reduced notably with the increase of 120590

3 which

means that the viscoelastic strain rate and viscoplastic strainrate change significantly when 120590

3increases For the lack

of more experiment data the further variation of modelparameters was unable to find out which needed furtherstudy

Based on the parameters in Tables 3 and 4 the com-ponents of elastic strain viscoelastic strain and viscoplasticstrain at any time can be calculated with (13) Tables 5 and 6show 120576

119890

11 120576V11989011 and 120576V119901

11of all loading levels when 119905 = 7 d it can

Mathematical Problems in Engineering 7

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 71424E minus 9R2 = 098984

93026E minus 319959E minus 346778E minus 146368E minus 6

0008

0009

0010

0011

0012

0013120576 1

1

P2P3P4

P1

Test dataFitting curve

(a) 119902 = 40 kPa

0 20 40 60 80 100 120 140 160 180t (h)

SSE = 27165E minus 8R2 = 099567

15502E minus 238505E minus 326029E minus 121015E minus 5

0010

0012

0014

0016

0018

0020

0022

0024

120576 11

P2P3P4

P1

Test dataFitting curve

(b) 119902 = 80 kPa

SSE = 85674E minus 8R2 = 0994

21595E minus 263737E minus 320369E minus 126044E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0016

0018

0020

0022

0024

0026

0028

0030

0032

0034

120576 11

P2P3P4

P1

Test dataFitting curve

(c) 119902 = 120 kPa

SSE = 15199E minus 7R2 = 099385

28845E minus 285110E minus 320264E minus 133395E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0020

0025

0030

0035

0040

0045

120576 11

P2P3P4

P1

Test dataFitting curve

(d) 119902 = 160 kPa

SSE = 26486E minus 7R2 = 099252

39406E minus 210413E minus 223826E minus 140897E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0030

0035

0040

0045

0050

0060

0055

120576 11

P2P3P4

P1

Test dataFitting curve

(e) 119902 = 200 kPa

0040

0045

0050

0060

0065

0075

0070

0055

120576 11

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 20786E minus 7R2 = 099572

49663E minus 211800E minus 223980E minus 150713E minus 5

P2P3P4

P1

Test dataFitting curve

(f) 119902 = 240 kPa

Figure 6 Continued

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 50043E minus 7R2 = 09928

66892E minus 215735E minus 220470E minus 146472E minus 5

00600060

00650065

00700070

00750075

00800080

00850085

00900090

00950095

120576 11

120576 11

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 280 kPa

Figure 6 Fitting curves of specimen 2 (1205903= 200 kPa)

001 10 100

02

04

06

08

10

12

14

16

Visc

ous s

trai

n (

)

Time (h)

VES q = 60kPaVPS q = 60kPaVES q = 120 kPa

VPS q = 120 kPaVES q = 180 kPaVPS q = 180 kPa

(a) Specimen 1

Visc

ous s

trai

n (

)

00

02

04

06

08

10

12

100Time (h)

VES q = 80kPaVPS q = 80kPaVES q = 160 kPa

VPS q = 160 kPaVES q = 240 kPaVPS q = 240 kPa

1 10

(b) Specimen 2

Figure 7 Development of computed viscous strain (VES viscoelastic strain VPS viscoplastic strain)

be found that the elastic strain accounted for the major partof the total strain but the viscoelastic strain and viscoplasticstrain increase with the increase of 120590

3 which takes about

15sim30 for each loading increment of the total and cannotbe neglected

Figure 7 shows several typical curves of the developmentof computed viscoelastic strain and viscoplastic strain ofspecimen 1 and specimen 2 It can be found that theviscoelastic strain increases quickly in the early stage andthen reaches steady in a short time (about 30 hours) whilethe viscoplastic strain increases the whole process whichindicates that the long-term deformation after consolidation

of soft clay foundation is mainly caused by the viscoplasticityrather than viscoelasticity

4 Derivation and Verification of UMAT

41 Derivation of UMAT ABAQUS is a FEM software whichhas powerful computing capabilities of strongly nonlinearproblems A user material subroutine (UMAT) is codedfor the proposed model on the platform of ABAQUS tostudy the elastic-viscoplastic consolidation properties of softclay According to ABAQUS the stiffness coefficient matrix

Mathematical Problems in Engineering 9

(Jacobian matrix) and the equation to update the stress andstrain should be defined in the UAMT The Jacobian matrixD is defined as follows

D =

120597Δ120590

120597Δ120576

(15)

where Δ120590 and Δ120576 denote the stress increment matrix andstrain increment matrix respectively

For the proposedmodel (4)sim(6) can be written as followsin 2D or 3D problems

120576119890

119899=

120583

1198640

A120590119899 (16a)

V119890119899=

1

1205781

A120590119899minus

1198661

1205781

120576V119890119899 (16b)

V119901119899=

1

1205782

⟨120601 (119865)⟩

120597119865

120597120590

(16c)

In (16a) (16b) and (16c) 120590119899 120576119899 120576V119890119899 and 120576V119901

119899denote the

Voigt matrix form of stress and strain component tensors Adenotes the flexibilitymatrix for plane strain problem and 3Dproblems and it can be written as follows

A =

[[[

[

1 minus120583 minus120583 0

minus120583 1 minus120583 0

minus120583 minus120583 1 0

0 0 0 2 (1 + 120583)

]]]

]

(17a)

A =

[[[[[[[

[

1 minus120583 minus120583 0 0 0

minus120583 1 minus120583 0 0 0

minus120583 minus120583 1 0 0 0

0 0 0 2 (1 + 120583) 0 0

0 0 0 0 2 (1 + 120583) 0

0 0 0 0 0 2 (1 + 120583)

]]]]]]]

]

(17b)

Suppose that the time step of the 119899th increment is Δ119905119899 so

the viscoelastic strain increment in the increment step can bewritten as follows

Δ120576V119890119899= Δ119905119899[(1 minus Θ)

V119890119899+ Θ

V119890119899+1] (18)

where Θ is differential coefficient 0 le Θ le 1 Suppose that119861119899= 1(1 + ΘΔ119905

11989911986411205781) C119899= 119861119899ΘΔ119905119899A1205781 (18) can be

written as follows

Δ120576V119890119899= 119861119899V119890119899Δ119905119899+ C119899Δ120590119899 (19)

The viscoplastic strain increment in the increment stepcan be written as follows

Δ120576V119901119899= Δ119905119899[(1 minus Θ)

V119901119899+ Θ

V119901119899+1] (20)

In order to calculate the viscoplastic strain rate at theend of the 119899th increment apply the Taylor series expansionand ignore the second order trace then we can obtain thefollowing equation

V119901119899+1

= V119901119899+

120597V119901119899

120597120590

120597120590

120597119905

Δ119905119899 (21)

Suppose thatH119899= 120597

V119901119899120597120590 (21) can be written as

V119901119899+1

= V119901119899+H119899Δ120590119899 (22)

Substituting (22) into (21) we can obtain the followingequation

Δ120576V119901119899= Δ119905119899V119901119899+ ΘΔ119905

119899H119899Δ120590119899 (23)

From (16c)H119899can be written as

H119899=

1

1205782

[⟨120601 (119865)⟩

120597a119879

120597120590

+

120597 ⟨120601 (119865)⟩

120597119865

aa119879] (24)

where a = 120597119865120597120590 According to (9) a can be written asfollows

a = 120597119865

120597120590

= 1198621a1+ 1198622a2 (25)

where 1198621= 2119902119872

2 1198622= 2119901 minus 119901

0 a1= 120597119902120597120590 a

2= 120597119901120597120590

From (25) the following equation can be derived

120597a119879

120597120590

=

1205971198621

120597119902

a1a1198791+ 1198621

120597a1198791

120597120590

+

1205971198622

120597119901

a2a1198792 (26)

where (120597a1198791120597120590) = (32119902)M

1minus(94119902

3

)M2 a1a1198791= (94119902

2

)M2

a2a1198792= (19)M

3 M1 M2 and M

3are matrices written as

follows

M1=

[[[[[[[

[

23 minus13 minus13 0 0 0

minus13 23 minus13 0 0 0

minus13 minus13 23 0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

]]]]]]]

]

M2=

[[[[[[

[

1198782

111198781111987822

1198781111987833

21198781111987812

21198781111987823

21198781111987813

1198781111987822

1198782

221198782211987833

21198782211987812

21198782211987823

21198782211987813

1198781111987833

1198782211987833

1198782

3321198783311987812

21198783311987823

21198783311987813

21198781111987812

21198782211987812

21198783311987812

41198782

1241198781211987823

41198781211987813

21198781111987823

21198782211987823

21198783311987823

41198781211987823

41198782

2341198782311987813

21198781111987813

21198782211987813

21198783311987813

41198781211987813

41198782311987813

41198782

13

]]]]]]

]

M3=

[[[[[[[

[

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]]]]]]]

]

(27)

Combining the above equations the value of H119899can

be derived Substituting H119899into (22) the viscoplastic strain

increment Δ120576V119901119899

is obtainedFor current increment Δ120590

119899is calculated by (28) as

follows

Δ120590119899= D119890Δ120576119890

119899 (28)

Substituting (19) and (23) into (28) we can obtain thefollowing equation

Δ120590119899= D (Δ120576

119899minus 119861Δ119905

119899V119890119899minus Δ119905119899V119901119899) (29)

10 Mathematical Problems in Engineering

Table 7 Material parameters of verification model

1198640(MPa) 119864

1(MPa) 120578

1(GPasdots) 120578

2(GPasdots) 119875

0(kPa) 119890

0120583 119872 120582 120581

10 20 1000 1000 50 108 035 0898 0102 0015

12059021205902

1205903

1205903

1205901

1205901

Figure 8 Sketch of verification model

Equation (29) is the relationship between stress incre-ment and strain increment in which matrix D is defined asfollows

D = [(D119890)minus1 + C119899+ ΘΔ119905

119899H119899]

minus1

(30)

Based on the above equations a user material subroutinewas coded for the proposed constitutive model on theplatform of ABAQUS

42 Verification of UMAT In order to verify the validity ofthe UMAT we study a soil column of triaxial compressionsuch as Figure 8 the size of the column is 01m times 01m times

02m Supposing that 1205901= 100 kPa 120590

2= 1205903= 50 kPa

and the model parameters are given in Table 7 The columnwas cut into one element the element type is C3D8 and theconsolidation effect was not considered here Embedding theUMAT coded in this paper on the platform of ABAQUS wecalculated the triaxial compression property of the columnthe results were summarized in Figure 9simFigure 10

From Figure 9(a) we can see the deformation of thecolumn exhibits apparent rheology the rheological deforma-tion takes about 37 of the whole deformation Figure 9(b)shows that the static yield surface gradually expanded withthe time until it reached steady state when it coincided withthe dynamic yield surface From Figures 9(c) and 9(d) wecan see that the viscoelastic strain and the viscoplastic strainboth gradually increased with time until they reached theirsteady state accordingly the viscoelastic strain rate and theviscoplastic strain rate were gradually deduced to nearly zeroThe above results fit the property of Nishihara model andthe hardening law of modified Cambridge model very wellwhich proved the correctness of the UMAT

In the proposed model 1205781 1205782 and 119901

0are the most

important parameters Figure 10 showed the sensitivity ofthese parameters From Figures 10(a) and 10(b) we can seethe final viscous deformation is a constant regardless of thedifferent values of 120578

1 1205782 but the time when it reaches steady

varies greatly so the accuracy of 1205781and 1205782is directly related

to the accuracy of computed displacement in the rheologicalprocess Figure 10(c) shows that the value of 119901

0has a great

influence on the viscoplastic strain the smaller 1199010is the

more rapidly the viscoplastic strain develops and the largerthe final viscoplastic strain occurs and so the accuracy of1199010is not only directly related to the accuracy of computed

displacement in the rheological process but also affected thefinal displacement which needs to be paid more attention

5 Analysis on Rheological Consolidation ofSoft Clay

51 Computed Model Consolidation problems can be classi-fied as small strain theory and finite strain theory accordingto the difference of geometric assumption Small straintheory assumed that the strain is little but the consolida-tion deformation is usually very large for deep soft clayfoundation Take dredger fill for example the deformationmay exceed 80 for which the small strain theory is nolonger suitable [14 15] As the pioneer researchers studiedon finite strain consolidation theory Mikasa [16] and Gib-son et al [17] derived their one-dimensional finite strainconsolidation equations respectively in 1960s and manyscholars developed further research on the topic in thefollowing decades Because of strong nonlinearity most ofexisting researches on finite strain consolidation adoptedsome simplified assumptionswithout considering rheologicalcharacteristics By means of ABAQUS this paper studied therheological consolidation of soft clay considering finite straineffect

The computed model is plane strain problem as shown inFigure 11 The soil foundation is homogeneous anisotropicsaturated and normally consolidated soft clay with a thick-ness of 100m and only the top boundary is permeable Inaddition the semilogarithmic empirical formula proposed byTavenas et al [18] was adopted to consider the variation ofpermeability in the process of consolidation as follows

119896V = 119896V0 sdot 10(119890minus1198900)119862119896 (31)

where 119896V0 is the initial permeability coefficients and 119896V is thepermeability coefficient in the consolidation process 119890

0is the

initial void ratio and 119890 is the void ratio in the consolidationprocess 119862

119896is the permeability index According to the

above work themodel parameters are summarized in Table 8approximately

52 Consolidation Property of Soft Clay From Figure 12 wecan see the differences of the consolidation process betweenfinite strain and small strain Figure 12(a) shows that for anylocation the excess pore pressure of finite strain is less thanthat of small strain at the same time and the difference isincreasing with the increase of time in beginning period

Mathematical Problems in Engineering 11

Table 8 Material parameters of computed model

1198640(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots) 120574sat (kNsdotm

minus3) 1198961199070(msdotsminus1) 119875

0(kPa) 119872 120582 120581 119890

0120583 119862

119896

25 115 150 24000 177 45119864 minus 9 50 0898 0102 0015 108 035 05

100000225

00200

00175

00150

00125

00100

Vert

ical

def

orm

atio

n (m

)

Time (s)10000 100000 1000000 1E7

(a)

40

50

60

70

80

90

100

110

120

1000Time (s)

10000 100000 1000000 1E7

Pcs

(kPa

)(b)

05

00

10

15

20

25

30

35

Viscoelastic strain

Visc

ous s

trai

n (

)

Viscoplastic strain

1000

Time (s)

10000 100000 1000000 1E71E7

(c)

Viscoelastic strain rateViscoplastic strain rate

0

2

4

6

8

10

12

14

16

1000Time (s)

10000 100000 1000000 1E71E7

Visc

ous s

trai

n ra

te (s

minus1)

(d)

Figure 9 Triaxial compression property of soil column

Figure 12(b) shows that the displacement of finite strain isless than that of small strain and the difference increasedsignificantly if the load is increasing For example when 119902 =200 kPa the differencemay take about 20while it is nearly 0when 119902= 50 kPa So it is essential to consider the error causedby small strain if the load is large and the compressibility ofthe soil is large

From Figure 13 we can see the influence of rheologyon the consolidation process Figure 13(a) shows that thedissipation of excess pore pressure is significantly slower ifthe rheological effect is considered for example when 119905 =1000 d the excess pore pressure is nearly 0 if the rheologicaleffect is not considered but it is still larger than 20 kPa ifthe rheological effect is considered Figure 13(a) shows that

the displacement is influenced by rheological effect too forexample in the early stage (119905 le 100 d) the displacement isnearly the same for two cases but the final displacementmay have a difference of more than 35 which indicatesthat the displacement in the early stage is mainly causedby consolidation while the displacement in the late stage ismainly caused by rheology

6 Conclusions

(1) An elastic-viscoplastic model developed to describethe time-dependent behavior of normally consoli-dated soft clay was presented in the paper on the

12 Mathematical Problems in Engineering

00

05

10

15

20

25

30Vi

scoe

lasti

c str

ain

()

100010 100Time (s)

10000 100000 1000000 1E7

1205781 = 10GPamiddots1205781 = 100GPamiddots1205781 = 1000GPamiddots

(a)

00

02

04

06

08

10

12

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

1205782 = 10GPamiddots1205782 = 100GPamiddots1205782 = 1000GPamiddots

(b)

00

04

08

12

16

20

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

P0 = 30kPaP0 = 40kPa

P0 = 50kPaP0 = 60kPa

(c)

Figure 10 Sensitivity of model parameters

O

Permeable

B = 5 mx

z

H = 10m

q = 100 kPa

Figure 11 Calculation model and mesh generation

framework of modified cam-clay elastic plastic modeland Perzynarsquos overstress viscoplastic theory

(2) A series of laboratory triaxial rheological tests ofNingbo soft clay with different confining pressure anddeviatoric stress are performed the results show thatthe presented constitutive model was suitable for therheological characteristic of Ningbo soft clay and it isachievable to determine the parameters of presentedmodel by curve fitting

(3) The analysis on the model parameters shows that thevalue of all parameters is related to confining pressureand deviatoric stress It may need further study on therelationship between parameters and the stress levelso as to use the model conveniently

Mathematical Problems in Engineering 13

0 20 40 60 80 100u (kPa)

0

2

4

6

8

10H

(m)

FS t = 100dSS t = 100dFS t = 500d

SS t = 500dFS t = 1000 dSS t = 1000 d

(a)

12

10

08

06

04

02

00

Def

orm

atio

n (m

)

Time (d)1000101 100 10000 100000

SS q= 100 kPaFS q= 50 kPaSS q= 50 kPa

SS q= 200 kPaFS q= 200 kPa

FS q= 100 kPa

(b)

Figure 12 Results comparison of large strain and small strain (SS small train FS finite strain)

00

2

4

6

8

10

H(m

)

20 40 60 80 100u (kPa)

t = 100d NRt = 100d Rt = 500d NR

t = 500d Rt = 1000 d NRt = 1000 d R

(a)

06

05

04

03

02

01

00

Def

orm

atio

n (m

)

Time (d)

NRR

1000101 100 10000 100000

(b)

Figure 13 Effects of rheology on calculated results (NR rheological effect not considered R rheological effect considered)

(4) In the proposed model the viscoelastic viscositycoefficient 120578

1 the viscoplastic viscosity coefficient 120578

2

and the initial yield surface 1199010are the most important

parameters 1205781and 120578

2have important influence on

the time of viscous strain developed and 1199010have

important influence on both the viscoplastic strainrate and the final viscoplastic strain

(5) The existence of rheology will significantly slow downthe dissipation of the excess pore pressure Consid-ering the rheological effect the total displacement

is larger and the displacement in the early stage ismainly caused by consolidation while in the late stageit is mainly caused by rheology

(6) For consolidation problems of high compressibilitysoft clay foundation great error will be caused bysmall strain assumption so finite strain effect shouldbe considered When finite strain effect is consideredthe consolidation process is developed more rapidlyand the computed consolidation displacement will beless than that of small strain

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0023

0024

0025

0026

0027

0 20 40 60 80 100 120 140 160 180

120576 11

SSE = 57349E minus 9R2 = 099069

t (h)

24105E minus 2P114547E minus 3P213321E minus 1P337098E minus 6P4

Test dataFitting curve

(a) 119902 = 30 kPa

0 20 40 60 80 100 120 140 160 180t (h)

0034

0036

0038

0040

0042

0044

120576 11

SSE = 58195E minus 8R2 = 098856

35852E minus 2P150523E minus 3P220259E minus 1P367168E minus 6P4

Test dataFitting curve

(b) 119902 = 60 kPa

SSE = 10771E minus 7R2 = 098854

0 20 40 60 80 100 120 140 160 180

t (h)

0046

0048

0050

0052

0054

0056

0058

120576 11

69284E minus 3P223128E minus 1P310127E minus 5P4

48144E minus 2P1

Test dataFitting curve

(c) 119902 = 90 kPa

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 1712E minus 7R2 = 099035

60822E minus 293268E minus 328662E minus 118749E minus 5

0055

0060

0070

0065

0075120576 1

1

P2P3P4

P1

Test dataFitting curve

(d) 119902 = 120 kPa

SSE = 61207E minus 7R2 = 098043

P1 76261E minus 2P2 12875E minus 3P3 35961E minus 1P4 26325E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0070

0080

0090

0075

0085

0095

120576 11

Test dataFitting curve

(e) 119902 = 150 kPa

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 872E minus 7R2 = 098046

89059E minus 215909E minus 237062E minus 128956E minus 5

0080

0090

0100

0110

0115

0085

0095

0105

120576 11

P2P3P4

P1

Test dataFitting curve

(f) 119902 = 180 kPa

Figure 5 Continued

6 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

0090

0100

0110

0120

0130

0125

0115

0095

0105

120576 11

SSE = 99965E minus 7R2 = 098620

10217E minus 119991E minus 228288E minus 129143E minus 5

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 210 kPa

Figure 5 Fitting curves of specimen 1 (1205903= 100 kPa)

Table 3 Model parameters of specimen 1 (1205903= 100 kPa)

1205901minus 1205903(kPa) 119864

0(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots)

30 249 1856 50160 1607360 251 1069 18993 1795090 249 1169 18198 21297120 247 1158 14544 24253150 236 1049 10497 25049180 236 1018 9891 22046210 235 945 12032 26240

Table 4 Model parameters of specimen 2 (1205903= 200 kPa)

1205901minus 1205903(kPa) 119864

0(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots)

40 1075 1804 13881 1222380 903 1870 25862 11747120 834 1694 29947 12591160 763 1692 30057 12704200 660 1729 26118 11732240 604 1831 27481 12823280 508 1602 28165 11589

1198751 1198752 1198753 and 119875

4 the value of the model parameters 119864

0 1198641

1205781 and 120578

2can be determined easily such as in Tables 3 and 4

34 Discussion Table 3 shows that the model parameters ofspecimen 1 (120590

3= 100 kPa) change slightly with the change

of deviatoric stress except the first stage Table 4 showsthat the model parameters of specimen 2 (120590

3= 200 kPa)

change slightly with the change of deviatoric stress except1198640 Compared with the model parameters of specimen 1 and

specimen 2 it can be found that both 1198640and 119864

1increase

notably with the increase of 1205903 which means that the elastic

strain reduces when 1205903increases and 120578

1increases notably

Table 5 Percentage of strain components of specimen 1 (1205903=

100 kPa 119905 = 7 d)

1205901minus 1205903(kPa) 120576

119890

11() 120576

119907119890

11() 120576

119907119901

11()

30 9206 556 23860 8530 1202 26890 8483 1221 296120 8298 1272 430150 8151 1376 473180 8109 1448 443210 8041 1573 385

Table 6 Percentage of strain components of specimen 2 (1205903=

200 kPa 119905 = 7 d)

1205901minus 1205903(kPa) 120576

119890

11() 120576

119907119890

11() 120576

119907119901

11()

40 7703 1653 64580 6774 1683 1543120 6677 1971 1353160 6713 1981 1306200 6951 1837 1212240 7096 1686 1217280 7397 1740 863

while 1205782is reduced notably with the increase of 120590

3 which

means that the viscoelastic strain rate and viscoplastic strainrate change significantly when 120590

3increases For the lack

of more experiment data the further variation of modelparameters was unable to find out which needed furtherstudy

Based on the parameters in Tables 3 and 4 the com-ponents of elastic strain viscoelastic strain and viscoplasticstrain at any time can be calculated with (13) Tables 5 and 6show 120576

119890

11 120576V11989011 and 120576V119901

11of all loading levels when 119905 = 7 d it can

Mathematical Problems in Engineering 7

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 71424E minus 9R2 = 098984

93026E minus 319959E minus 346778E minus 146368E minus 6

0008

0009

0010

0011

0012

0013120576 1

1

P2P3P4

P1

Test dataFitting curve

(a) 119902 = 40 kPa

0 20 40 60 80 100 120 140 160 180t (h)

SSE = 27165E minus 8R2 = 099567

15502E minus 238505E minus 326029E minus 121015E minus 5

0010

0012

0014

0016

0018

0020

0022

0024

120576 11

P2P3P4

P1

Test dataFitting curve

(b) 119902 = 80 kPa

SSE = 85674E minus 8R2 = 0994

21595E minus 263737E minus 320369E minus 126044E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0016

0018

0020

0022

0024

0026

0028

0030

0032

0034

120576 11

P2P3P4

P1

Test dataFitting curve

(c) 119902 = 120 kPa

SSE = 15199E minus 7R2 = 099385

28845E minus 285110E minus 320264E minus 133395E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0020

0025

0030

0035

0040

0045

120576 11

P2P3P4

P1

Test dataFitting curve

(d) 119902 = 160 kPa

SSE = 26486E minus 7R2 = 099252

39406E minus 210413E minus 223826E minus 140897E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0030

0035

0040

0045

0050

0060

0055

120576 11

P2P3P4

P1

Test dataFitting curve

(e) 119902 = 200 kPa

0040

0045

0050

0060

0065

0075

0070

0055

120576 11

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 20786E minus 7R2 = 099572

49663E minus 211800E minus 223980E minus 150713E minus 5

P2P3P4

P1

Test dataFitting curve

(f) 119902 = 240 kPa

Figure 6 Continued

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 50043E minus 7R2 = 09928

66892E minus 215735E minus 220470E minus 146472E minus 5

00600060

00650065

00700070

00750075

00800080

00850085

00900090

00950095

120576 11

120576 11

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 280 kPa

Figure 6 Fitting curves of specimen 2 (1205903= 200 kPa)

001 10 100

02

04

06

08

10

12

14

16

Visc

ous s

trai

n (

)

Time (h)

VES q = 60kPaVPS q = 60kPaVES q = 120 kPa

VPS q = 120 kPaVES q = 180 kPaVPS q = 180 kPa

(a) Specimen 1

Visc

ous s

trai

n (

)

00

02

04

06

08

10

12

100Time (h)

VES q = 80kPaVPS q = 80kPaVES q = 160 kPa

VPS q = 160 kPaVES q = 240 kPaVPS q = 240 kPa

1 10

(b) Specimen 2

Figure 7 Development of computed viscous strain (VES viscoelastic strain VPS viscoplastic strain)

be found that the elastic strain accounted for the major partof the total strain but the viscoelastic strain and viscoplasticstrain increase with the increase of 120590

3 which takes about

15sim30 for each loading increment of the total and cannotbe neglected

Figure 7 shows several typical curves of the developmentof computed viscoelastic strain and viscoplastic strain ofspecimen 1 and specimen 2 It can be found that theviscoelastic strain increases quickly in the early stage andthen reaches steady in a short time (about 30 hours) whilethe viscoplastic strain increases the whole process whichindicates that the long-term deformation after consolidation

of soft clay foundation is mainly caused by the viscoplasticityrather than viscoelasticity

4 Derivation and Verification of UMAT

41 Derivation of UMAT ABAQUS is a FEM software whichhas powerful computing capabilities of strongly nonlinearproblems A user material subroutine (UMAT) is codedfor the proposed model on the platform of ABAQUS tostudy the elastic-viscoplastic consolidation properties of softclay According to ABAQUS the stiffness coefficient matrix

Mathematical Problems in Engineering 9

(Jacobian matrix) and the equation to update the stress andstrain should be defined in the UAMT The Jacobian matrixD is defined as follows

D =

120597Δ120590

120597Δ120576

(15)

where Δ120590 and Δ120576 denote the stress increment matrix andstrain increment matrix respectively

For the proposedmodel (4)sim(6) can be written as followsin 2D or 3D problems

120576119890

119899=

120583

1198640

A120590119899 (16a)

V119890119899=

1

1205781

A120590119899minus

1198661

1205781

120576V119890119899 (16b)

V119901119899=

1

1205782

⟨120601 (119865)⟩

120597119865

120597120590

(16c)

In (16a) (16b) and (16c) 120590119899 120576119899 120576V119890119899 and 120576V119901

119899denote the

Voigt matrix form of stress and strain component tensors Adenotes the flexibilitymatrix for plane strain problem and 3Dproblems and it can be written as follows

A =

[[[

[

1 minus120583 minus120583 0

minus120583 1 minus120583 0

minus120583 minus120583 1 0

0 0 0 2 (1 + 120583)

]]]

]

(17a)

A =

[[[[[[[

[

1 minus120583 minus120583 0 0 0

minus120583 1 minus120583 0 0 0

minus120583 minus120583 1 0 0 0

0 0 0 2 (1 + 120583) 0 0

0 0 0 0 2 (1 + 120583) 0

0 0 0 0 0 2 (1 + 120583)

]]]]]]]

]

(17b)

Suppose that the time step of the 119899th increment is Δ119905119899 so

the viscoelastic strain increment in the increment step can bewritten as follows

Δ120576V119890119899= Δ119905119899[(1 minus Θ)

V119890119899+ Θ

V119890119899+1] (18)

where Θ is differential coefficient 0 le Θ le 1 Suppose that119861119899= 1(1 + ΘΔ119905

11989911986411205781) C119899= 119861119899ΘΔ119905119899A1205781 (18) can be

written as follows

Δ120576V119890119899= 119861119899V119890119899Δ119905119899+ C119899Δ120590119899 (19)

The viscoplastic strain increment in the increment stepcan be written as follows

Δ120576V119901119899= Δ119905119899[(1 minus Θ)

V119901119899+ Θ

V119901119899+1] (20)

In order to calculate the viscoplastic strain rate at theend of the 119899th increment apply the Taylor series expansionand ignore the second order trace then we can obtain thefollowing equation

V119901119899+1

= V119901119899+

120597V119901119899

120597120590

120597120590

120597119905

Δ119905119899 (21)

Suppose thatH119899= 120597

V119901119899120597120590 (21) can be written as

V119901119899+1

= V119901119899+H119899Δ120590119899 (22)

Substituting (22) into (21) we can obtain the followingequation

Δ120576V119901119899= Δ119905119899V119901119899+ ΘΔ119905

119899H119899Δ120590119899 (23)

From (16c)H119899can be written as

H119899=

1

1205782

[⟨120601 (119865)⟩

120597a119879

120597120590

+

120597 ⟨120601 (119865)⟩

120597119865

aa119879] (24)

where a = 120597119865120597120590 According to (9) a can be written asfollows

a = 120597119865

120597120590

= 1198621a1+ 1198622a2 (25)

where 1198621= 2119902119872

2 1198622= 2119901 minus 119901

0 a1= 120597119902120597120590 a

2= 120597119901120597120590

From (25) the following equation can be derived

120597a119879

120597120590

=

1205971198621

120597119902

a1a1198791+ 1198621

120597a1198791

120597120590

+

1205971198622

120597119901

a2a1198792 (26)

where (120597a1198791120597120590) = (32119902)M

1minus(94119902

3

)M2 a1a1198791= (94119902

2

)M2

a2a1198792= (19)M

3 M1 M2 and M

3are matrices written as

follows

M1=

[[[[[[[

[

23 minus13 minus13 0 0 0

minus13 23 minus13 0 0 0

minus13 minus13 23 0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

]]]]]]]

]

M2=

[[[[[[

[

1198782

111198781111987822

1198781111987833

21198781111987812

21198781111987823

21198781111987813

1198781111987822

1198782

221198782211987833

21198782211987812

21198782211987823

21198782211987813

1198781111987833

1198782211987833

1198782

3321198783311987812

21198783311987823

21198783311987813

21198781111987812

21198782211987812

21198783311987812

41198782

1241198781211987823

41198781211987813

21198781111987823

21198782211987823

21198783311987823

41198781211987823

41198782

2341198782311987813

21198781111987813

21198782211987813

21198783311987813

41198781211987813

41198782311987813

41198782

13

]]]]]]

]

M3=

[[[[[[[

[

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]]]]]]]

]

(27)

Combining the above equations the value of H119899can

be derived Substituting H119899into (22) the viscoplastic strain

increment Δ120576V119901119899

is obtainedFor current increment Δ120590

119899is calculated by (28) as

follows

Δ120590119899= D119890Δ120576119890

119899 (28)

Substituting (19) and (23) into (28) we can obtain thefollowing equation

Δ120590119899= D (Δ120576

119899minus 119861Δ119905

119899V119890119899minus Δ119905119899V119901119899) (29)

10 Mathematical Problems in Engineering

Table 7 Material parameters of verification model

1198640(MPa) 119864

1(MPa) 120578

1(GPasdots) 120578

2(GPasdots) 119875

0(kPa) 119890

0120583 119872 120582 120581

10 20 1000 1000 50 108 035 0898 0102 0015

12059021205902

1205903

1205903

1205901

1205901

Figure 8 Sketch of verification model

Equation (29) is the relationship between stress incre-ment and strain increment in which matrix D is defined asfollows

D = [(D119890)minus1 + C119899+ ΘΔ119905

119899H119899]

minus1

(30)

Based on the above equations a user material subroutinewas coded for the proposed constitutive model on theplatform of ABAQUS

42 Verification of UMAT In order to verify the validity ofthe UMAT we study a soil column of triaxial compressionsuch as Figure 8 the size of the column is 01m times 01m times

02m Supposing that 1205901= 100 kPa 120590

2= 1205903= 50 kPa

and the model parameters are given in Table 7 The columnwas cut into one element the element type is C3D8 and theconsolidation effect was not considered here Embedding theUMAT coded in this paper on the platform of ABAQUS wecalculated the triaxial compression property of the columnthe results were summarized in Figure 9simFigure 10

From Figure 9(a) we can see the deformation of thecolumn exhibits apparent rheology the rheological deforma-tion takes about 37 of the whole deformation Figure 9(b)shows that the static yield surface gradually expanded withthe time until it reached steady state when it coincided withthe dynamic yield surface From Figures 9(c) and 9(d) wecan see that the viscoelastic strain and the viscoplastic strainboth gradually increased with time until they reached theirsteady state accordingly the viscoelastic strain rate and theviscoplastic strain rate were gradually deduced to nearly zeroThe above results fit the property of Nishihara model andthe hardening law of modified Cambridge model very wellwhich proved the correctness of the UMAT

In the proposed model 1205781 1205782 and 119901

0are the most

important parameters Figure 10 showed the sensitivity ofthese parameters From Figures 10(a) and 10(b) we can seethe final viscous deformation is a constant regardless of thedifferent values of 120578

1 1205782 but the time when it reaches steady

varies greatly so the accuracy of 1205781and 1205782is directly related

to the accuracy of computed displacement in the rheologicalprocess Figure 10(c) shows that the value of 119901

0has a great

influence on the viscoplastic strain the smaller 1199010is the

more rapidly the viscoplastic strain develops and the largerthe final viscoplastic strain occurs and so the accuracy of1199010is not only directly related to the accuracy of computed

displacement in the rheological process but also affected thefinal displacement which needs to be paid more attention

5 Analysis on Rheological Consolidation ofSoft Clay

51 Computed Model Consolidation problems can be classi-fied as small strain theory and finite strain theory accordingto the difference of geometric assumption Small straintheory assumed that the strain is little but the consolida-tion deformation is usually very large for deep soft clayfoundation Take dredger fill for example the deformationmay exceed 80 for which the small strain theory is nolonger suitable [14 15] As the pioneer researchers studiedon finite strain consolidation theory Mikasa [16] and Gib-son et al [17] derived their one-dimensional finite strainconsolidation equations respectively in 1960s and manyscholars developed further research on the topic in thefollowing decades Because of strong nonlinearity most ofexisting researches on finite strain consolidation adoptedsome simplified assumptionswithout considering rheologicalcharacteristics By means of ABAQUS this paper studied therheological consolidation of soft clay considering finite straineffect

The computed model is plane strain problem as shown inFigure 11 The soil foundation is homogeneous anisotropicsaturated and normally consolidated soft clay with a thick-ness of 100m and only the top boundary is permeable Inaddition the semilogarithmic empirical formula proposed byTavenas et al [18] was adopted to consider the variation ofpermeability in the process of consolidation as follows

119896V = 119896V0 sdot 10(119890minus1198900)119862119896 (31)

where 119896V0 is the initial permeability coefficients and 119896V is thepermeability coefficient in the consolidation process 119890

0is the

initial void ratio and 119890 is the void ratio in the consolidationprocess 119862

119896is the permeability index According to the

above work themodel parameters are summarized in Table 8approximately

52 Consolidation Property of Soft Clay From Figure 12 wecan see the differences of the consolidation process betweenfinite strain and small strain Figure 12(a) shows that for anylocation the excess pore pressure of finite strain is less thanthat of small strain at the same time and the difference isincreasing with the increase of time in beginning period

Mathematical Problems in Engineering 11

Table 8 Material parameters of computed model

1198640(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots) 120574sat (kNsdotm

minus3) 1198961199070(msdotsminus1) 119875

0(kPa) 119872 120582 120581 119890

0120583 119862

119896

25 115 150 24000 177 45119864 minus 9 50 0898 0102 0015 108 035 05

100000225

00200

00175

00150

00125

00100

Vert

ical

def

orm

atio

n (m

)

Time (s)10000 100000 1000000 1E7

(a)

40

50

60

70

80

90

100

110

120

1000Time (s)

10000 100000 1000000 1E7

Pcs

(kPa

)(b)

05

00

10

15

20

25

30

35

Viscoelastic strain

Visc

ous s

trai

n (

)

Viscoplastic strain

1000

Time (s)

10000 100000 1000000 1E71E7

(c)

Viscoelastic strain rateViscoplastic strain rate

0

2

4

6

8

10

12

14

16

1000Time (s)

10000 100000 1000000 1E71E7

Visc

ous s

trai

n ra

te (s

minus1)

(d)

Figure 9 Triaxial compression property of soil column

Figure 12(b) shows that the displacement of finite strain isless than that of small strain and the difference increasedsignificantly if the load is increasing For example when 119902 =200 kPa the differencemay take about 20while it is nearly 0when 119902= 50 kPa So it is essential to consider the error causedby small strain if the load is large and the compressibility ofthe soil is large

From Figure 13 we can see the influence of rheologyon the consolidation process Figure 13(a) shows that thedissipation of excess pore pressure is significantly slower ifthe rheological effect is considered for example when 119905 =1000 d the excess pore pressure is nearly 0 if the rheologicaleffect is not considered but it is still larger than 20 kPa ifthe rheological effect is considered Figure 13(a) shows that

the displacement is influenced by rheological effect too forexample in the early stage (119905 le 100 d) the displacement isnearly the same for two cases but the final displacementmay have a difference of more than 35 which indicatesthat the displacement in the early stage is mainly causedby consolidation while the displacement in the late stage ismainly caused by rheology

6 Conclusions

(1) An elastic-viscoplastic model developed to describethe time-dependent behavior of normally consoli-dated soft clay was presented in the paper on the

12 Mathematical Problems in Engineering

00

05

10

15

20

25

30Vi

scoe

lasti

c str

ain

()

100010 100Time (s)

10000 100000 1000000 1E7

1205781 = 10GPamiddots1205781 = 100GPamiddots1205781 = 1000GPamiddots

(a)

00

02

04

06

08

10

12

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

1205782 = 10GPamiddots1205782 = 100GPamiddots1205782 = 1000GPamiddots

(b)

00

04

08

12

16

20

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

P0 = 30kPaP0 = 40kPa

P0 = 50kPaP0 = 60kPa

(c)

Figure 10 Sensitivity of model parameters

O

Permeable

B = 5 mx

z

H = 10m

q = 100 kPa

Figure 11 Calculation model and mesh generation

framework of modified cam-clay elastic plastic modeland Perzynarsquos overstress viscoplastic theory

(2) A series of laboratory triaxial rheological tests ofNingbo soft clay with different confining pressure anddeviatoric stress are performed the results show thatthe presented constitutive model was suitable for therheological characteristic of Ningbo soft clay and it isachievable to determine the parameters of presentedmodel by curve fitting

(3) The analysis on the model parameters shows that thevalue of all parameters is related to confining pressureand deviatoric stress It may need further study on therelationship between parameters and the stress levelso as to use the model conveniently

Mathematical Problems in Engineering 13

0 20 40 60 80 100u (kPa)

0

2

4

6

8

10H

(m)

FS t = 100dSS t = 100dFS t = 500d

SS t = 500dFS t = 1000 dSS t = 1000 d

(a)

12

10

08

06

04

02

00

Def

orm

atio

n (m

)

Time (d)1000101 100 10000 100000

SS q= 100 kPaFS q= 50 kPaSS q= 50 kPa

SS q= 200 kPaFS q= 200 kPa

FS q= 100 kPa

(b)

Figure 12 Results comparison of large strain and small strain (SS small train FS finite strain)

00

2

4

6

8

10

H(m

)

20 40 60 80 100u (kPa)

t = 100d NRt = 100d Rt = 500d NR

t = 500d Rt = 1000 d NRt = 1000 d R

(a)

06

05

04

03

02

01

00

Def

orm

atio

n (m

)

Time (d)

NRR

1000101 100 10000 100000

(b)

Figure 13 Effects of rheology on calculated results (NR rheological effect not considered R rheological effect considered)

(4) In the proposed model the viscoelastic viscositycoefficient 120578

1 the viscoplastic viscosity coefficient 120578

2

and the initial yield surface 1199010are the most important

parameters 1205781and 120578

2have important influence on

the time of viscous strain developed and 1199010have

important influence on both the viscoplastic strainrate and the final viscoplastic strain

(5) The existence of rheology will significantly slow downthe dissipation of the excess pore pressure Consid-ering the rheological effect the total displacement

is larger and the displacement in the early stage ismainly caused by consolidation while in the late stageit is mainly caused by rheology

(6) For consolidation problems of high compressibilitysoft clay foundation great error will be caused bysmall strain assumption so finite strain effect shouldbe considered When finite strain effect is consideredthe consolidation process is developed more rapidlyand the computed consolidation displacement will beless than that of small strain

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

0090

0100

0110

0120

0130

0125

0115

0095

0105

120576 11

SSE = 99965E minus 7R2 = 098620

10217E minus 119991E minus 228288E minus 129143E minus 5

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 210 kPa

Figure 5 Fitting curves of specimen 1 (1205903= 100 kPa)

Table 3 Model parameters of specimen 1 (1205903= 100 kPa)

1205901minus 1205903(kPa) 119864

0(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots)

30 249 1856 50160 1607360 251 1069 18993 1795090 249 1169 18198 21297120 247 1158 14544 24253150 236 1049 10497 25049180 236 1018 9891 22046210 235 945 12032 26240

Table 4 Model parameters of specimen 2 (1205903= 200 kPa)

1205901minus 1205903(kPa) 119864

0(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots)

40 1075 1804 13881 1222380 903 1870 25862 11747120 834 1694 29947 12591160 763 1692 30057 12704200 660 1729 26118 11732240 604 1831 27481 12823280 508 1602 28165 11589

1198751 1198752 1198753 and 119875

4 the value of the model parameters 119864

0 1198641

1205781 and 120578

2can be determined easily such as in Tables 3 and 4

34 Discussion Table 3 shows that the model parameters ofspecimen 1 (120590

3= 100 kPa) change slightly with the change

of deviatoric stress except the first stage Table 4 showsthat the model parameters of specimen 2 (120590

3= 200 kPa)

change slightly with the change of deviatoric stress except1198640 Compared with the model parameters of specimen 1 and

specimen 2 it can be found that both 1198640and 119864

1increase

notably with the increase of 1205903 which means that the elastic

strain reduces when 1205903increases and 120578

1increases notably

Table 5 Percentage of strain components of specimen 1 (1205903=

100 kPa 119905 = 7 d)

1205901minus 1205903(kPa) 120576

119890

11() 120576

119907119890

11() 120576

119907119901

11()

30 9206 556 23860 8530 1202 26890 8483 1221 296120 8298 1272 430150 8151 1376 473180 8109 1448 443210 8041 1573 385

Table 6 Percentage of strain components of specimen 2 (1205903=

200 kPa 119905 = 7 d)

1205901minus 1205903(kPa) 120576

119890

11() 120576

119907119890

11() 120576

119907119901

11()

40 7703 1653 64580 6774 1683 1543120 6677 1971 1353160 6713 1981 1306200 6951 1837 1212240 7096 1686 1217280 7397 1740 863

while 1205782is reduced notably with the increase of 120590

3 which

means that the viscoelastic strain rate and viscoplastic strainrate change significantly when 120590

3increases For the lack

of more experiment data the further variation of modelparameters was unable to find out which needed furtherstudy

Based on the parameters in Tables 3 and 4 the com-ponents of elastic strain viscoelastic strain and viscoplasticstrain at any time can be calculated with (13) Tables 5 and 6show 120576

119890

11 120576V11989011 and 120576V119901

11of all loading levels when 119905 = 7 d it can

Mathematical Problems in Engineering 7

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 71424E minus 9R2 = 098984

93026E minus 319959E minus 346778E minus 146368E minus 6

0008

0009

0010

0011

0012

0013120576 1

1

P2P3P4

P1

Test dataFitting curve

(a) 119902 = 40 kPa

0 20 40 60 80 100 120 140 160 180t (h)

SSE = 27165E minus 8R2 = 099567

15502E minus 238505E minus 326029E minus 121015E minus 5

0010

0012

0014

0016

0018

0020

0022

0024

120576 11

P2P3P4

P1

Test dataFitting curve

(b) 119902 = 80 kPa

SSE = 85674E minus 8R2 = 0994

21595E minus 263737E minus 320369E minus 126044E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0016

0018

0020

0022

0024

0026

0028

0030

0032

0034

120576 11

P2P3P4

P1

Test dataFitting curve

(c) 119902 = 120 kPa

SSE = 15199E minus 7R2 = 099385

28845E minus 285110E minus 320264E minus 133395E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0020

0025

0030

0035

0040

0045

120576 11

P2P3P4

P1

Test dataFitting curve

(d) 119902 = 160 kPa

SSE = 26486E minus 7R2 = 099252

39406E minus 210413E minus 223826E minus 140897E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0030

0035

0040

0045

0050

0060

0055

120576 11

P2P3P4

P1

Test dataFitting curve

(e) 119902 = 200 kPa

0040

0045

0050

0060

0065

0075

0070

0055

120576 11

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 20786E minus 7R2 = 099572

49663E minus 211800E minus 223980E minus 150713E minus 5

P2P3P4

P1

Test dataFitting curve

(f) 119902 = 240 kPa

Figure 6 Continued

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 50043E minus 7R2 = 09928

66892E minus 215735E minus 220470E minus 146472E minus 5

00600060

00650065

00700070

00750075

00800080

00850085

00900090

00950095

120576 11

120576 11

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 280 kPa

Figure 6 Fitting curves of specimen 2 (1205903= 200 kPa)

001 10 100

02

04

06

08

10

12

14

16

Visc

ous s

trai

n (

)

Time (h)

VES q = 60kPaVPS q = 60kPaVES q = 120 kPa

VPS q = 120 kPaVES q = 180 kPaVPS q = 180 kPa

(a) Specimen 1

Visc

ous s

trai

n (

)

00

02

04

06

08

10

12

100Time (h)

VES q = 80kPaVPS q = 80kPaVES q = 160 kPa

VPS q = 160 kPaVES q = 240 kPaVPS q = 240 kPa

1 10

(b) Specimen 2

Figure 7 Development of computed viscous strain (VES viscoelastic strain VPS viscoplastic strain)

be found that the elastic strain accounted for the major partof the total strain but the viscoelastic strain and viscoplasticstrain increase with the increase of 120590

3 which takes about

15sim30 for each loading increment of the total and cannotbe neglected

Figure 7 shows several typical curves of the developmentof computed viscoelastic strain and viscoplastic strain ofspecimen 1 and specimen 2 It can be found that theviscoelastic strain increases quickly in the early stage andthen reaches steady in a short time (about 30 hours) whilethe viscoplastic strain increases the whole process whichindicates that the long-term deformation after consolidation

of soft clay foundation is mainly caused by the viscoplasticityrather than viscoelasticity

4 Derivation and Verification of UMAT

41 Derivation of UMAT ABAQUS is a FEM software whichhas powerful computing capabilities of strongly nonlinearproblems A user material subroutine (UMAT) is codedfor the proposed model on the platform of ABAQUS tostudy the elastic-viscoplastic consolidation properties of softclay According to ABAQUS the stiffness coefficient matrix

Mathematical Problems in Engineering 9

(Jacobian matrix) and the equation to update the stress andstrain should be defined in the UAMT The Jacobian matrixD is defined as follows

D =

120597Δ120590

120597Δ120576

(15)

where Δ120590 and Δ120576 denote the stress increment matrix andstrain increment matrix respectively

For the proposedmodel (4)sim(6) can be written as followsin 2D or 3D problems

120576119890

119899=

120583

1198640

A120590119899 (16a)

V119890119899=

1

1205781

A120590119899minus

1198661

1205781

120576V119890119899 (16b)

V119901119899=

1

1205782

⟨120601 (119865)⟩

120597119865

120597120590

(16c)

In (16a) (16b) and (16c) 120590119899 120576119899 120576V119890119899 and 120576V119901

119899denote the

Voigt matrix form of stress and strain component tensors Adenotes the flexibilitymatrix for plane strain problem and 3Dproblems and it can be written as follows

A =

[[[

[

1 minus120583 minus120583 0

minus120583 1 minus120583 0

minus120583 minus120583 1 0

0 0 0 2 (1 + 120583)

]]]

]

(17a)

A =

[[[[[[[

[

1 minus120583 minus120583 0 0 0

minus120583 1 minus120583 0 0 0

minus120583 minus120583 1 0 0 0

0 0 0 2 (1 + 120583) 0 0

0 0 0 0 2 (1 + 120583) 0

0 0 0 0 0 2 (1 + 120583)

]]]]]]]

]

(17b)

Suppose that the time step of the 119899th increment is Δ119905119899 so

the viscoelastic strain increment in the increment step can bewritten as follows

Δ120576V119890119899= Δ119905119899[(1 minus Θ)

V119890119899+ Θ

V119890119899+1] (18)

where Θ is differential coefficient 0 le Θ le 1 Suppose that119861119899= 1(1 + ΘΔ119905

11989911986411205781) C119899= 119861119899ΘΔ119905119899A1205781 (18) can be

written as follows

Δ120576V119890119899= 119861119899V119890119899Δ119905119899+ C119899Δ120590119899 (19)

The viscoplastic strain increment in the increment stepcan be written as follows

Δ120576V119901119899= Δ119905119899[(1 minus Θ)

V119901119899+ Θ

V119901119899+1] (20)

In order to calculate the viscoplastic strain rate at theend of the 119899th increment apply the Taylor series expansionand ignore the second order trace then we can obtain thefollowing equation

V119901119899+1

= V119901119899+

120597V119901119899

120597120590

120597120590

120597119905

Δ119905119899 (21)

Suppose thatH119899= 120597

V119901119899120597120590 (21) can be written as

V119901119899+1

= V119901119899+H119899Δ120590119899 (22)

Substituting (22) into (21) we can obtain the followingequation

Δ120576V119901119899= Δ119905119899V119901119899+ ΘΔ119905

119899H119899Δ120590119899 (23)

From (16c)H119899can be written as

H119899=

1

1205782

[⟨120601 (119865)⟩

120597a119879

120597120590

+

120597 ⟨120601 (119865)⟩

120597119865

aa119879] (24)

where a = 120597119865120597120590 According to (9) a can be written asfollows

a = 120597119865

120597120590

= 1198621a1+ 1198622a2 (25)

where 1198621= 2119902119872

2 1198622= 2119901 minus 119901

0 a1= 120597119902120597120590 a

2= 120597119901120597120590

From (25) the following equation can be derived

120597a119879

120597120590

=

1205971198621

120597119902

a1a1198791+ 1198621

120597a1198791

120597120590

+

1205971198622

120597119901

a2a1198792 (26)

where (120597a1198791120597120590) = (32119902)M

1minus(94119902

3

)M2 a1a1198791= (94119902

2

)M2

a2a1198792= (19)M

3 M1 M2 and M

3are matrices written as

follows

M1=

[[[[[[[

[

23 minus13 minus13 0 0 0

minus13 23 minus13 0 0 0

minus13 minus13 23 0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

]]]]]]]

]

M2=

[[[[[[

[

1198782

111198781111987822

1198781111987833

21198781111987812

21198781111987823

21198781111987813

1198781111987822

1198782

221198782211987833

21198782211987812

21198782211987823

21198782211987813

1198781111987833

1198782211987833

1198782

3321198783311987812

21198783311987823

21198783311987813

21198781111987812

21198782211987812

21198783311987812

41198782

1241198781211987823

41198781211987813

21198781111987823

21198782211987823

21198783311987823

41198781211987823

41198782

2341198782311987813

21198781111987813

21198782211987813

21198783311987813

41198781211987813

41198782311987813

41198782

13

]]]]]]

]

M3=

[[[[[[[

[

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]]]]]]]

]

(27)

Combining the above equations the value of H119899can

be derived Substituting H119899into (22) the viscoplastic strain

increment Δ120576V119901119899

is obtainedFor current increment Δ120590

119899is calculated by (28) as

follows

Δ120590119899= D119890Δ120576119890

119899 (28)

Substituting (19) and (23) into (28) we can obtain thefollowing equation

Δ120590119899= D (Δ120576

119899minus 119861Δ119905

119899V119890119899minus Δ119905119899V119901119899) (29)

10 Mathematical Problems in Engineering

Table 7 Material parameters of verification model

1198640(MPa) 119864

1(MPa) 120578

1(GPasdots) 120578

2(GPasdots) 119875

0(kPa) 119890

0120583 119872 120582 120581

10 20 1000 1000 50 108 035 0898 0102 0015

12059021205902

1205903

1205903

1205901

1205901

Figure 8 Sketch of verification model

Equation (29) is the relationship between stress incre-ment and strain increment in which matrix D is defined asfollows

D = [(D119890)minus1 + C119899+ ΘΔ119905

119899H119899]

minus1

(30)

Based on the above equations a user material subroutinewas coded for the proposed constitutive model on theplatform of ABAQUS

42 Verification of UMAT In order to verify the validity ofthe UMAT we study a soil column of triaxial compressionsuch as Figure 8 the size of the column is 01m times 01m times

02m Supposing that 1205901= 100 kPa 120590

2= 1205903= 50 kPa

and the model parameters are given in Table 7 The columnwas cut into one element the element type is C3D8 and theconsolidation effect was not considered here Embedding theUMAT coded in this paper on the platform of ABAQUS wecalculated the triaxial compression property of the columnthe results were summarized in Figure 9simFigure 10

From Figure 9(a) we can see the deformation of thecolumn exhibits apparent rheology the rheological deforma-tion takes about 37 of the whole deformation Figure 9(b)shows that the static yield surface gradually expanded withthe time until it reached steady state when it coincided withthe dynamic yield surface From Figures 9(c) and 9(d) wecan see that the viscoelastic strain and the viscoplastic strainboth gradually increased with time until they reached theirsteady state accordingly the viscoelastic strain rate and theviscoplastic strain rate were gradually deduced to nearly zeroThe above results fit the property of Nishihara model andthe hardening law of modified Cambridge model very wellwhich proved the correctness of the UMAT

In the proposed model 1205781 1205782 and 119901

0are the most

important parameters Figure 10 showed the sensitivity ofthese parameters From Figures 10(a) and 10(b) we can seethe final viscous deformation is a constant regardless of thedifferent values of 120578

1 1205782 but the time when it reaches steady

varies greatly so the accuracy of 1205781and 1205782is directly related

to the accuracy of computed displacement in the rheologicalprocess Figure 10(c) shows that the value of 119901

0has a great

influence on the viscoplastic strain the smaller 1199010is the

more rapidly the viscoplastic strain develops and the largerthe final viscoplastic strain occurs and so the accuracy of1199010is not only directly related to the accuracy of computed

displacement in the rheological process but also affected thefinal displacement which needs to be paid more attention

5 Analysis on Rheological Consolidation ofSoft Clay

51 Computed Model Consolidation problems can be classi-fied as small strain theory and finite strain theory accordingto the difference of geometric assumption Small straintheory assumed that the strain is little but the consolida-tion deformation is usually very large for deep soft clayfoundation Take dredger fill for example the deformationmay exceed 80 for which the small strain theory is nolonger suitable [14 15] As the pioneer researchers studiedon finite strain consolidation theory Mikasa [16] and Gib-son et al [17] derived their one-dimensional finite strainconsolidation equations respectively in 1960s and manyscholars developed further research on the topic in thefollowing decades Because of strong nonlinearity most ofexisting researches on finite strain consolidation adoptedsome simplified assumptionswithout considering rheologicalcharacteristics By means of ABAQUS this paper studied therheological consolidation of soft clay considering finite straineffect

The computed model is plane strain problem as shown inFigure 11 The soil foundation is homogeneous anisotropicsaturated and normally consolidated soft clay with a thick-ness of 100m and only the top boundary is permeable Inaddition the semilogarithmic empirical formula proposed byTavenas et al [18] was adopted to consider the variation ofpermeability in the process of consolidation as follows

119896V = 119896V0 sdot 10(119890minus1198900)119862119896 (31)

where 119896V0 is the initial permeability coefficients and 119896V is thepermeability coefficient in the consolidation process 119890

0is the

initial void ratio and 119890 is the void ratio in the consolidationprocess 119862

119896is the permeability index According to the

above work themodel parameters are summarized in Table 8approximately

52 Consolidation Property of Soft Clay From Figure 12 wecan see the differences of the consolidation process betweenfinite strain and small strain Figure 12(a) shows that for anylocation the excess pore pressure of finite strain is less thanthat of small strain at the same time and the difference isincreasing with the increase of time in beginning period

Mathematical Problems in Engineering 11

Table 8 Material parameters of computed model

1198640(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots) 120574sat (kNsdotm

minus3) 1198961199070(msdotsminus1) 119875

0(kPa) 119872 120582 120581 119890

0120583 119862

119896

25 115 150 24000 177 45119864 minus 9 50 0898 0102 0015 108 035 05

100000225

00200

00175

00150

00125

00100

Vert

ical

def

orm

atio

n (m

)

Time (s)10000 100000 1000000 1E7

(a)

40

50

60

70

80

90

100

110

120

1000Time (s)

10000 100000 1000000 1E7

Pcs

(kPa

)(b)

05

00

10

15

20

25

30

35

Viscoelastic strain

Visc

ous s

trai

n (

)

Viscoplastic strain

1000

Time (s)

10000 100000 1000000 1E71E7

(c)

Viscoelastic strain rateViscoplastic strain rate

0

2

4

6

8

10

12

14

16

1000Time (s)

10000 100000 1000000 1E71E7

Visc

ous s

trai

n ra

te (s

minus1)

(d)

Figure 9 Triaxial compression property of soil column

Figure 12(b) shows that the displacement of finite strain isless than that of small strain and the difference increasedsignificantly if the load is increasing For example when 119902 =200 kPa the differencemay take about 20while it is nearly 0when 119902= 50 kPa So it is essential to consider the error causedby small strain if the load is large and the compressibility ofthe soil is large

From Figure 13 we can see the influence of rheologyon the consolidation process Figure 13(a) shows that thedissipation of excess pore pressure is significantly slower ifthe rheological effect is considered for example when 119905 =1000 d the excess pore pressure is nearly 0 if the rheologicaleffect is not considered but it is still larger than 20 kPa ifthe rheological effect is considered Figure 13(a) shows that

the displacement is influenced by rheological effect too forexample in the early stage (119905 le 100 d) the displacement isnearly the same for two cases but the final displacementmay have a difference of more than 35 which indicatesthat the displacement in the early stage is mainly causedby consolidation while the displacement in the late stage ismainly caused by rheology

6 Conclusions

(1) An elastic-viscoplastic model developed to describethe time-dependent behavior of normally consoli-dated soft clay was presented in the paper on the

12 Mathematical Problems in Engineering

00

05

10

15

20

25

30Vi

scoe

lasti

c str

ain

()

100010 100Time (s)

10000 100000 1000000 1E7

1205781 = 10GPamiddots1205781 = 100GPamiddots1205781 = 1000GPamiddots

(a)

00

02

04

06

08

10

12

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

1205782 = 10GPamiddots1205782 = 100GPamiddots1205782 = 1000GPamiddots

(b)

00

04

08

12

16

20

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

P0 = 30kPaP0 = 40kPa

P0 = 50kPaP0 = 60kPa

(c)

Figure 10 Sensitivity of model parameters

O

Permeable

B = 5 mx

z

H = 10m

q = 100 kPa

Figure 11 Calculation model and mesh generation

framework of modified cam-clay elastic plastic modeland Perzynarsquos overstress viscoplastic theory

(2) A series of laboratory triaxial rheological tests ofNingbo soft clay with different confining pressure anddeviatoric stress are performed the results show thatthe presented constitutive model was suitable for therheological characteristic of Ningbo soft clay and it isachievable to determine the parameters of presentedmodel by curve fitting

(3) The analysis on the model parameters shows that thevalue of all parameters is related to confining pressureand deviatoric stress It may need further study on therelationship between parameters and the stress levelso as to use the model conveniently

Mathematical Problems in Engineering 13

0 20 40 60 80 100u (kPa)

0

2

4

6

8

10H

(m)

FS t = 100dSS t = 100dFS t = 500d

SS t = 500dFS t = 1000 dSS t = 1000 d

(a)

12

10

08

06

04

02

00

Def

orm

atio

n (m

)

Time (d)1000101 100 10000 100000

SS q= 100 kPaFS q= 50 kPaSS q= 50 kPa

SS q= 200 kPaFS q= 200 kPa

FS q= 100 kPa

(b)

Figure 12 Results comparison of large strain and small strain (SS small train FS finite strain)

00

2

4

6

8

10

H(m

)

20 40 60 80 100u (kPa)

t = 100d NRt = 100d Rt = 500d NR

t = 500d Rt = 1000 d NRt = 1000 d R

(a)

06

05

04

03

02

01

00

Def

orm

atio

n (m

)

Time (d)

NRR

1000101 100 10000 100000

(b)

Figure 13 Effects of rheology on calculated results (NR rheological effect not considered R rheological effect considered)

(4) In the proposed model the viscoelastic viscositycoefficient 120578

1 the viscoplastic viscosity coefficient 120578

2

and the initial yield surface 1199010are the most important

parameters 1205781and 120578

2have important influence on

the time of viscous strain developed and 1199010have

important influence on both the viscoplastic strainrate and the final viscoplastic strain

(5) The existence of rheology will significantly slow downthe dissipation of the excess pore pressure Consid-ering the rheological effect the total displacement

is larger and the displacement in the early stage ismainly caused by consolidation while in the late stageit is mainly caused by rheology

(6) For consolidation problems of high compressibilitysoft clay foundation great error will be caused bysmall strain assumption so finite strain effect shouldbe considered When finite strain effect is consideredthe consolidation process is developed more rapidlyand the computed consolidation displacement will beless than that of small strain

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 71424E minus 9R2 = 098984

93026E minus 319959E minus 346778E minus 146368E minus 6

0008

0009

0010

0011

0012

0013120576 1

1

P2P3P4

P1

Test dataFitting curve

(a) 119902 = 40 kPa

0 20 40 60 80 100 120 140 160 180t (h)

SSE = 27165E minus 8R2 = 099567

15502E minus 238505E minus 326029E minus 121015E minus 5

0010

0012

0014

0016

0018

0020

0022

0024

120576 11

P2P3P4

P1

Test dataFitting curve

(b) 119902 = 80 kPa

SSE = 85674E minus 8R2 = 0994

21595E minus 263737E minus 320369E minus 126044E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0016

0018

0020

0022

0024

0026

0028

0030

0032

0034

120576 11

P2P3P4

P1

Test dataFitting curve

(c) 119902 = 120 kPa

SSE = 15199E minus 7R2 = 099385

28845E minus 285110E minus 320264E minus 133395E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0020

0025

0030

0035

0040

0045

120576 11

P2P3P4

P1

Test dataFitting curve

(d) 119902 = 160 kPa

SSE = 26486E minus 7R2 = 099252

39406E minus 210413E minus 223826E minus 140897E minus 5

0 20 40 60 80 100 120 140 160 180

t (h)

0030

0035

0040

0045

0050

0060

0055

120576 11

P2P3P4

P1

Test dataFitting curve

(e) 119902 = 200 kPa

0040

0045

0050

0060

0065

0075

0070

0055

120576 11

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 20786E minus 7R2 = 099572

49663E minus 211800E minus 223980E minus 150713E minus 5

P2P3P4

P1

Test dataFitting curve

(f) 119902 = 240 kPa

Figure 6 Continued

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 50043E minus 7R2 = 09928

66892E minus 215735E minus 220470E minus 146472E minus 5

00600060

00650065

00700070

00750075

00800080

00850085

00900090

00950095

120576 11

120576 11

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 280 kPa

Figure 6 Fitting curves of specimen 2 (1205903= 200 kPa)

001 10 100

02

04

06

08

10

12

14

16

Visc

ous s

trai

n (

)

Time (h)

VES q = 60kPaVPS q = 60kPaVES q = 120 kPa

VPS q = 120 kPaVES q = 180 kPaVPS q = 180 kPa

(a) Specimen 1

Visc

ous s

trai

n (

)

00

02

04

06

08

10

12

100Time (h)

VES q = 80kPaVPS q = 80kPaVES q = 160 kPa

VPS q = 160 kPaVES q = 240 kPaVPS q = 240 kPa

1 10

(b) Specimen 2

Figure 7 Development of computed viscous strain (VES viscoelastic strain VPS viscoplastic strain)

be found that the elastic strain accounted for the major partof the total strain but the viscoelastic strain and viscoplasticstrain increase with the increase of 120590

3 which takes about

15sim30 for each loading increment of the total and cannotbe neglected

Figure 7 shows several typical curves of the developmentof computed viscoelastic strain and viscoplastic strain ofspecimen 1 and specimen 2 It can be found that theviscoelastic strain increases quickly in the early stage andthen reaches steady in a short time (about 30 hours) whilethe viscoplastic strain increases the whole process whichindicates that the long-term deformation after consolidation

of soft clay foundation is mainly caused by the viscoplasticityrather than viscoelasticity

4 Derivation and Verification of UMAT

41 Derivation of UMAT ABAQUS is a FEM software whichhas powerful computing capabilities of strongly nonlinearproblems A user material subroutine (UMAT) is codedfor the proposed model on the platform of ABAQUS tostudy the elastic-viscoplastic consolidation properties of softclay According to ABAQUS the stiffness coefficient matrix

Mathematical Problems in Engineering 9

(Jacobian matrix) and the equation to update the stress andstrain should be defined in the UAMT The Jacobian matrixD is defined as follows

D =

120597Δ120590

120597Δ120576

(15)

where Δ120590 and Δ120576 denote the stress increment matrix andstrain increment matrix respectively

For the proposedmodel (4)sim(6) can be written as followsin 2D or 3D problems

120576119890

119899=

120583

1198640

A120590119899 (16a)

V119890119899=

1

1205781

A120590119899minus

1198661

1205781

120576V119890119899 (16b)

V119901119899=

1

1205782

⟨120601 (119865)⟩

120597119865

120597120590

(16c)

In (16a) (16b) and (16c) 120590119899 120576119899 120576V119890119899 and 120576V119901

119899denote the

Voigt matrix form of stress and strain component tensors Adenotes the flexibilitymatrix for plane strain problem and 3Dproblems and it can be written as follows

A =

[[[

[

1 minus120583 minus120583 0

minus120583 1 minus120583 0

minus120583 minus120583 1 0

0 0 0 2 (1 + 120583)

]]]

]

(17a)

A =

[[[[[[[

[

1 minus120583 minus120583 0 0 0

minus120583 1 minus120583 0 0 0

minus120583 minus120583 1 0 0 0

0 0 0 2 (1 + 120583) 0 0

0 0 0 0 2 (1 + 120583) 0

0 0 0 0 0 2 (1 + 120583)

]]]]]]]

]

(17b)

Suppose that the time step of the 119899th increment is Δ119905119899 so

the viscoelastic strain increment in the increment step can bewritten as follows

Δ120576V119890119899= Δ119905119899[(1 minus Θ)

V119890119899+ Θ

V119890119899+1] (18)

where Θ is differential coefficient 0 le Θ le 1 Suppose that119861119899= 1(1 + ΘΔ119905

11989911986411205781) C119899= 119861119899ΘΔ119905119899A1205781 (18) can be

written as follows

Δ120576V119890119899= 119861119899V119890119899Δ119905119899+ C119899Δ120590119899 (19)

The viscoplastic strain increment in the increment stepcan be written as follows

Δ120576V119901119899= Δ119905119899[(1 minus Θ)

V119901119899+ Θ

V119901119899+1] (20)

In order to calculate the viscoplastic strain rate at theend of the 119899th increment apply the Taylor series expansionand ignore the second order trace then we can obtain thefollowing equation

V119901119899+1

= V119901119899+

120597V119901119899

120597120590

120597120590

120597119905

Δ119905119899 (21)

Suppose thatH119899= 120597

V119901119899120597120590 (21) can be written as

V119901119899+1

= V119901119899+H119899Δ120590119899 (22)

Substituting (22) into (21) we can obtain the followingequation

Δ120576V119901119899= Δ119905119899V119901119899+ ΘΔ119905

119899H119899Δ120590119899 (23)

From (16c)H119899can be written as

H119899=

1

1205782

[⟨120601 (119865)⟩

120597a119879

120597120590

+

120597 ⟨120601 (119865)⟩

120597119865

aa119879] (24)

where a = 120597119865120597120590 According to (9) a can be written asfollows

a = 120597119865

120597120590

= 1198621a1+ 1198622a2 (25)

where 1198621= 2119902119872

2 1198622= 2119901 minus 119901

0 a1= 120597119902120597120590 a

2= 120597119901120597120590

From (25) the following equation can be derived

120597a119879

120597120590

=

1205971198621

120597119902

a1a1198791+ 1198621

120597a1198791

120597120590

+

1205971198622

120597119901

a2a1198792 (26)

where (120597a1198791120597120590) = (32119902)M

1minus(94119902

3

)M2 a1a1198791= (94119902

2

)M2

a2a1198792= (19)M

3 M1 M2 and M

3are matrices written as

follows

M1=

[[[[[[[

[

23 minus13 minus13 0 0 0

minus13 23 minus13 0 0 0

minus13 minus13 23 0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

]]]]]]]

]

M2=

[[[[[[

[

1198782

111198781111987822

1198781111987833

21198781111987812

21198781111987823

21198781111987813

1198781111987822

1198782

221198782211987833

21198782211987812

21198782211987823

21198782211987813

1198781111987833

1198782211987833

1198782

3321198783311987812

21198783311987823

21198783311987813

21198781111987812

21198782211987812

21198783311987812

41198782

1241198781211987823

41198781211987813

21198781111987823

21198782211987823

21198783311987823

41198781211987823

41198782

2341198782311987813

21198781111987813

21198782211987813

21198783311987813

41198781211987813

41198782311987813

41198782

13

]]]]]]

]

M3=

[[[[[[[

[

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]]]]]]]

]

(27)

Combining the above equations the value of H119899can

be derived Substituting H119899into (22) the viscoplastic strain

increment Δ120576V119901119899

is obtainedFor current increment Δ120590

119899is calculated by (28) as

follows

Δ120590119899= D119890Δ120576119890

119899 (28)

Substituting (19) and (23) into (28) we can obtain thefollowing equation

Δ120590119899= D (Δ120576

119899minus 119861Δ119905

119899V119890119899minus Δ119905119899V119901119899) (29)

10 Mathematical Problems in Engineering

Table 7 Material parameters of verification model

1198640(MPa) 119864

1(MPa) 120578

1(GPasdots) 120578

2(GPasdots) 119875

0(kPa) 119890

0120583 119872 120582 120581

10 20 1000 1000 50 108 035 0898 0102 0015

12059021205902

1205903

1205903

1205901

1205901

Figure 8 Sketch of verification model

Equation (29) is the relationship between stress incre-ment and strain increment in which matrix D is defined asfollows

D = [(D119890)minus1 + C119899+ ΘΔ119905

119899H119899]

minus1

(30)

Based on the above equations a user material subroutinewas coded for the proposed constitutive model on theplatform of ABAQUS

42 Verification of UMAT In order to verify the validity ofthe UMAT we study a soil column of triaxial compressionsuch as Figure 8 the size of the column is 01m times 01m times

02m Supposing that 1205901= 100 kPa 120590

2= 1205903= 50 kPa

and the model parameters are given in Table 7 The columnwas cut into one element the element type is C3D8 and theconsolidation effect was not considered here Embedding theUMAT coded in this paper on the platform of ABAQUS wecalculated the triaxial compression property of the columnthe results were summarized in Figure 9simFigure 10

From Figure 9(a) we can see the deformation of thecolumn exhibits apparent rheology the rheological deforma-tion takes about 37 of the whole deformation Figure 9(b)shows that the static yield surface gradually expanded withthe time until it reached steady state when it coincided withthe dynamic yield surface From Figures 9(c) and 9(d) wecan see that the viscoelastic strain and the viscoplastic strainboth gradually increased with time until they reached theirsteady state accordingly the viscoelastic strain rate and theviscoplastic strain rate were gradually deduced to nearly zeroThe above results fit the property of Nishihara model andthe hardening law of modified Cambridge model very wellwhich proved the correctness of the UMAT

In the proposed model 1205781 1205782 and 119901

0are the most

important parameters Figure 10 showed the sensitivity ofthese parameters From Figures 10(a) and 10(b) we can seethe final viscous deformation is a constant regardless of thedifferent values of 120578

1 1205782 but the time when it reaches steady

varies greatly so the accuracy of 1205781and 1205782is directly related

to the accuracy of computed displacement in the rheologicalprocess Figure 10(c) shows that the value of 119901

0has a great

influence on the viscoplastic strain the smaller 1199010is the

more rapidly the viscoplastic strain develops and the largerthe final viscoplastic strain occurs and so the accuracy of1199010is not only directly related to the accuracy of computed

displacement in the rheological process but also affected thefinal displacement which needs to be paid more attention

5 Analysis on Rheological Consolidation ofSoft Clay

51 Computed Model Consolidation problems can be classi-fied as small strain theory and finite strain theory accordingto the difference of geometric assumption Small straintheory assumed that the strain is little but the consolida-tion deformation is usually very large for deep soft clayfoundation Take dredger fill for example the deformationmay exceed 80 for which the small strain theory is nolonger suitable [14 15] As the pioneer researchers studiedon finite strain consolidation theory Mikasa [16] and Gib-son et al [17] derived their one-dimensional finite strainconsolidation equations respectively in 1960s and manyscholars developed further research on the topic in thefollowing decades Because of strong nonlinearity most ofexisting researches on finite strain consolidation adoptedsome simplified assumptionswithout considering rheologicalcharacteristics By means of ABAQUS this paper studied therheological consolidation of soft clay considering finite straineffect

The computed model is plane strain problem as shown inFigure 11 The soil foundation is homogeneous anisotropicsaturated and normally consolidated soft clay with a thick-ness of 100m and only the top boundary is permeable Inaddition the semilogarithmic empirical formula proposed byTavenas et al [18] was adopted to consider the variation ofpermeability in the process of consolidation as follows

119896V = 119896V0 sdot 10(119890minus1198900)119862119896 (31)

where 119896V0 is the initial permeability coefficients and 119896V is thepermeability coefficient in the consolidation process 119890

0is the

initial void ratio and 119890 is the void ratio in the consolidationprocess 119862

119896is the permeability index According to the

above work themodel parameters are summarized in Table 8approximately

52 Consolidation Property of Soft Clay From Figure 12 wecan see the differences of the consolidation process betweenfinite strain and small strain Figure 12(a) shows that for anylocation the excess pore pressure of finite strain is less thanthat of small strain at the same time and the difference isincreasing with the increase of time in beginning period

Mathematical Problems in Engineering 11

Table 8 Material parameters of computed model

1198640(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots) 120574sat (kNsdotm

minus3) 1198961199070(msdotsminus1) 119875

0(kPa) 119872 120582 120581 119890

0120583 119862

119896

25 115 150 24000 177 45119864 minus 9 50 0898 0102 0015 108 035 05

100000225

00200

00175

00150

00125

00100

Vert

ical

def

orm

atio

n (m

)

Time (s)10000 100000 1000000 1E7

(a)

40

50

60

70

80

90

100

110

120

1000Time (s)

10000 100000 1000000 1E7

Pcs

(kPa

)(b)

05

00

10

15

20

25

30

35

Viscoelastic strain

Visc

ous s

trai

n (

)

Viscoplastic strain

1000

Time (s)

10000 100000 1000000 1E71E7

(c)

Viscoelastic strain rateViscoplastic strain rate

0

2

4

6

8

10

12

14

16

1000Time (s)

10000 100000 1000000 1E71E7

Visc

ous s

trai

n ra

te (s

minus1)

(d)

Figure 9 Triaxial compression property of soil column

Figure 12(b) shows that the displacement of finite strain isless than that of small strain and the difference increasedsignificantly if the load is increasing For example when 119902 =200 kPa the differencemay take about 20while it is nearly 0when 119902= 50 kPa So it is essential to consider the error causedby small strain if the load is large and the compressibility ofthe soil is large

From Figure 13 we can see the influence of rheologyon the consolidation process Figure 13(a) shows that thedissipation of excess pore pressure is significantly slower ifthe rheological effect is considered for example when 119905 =1000 d the excess pore pressure is nearly 0 if the rheologicaleffect is not considered but it is still larger than 20 kPa ifthe rheological effect is considered Figure 13(a) shows that

the displacement is influenced by rheological effect too forexample in the early stage (119905 le 100 d) the displacement isnearly the same for two cases but the final displacementmay have a difference of more than 35 which indicatesthat the displacement in the early stage is mainly causedby consolidation while the displacement in the late stage ismainly caused by rheology

6 Conclusions

(1) An elastic-viscoplastic model developed to describethe time-dependent behavior of normally consoli-dated soft clay was presented in the paper on the

12 Mathematical Problems in Engineering

00

05

10

15

20

25

30Vi

scoe

lasti

c str

ain

()

100010 100Time (s)

10000 100000 1000000 1E7

1205781 = 10GPamiddots1205781 = 100GPamiddots1205781 = 1000GPamiddots

(a)

00

02

04

06

08

10

12

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

1205782 = 10GPamiddots1205782 = 100GPamiddots1205782 = 1000GPamiddots

(b)

00

04

08

12

16

20

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

P0 = 30kPaP0 = 40kPa

P0 = 50kPaP0 = 60kPa

(c)

Figure 10 Sensitivity of model parameters

O

Permeable

B = 5 mx

z

H = 10m

q = 100 kPa

Figure 11 Calculation model and mesh generation

framework of modified cam-clay elastic plastic modeland Perzynarsquos overstress viscoplastic theory

(2) A series of laboratory triaxial rheological tests ofNingbo soft clay with different confining pressure anddeviatoric stress are performed the results show thatthe presented constitutive model was suitable for therheological characteristic of Ningbo soft clay and it isachievable to determine the parameters of presentedmodel by curve fitting

(3) The analysis on the model parameters shows that thevalue of all parameters is related to confining pressureand deviatoric stress It may need further study on therelationship between parameters and the stress levelso as to use the model conveniently

Mathematical Problems in Engineering 13

0 20 40 60 80 100u (kPa)

0

2

4

6

8

10H

(m)

FS t = 100dSS t = 100dFS t = 500d

SS t = 500dFS t = 1000 dSS t = 1000 d

(a)

12

10

08

06

04

02

00

Def

orm

atio

n (m

)

Time (d)1000101 100 10000 100000

SS q= 100 kPaFS q= 50 kPaSS q= 50 kPa

SS q= 200 kPaFS q= 200 kPa

FS q= 100 kPa

(b)

Figure 12 Results comparison of large strain and small strain (SS small train FS finite strain)

00

2

4

6

8

10

H(m

)

20 40 60 80 100u (kPa)

t = 100d NRt = 100d Rt = 500d NR

t = 500d Rt = 1000 d NRt = 1000 d R

(a)

06

05

04

03

02

01

00

Def

orm

atio

n (m

)

Time (d)

NRR

1000101 100 10000 100000

(b)

Figure 13 Effects of rheology on calculated results (NR rheological effect not considered R rheological effect considered)

(4) In the proposed model the viscoelastic viscositycoefficient 120578

1 the viscoplastic viscosity coefficient 120578

2

and the initial yield surface 1199010are the most important

parameters 1205781and 120578

2have important influence on

the time of viscous strain developed and 1199010have

important influence on both the viscoplastic strainrate and the final viscoplastic strain

(5) The existence of rheology will significantly slow downthe dissipation of the excess pore pressure Consid-ering the rheological effect the total displacement

is larger and the displacement in the early stage ismainly caused by consolidation while in the late stageit is mainly caused by rheology

(6) For consolidation problems of high compressibilitysoft clay foundation great error will be caused bysmall strain assumption so finite strain effect shouldbe considered When finite strain effect is consideredthe consolidation process is developed more rapidlyand the computed consolidation displacement will beless than that of small strain

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160 180

t (h)

SSE = 50043E minus 7R2 = 09928

66892E minus 215735E minus 220470E minus 146472E minus 5

00600060

00650065

00700070

00750075

00800080

00850085

00900090

00950095

120576 11

120576 11

Test dataFitting curve

P2P3P4

P1

(g) 119902 = 280 kPa

Figure 6 Fitting curves of specimen 2 (1205903= 200 kPa)

001 10 100

02

04

06

08

10

12

14

16

Visc

ous s

trai

n (

)

Time (h)

VES q = 60kPaVPS q = 60kPaVES q = 120 kPa

VPS q = 120 kPaVES q = 180 kPaVPS q = 180 kPa

(a) Specimen 1

Visc

ous s

trai

n (

)

00

02

04

06

08

10

12

100Time (h)

VES q = 80kPaVPS q = 80kPaVES q = 160 kPa

VPS q = 160 kPaVES q = 240 kPaVPS q = 240 kPa

1 10

(b) Specimen 2

Figure 7 Development of computed viscous strain (VES viscoelastic strain VPS viscoplastic strain)

be found that the elastic strain accounted for the major partof the total strain but the viscoelastic strain and viscoplasticstrain increase with the increase of 120590

3 which takes about

15sim30 for each loading increment of the total and cannotbe neglected

Figure 7 shows several typical curves of the developmentof computed viscoelastic strain and viscoplastic strain ofspecimen 1 and specimen 2 It can be found that theviscoelastic strain increases quickly in the early stage andthen reaches steady in a short time (about 30 hours) whilethe viscoplastic strain increases the whole process whichindicates that the long-term deformation after consolidation

of soft clay foundation is mainly caused by the viscoplasticityrather than viscoelasticity

4 Derivation and Verification of UMAT

41 Derivation of UMAT ABAQUS is a FEM software whichhas powerful computing capabilities of strongly nonlinearproblems A user material subroutine (UMAT) is codedfor the proposed model on the platform of ABAQUS tostudy the elastic-viscoplastic consolidation properties of softclay According to ABAQUS the stiffness coefficient matrix

Mathematical Problems in Engineering 9

(Jacobian matrix) and the equation to update the stress andstrain should be defined in the UAMT The Jacobian matrixD is defined as follows

D =

120597Δ120590

120597Δ120576

(15)

where Δ120590 and Δ120576 denote the stress increment matrix andstrain increment matrix respectively

For the proposedmodel (4)sim(6) can be written as followsin 2D or 3D problems

120576119890

119899=

120583

1198640

A120590119899 (16a)

V119890119899=

1

1205781

A120590119899minus

1198661

1205781

120576V119890119899 (16b)

V119901119899=

1

1205782

⟨120601 (119865)⟩

120597119865

120597120590

(16c)

In (16a) (16b) and (16c) 120590119899 120576119899 120576V119890119899 and 120576V119901

119899denote the

Voigt matrix form of stress and strain component tensors Adenotes the flexibilitymatrix for plane strain problem and 3Dproblems and it can be written as follows

A =

[[[

[

1 minus120583 minus120583 0

minus120583 1 minus120583 0

minus120583 minus120583 1 0

0 0 0 2 (1 + 120583)

]]]

]

(17a)

A =

[[[[[[[

[

1 minus120583 minus120583 0 0 0

minus120583 1 minus120583 0 0 0

minus120583 minus120583 1 0 0 0

0 0 0 2 (1 + 120583) 0 0

0 0 0 0 2 (1 + 120583) 0

0 0 0 0 0 2 (1 + 120583)

]]]]]]]

]

(17b)

Suppose that the time step of the 119899th increment is Δ119905119899 so

the viscoelastic strain increment in the increment step can bewritten as follows

Δ120576V119890119899= Δ119905119899[(1 minus Θ)

V119890119899+ Θ

V119890119899+1] (18)

where Θ is differential coefficient 0 le Θ le 1 Suppose that119861119899= 1(1 + ΘΔ119905

11989911986411205781) C119899= 119861119899ΘΔ119905119899A1205781 (18) can be

written as follows

Δ120576V119890119899= 119861119899V119890119899Δ119905119899+ C119899Δ120590119899 (19)

The viscoplastic strain increment in the increment stepcan be written as follows

Δ120576V119901119899= Δ119905119899[(1 minus Θ)

V119901119899+ Θ

V119901119899+1] (20)

In order to calculate the viscoplastic strain rate at theend of the 119899th increment apply the Taylor series expansionand ignore the second order trace then we can obtain thefollowing equation

V119901119899+1

= V119901119899+

120597V119901119899

120597120590

120597120590

120597119905

Δ119905119899 (21)

Suppose thatH119899= 120597

V119901119899120597120590 (21) can be written as

V119901119899+1

= V119901119899+H119899Δ120590119899 (22)

Substituting (22) into (21) we can obtain the followingequation

Δ120576V119901119899= Δ119905119899V119901119899+ ΘΔ119905

119899H119899Δ120590119899 (23)

From (16c)H119899can be written as

H119899=

1

1205782

[⟨120601 (119865)⟩

120597a119879

120597120590

+

120597 ⟨120601 (119865)⟩

120597119865

aa119879] (24)

where a = 120597119865120597120590 According to (9) a can be written asfollows

a = 120597119865

120597120590

= 1198621a1+ 1198622a2 (25)

where 1198621= 2119902119872

2 1198622= 2119901 minus 119901

0 a1= 120597119902120597120590 a

2= 120597119901120597120590

From (25) the following equation can be derived

120597a119879

120597120590

=

1205971198621

120597119902

a1a1198791+ 1198621

120597a1198791

120597120590

+

1205971198622

120597119901

a2a1198792 (26)

where (120597a1198791120597120590) = (32119902)M

1minus(94119902

3

)M2 a1a1198791= (94119902

2

)M2

a2a1198792= (19)M

3 M1 M2 and M

3are matrices written as

follows

M1=

[[[[[[[

[

23 minus13 minus13 0 0 0

minus13 23 minus13 0 0 0

minus13 minus13 23 0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

]]]]]]]

]

M2=

[[[[[[

[

1198782

111198781111987822

1198781111987833

21198781111987812

21198781111987823

21198781111987813

1198781111987822

1198782

221198782211987833

21198782211987812

21198782211987823

21198782211987813

1198781111987833

1198782211987833

1198782

3321198783311987812

21198783311987823

21198783311987813

21198781111987812

21198782211987812

21198783311987812

41198782

1241198781211987823

41198781211987813

21198781111987823

21198782211987823

21198783311987823

41198781211987823

41198782

2341198782311987813

21198781111987813

21198782211987813

21198783311987813

41198781211987813

41198782311987813

41198782

13

]]]]]]

]

M3=

[[[[[[[

[

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]]]]]]]

]

(27)

Combining the above equations the value of H119899can

be derived Substituting H119899into (22) the viscoplastic strain

increment Δ120576V119901119899

is obtainedFor current increment Δ120590

119899is calculated by (28) as

follows

Δ120590119899= D119890Δ120576119890

119899 (28)

Substituting (19) and (23) into (28) we can obtain thefollowing equation

Δ120590119899= D (Δ120576

119899minus 119861Δ119905

119899V119890119899minus Δ119905119899V119901119899) (29)

10 Mathematical Problems in Engineering

Table 7 Material parameters of verification model

1198640(MPa) 119864

1(MPa) 120578

1(GPasdots) 120578

2(GPasdots) 119875

0(kPa) 119890

0120583 119872 120582 120581

10 20 1000 1000 50 108 035 0898 0102 0015

12059021205902

1205903

1205903

1205901

1205901

Figure 8 Sketch of verification model

Equation (29) is the relationship between stress incre-ment and strain increment in which matrix D is defined asfollows

D = [(D119890)minus1 + C119899+ ΘΔ119905

119899H119899]

minus1

(30)

Based on the above equations a user material subroutinewas coded for the proposed constitutive model on theplatform of ABAQUS

42 Verification of UMAT In order to verify the validity ofthe UMAT we study a soil column of triaxial compressionsuch as Figure 8 the size of the column is 01m times 01m times

02m Supposing that 1205901= 100 kPa 120590

2= 1205903= 50 kPa

and the model parameters are given in Table 7 The columnwas cut into one element the element type is C3D8 and theconsolidation effect was not considered here Embedding theUMAT coded in this paper on the platform of ABAQUS wecalculated the triaxial compression property of the columnthe results were summarized in Figure 9simFigure 10

From Figure 9(a) we can see the deformation of thecolumn exhibits apparent rheology the rheological deforma-tion takes about 37 of the whole deformation Figure 9(b)shows that the static yield surface gradually expanded withthe time until it reached steady state when it coincided withthe dynamic yield surface From Figures 9(c) and 9(d) wecan see that the viscoelastic strain and the viscoplastic strainboth gradually increased with time until they reached theirsteady state accordingly the viscoelastic strain rate and theviscoplastic strain rate were gradually deduced to nearly zeroThe above results fit the property of Nishihara model andthe hardening law of modified Cambridge model very wellwhich proved the correctness of the UMAT

In the proposed model 1205781 1205782 and 119901

0are the most

important parameters Figure 10 showed the sensitivity ofthese parameters From Figures 10(a) and 10(b) we can seethe final viscous deformation is a constant regardless of thedifferent values of 120578

1 1205782 but the time when it reaches steady

varies greatly so the accuracy of 1205781and 1205782is directly related

to the accuracy of computed displacement in the rheologicalprocess Figure 10(c) shows that the value of 119901

0has a great

influence on the viscoplastic strain the smaller 1199010is the

more rapidly the viscoplastic strain develops and the largerthe final viscoplastic strain occurs and so the accuracy of1199010is not only directly related to the accuracy of computed

displacement in the rheological process but also affected thefinal displacement which needs to be paid more attention

5 Analysis on Rheological Consolidation ofSoft Clay

51 Computed Model Consolidation problems can be classi-fied as small strain theory and finite strain theory accordingto the difference of geometric assumption Small straintheory assumed that the strain is little but the consolida-tion deformation is usually very large for deep soft clayfoundation Take dredger fill for example the deformationmay exceed 80 for which the small strain theory is nolonger suitable [14 15] As the pioneer researchers studiedon finite strain consolidation theory Mikasa [16] and Gib-son et al [17] derived their one-dimensional finite strainconsolidation equations respectively in 1960s and manyscholars developed further research on the topic in thefollowing decades Because of strong nonlinearity most ofexisting researches on finite strain consolidation adoptedsome simplified assumptionswithout considering rheologicalcharacteristics By means of ABAQUS this paper studied therheological consolidation of soft clay considering finite straineffect

The computed model is plane strain problem as shown inFigure 11 The soil foundation is homogeneous anisotropicsaturated and normally consolidated soft clay with a thick-ness of 100m and only the top boundary is permeable Inaddition the semilogarithmic empirical formula proposed byTavenas et al [18] was adopted to consider the variation ofpermeability in the process of consolidation as follows

119896V = 119896V0 sdot 10(119890minus1198900)119862119896 (31)

where 119896V0 is the initial permeability coefficients and 119896V is thepermeability coefficient in the consolidation process 119890

0is the

initial void ratio and 119890 is the void ratio in the consolidationprocess 119862

119896is the permeability index According to the

above work themodel parameters are summarized in Table 8approximately

52 Consolidation Property of Soft Clay From Figure 12 wecan see the differences of the consolidation process betweenfinite strain and small strain Figure 12(a) shows that for anylocation the excess pore pressure of finite strain is less thanthat of small strain at the same time and the difference isincreasing with the increase of time in beginning period

Mathematical Problems in Engineering 11

Table 8 Material parameters of computed model

1198640(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots) 120574sat (kNsdotm

minus3) 1198961199070(msdotsminus1) 119875

0(kPa) 119872 120582 120581 119890

0120583 119862

119896

25 115 150 24000 177 45119864 minus 9 50 0898 0102 0015 108 035 05

100000225

00200

00175

00150

00125

00100

Vert

ical

def

orm

atio

n (m

)

Time (s)10000 100000 1000000 1E7

(a)

40

50

60

70

80

90

100

110

120

1000Time (s)

10000 100000 1000000 1E7

Pcs

(kPa

)(b)

05

00

10

15

20

25

30

35

Viscoelastic strain

Visc

ous s

trai

n (

)

Viscoplastic strain

1000

Time (s)

10000 100000 1000000 1E71E7

(c)

Viscoelastic strain rateViscoplastic strain rate

0

2

4

6

8

10

12

14

16

1000Time (s)

10000 100000 1000000 1E71E7

Visc

ous s

trai

n ra

te (s

minus1)

(d)

Figure 9 Triaxial compression property of soil column

Figure 12(b) shows that the displacement of finite strain isless than that of small strain and the difference increasedsignificantly if the load is increasing For example when 119902 =200 kPa the differencemay take about 20while it is nearly 0when 119902= 50 kPa So it is essential to consider the error causedby small strain if the load is large and the compressibility ofthe soil is large

From Figure 13 we can see the influence of rheologyon the consolidation process Figure 13(a) shows that thedissipation of excess pore pressure is significantly slower ifthe rheological effect is considered for example when 119905 =1000 d the excess pore pressure is nearly 0 if the rheologicaleffect is not considered but it is still larger than 20 kPa ifthe rheological effect is considered Figure 13(a) shows that

the displacement is influenced by rheological effect too forexample in the early stage (119905 le 100 d) the displacement isnearly the same for two cases but the final displacementmay have a difference of more than 35 which indicatesthat the displacement in the early stage is mainly causedby consolidation while the displacement in the late stage ismainly caused by rheology

6 Conclusions

(1) An elastic-viscoplastic model developed to describethe time-dependent behavior of normally consoli-dated soft clay was presented in the paper on the

12 Mathematical Problems in Engineering

00

05

10

15

20

25

30Vi

scoe

lasti

c str

ain

()

100010 100Time (s)

10000 100000 1000000 1E7

1205781 = 10GPamiddots1205781 = 100GPamiddots1205781 = 1000GPamiddots

(a)

00

02

04

06

08

10

12

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

1205782 = 10GPamiddots1205782 = 100GPamiddots1205782 = 1000GPamiddots

(b)

00

04

08

12

16

20

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

P0 = 30kPaP0 = 40kPa

P0 = 50kPaP0 = 60kPa

(c)

Figure 10 Sensitivity of model parameters

O

Permeable

B = 5 mx

z

H = 10m

q = 100 kPa

Figure 11 Calculation model and mesh generation

framework of modified cam-clay elastic plastic modeland Perzynarsquos overstress viscoplastic theory

(2) A series of laboratory triaxial rheological tests ofNingbo soft clay with different confining pressure anddeviatoric stress are performed the results show thatthe presented constitutive model was suitable for therheological characteristic of Ningbo soft clay and it isachievable to determine the parameters of presentedmodel by curve fitting

(3) The analysis on the model parameters shows that thevalue of all parameters is related to confining pressureand deviatoric stress It may need further study on therelationship between parameters and the stress levelso as to use the model conveniently

Mathematical Problems in Engineering 13

0 20 40 60 80 100u (kPa)

0

2

4

6

8

10H

(m)

FS t = 100dSS t = 100dFS t = 500d

SS t = 500dFS t = 1000 dSS t = 1000 d

(a)

12

10

08

06

04

02

00

Def

orm

atio

n (m

)

Time (d)1000101 100 10000 100000

SS q= 100 kPaFS q= 50 kPaSS q= 50 kPa

SS q= 200 kPaFS q= 200 kPa

FS q= 100 kPa

(b)

Figure 12 Results comparison of large strain and small strain (SS small train FS finite strain)

00

2

4

6

8

10

H(m

)

20 40 60 80 100u (kPa)

t = 100d NRt = 100d Rt = 500d NR

t = 500d Rt = 1000 d NRt = 1000 d R

(a)

06

05

04

03

02

01

00

Def

orm

atio

n (m

)

Time (d)

NRR

1000101 100 10000 100000

(b)

Figure 13 Effects of rheology on calculated results (NR rheological effect not considered R rheological effect considered)

(4) In the proposed model the viscoelastic viscositycoefficient 120578

1 the viscoplastic viscosity coefficient 120578

2

and the initial yield surface 1199010are the most important

parameters 1205781and 120578

2have important influence on

the time of viscous strain developed and 1199010have

important influence on both the viscoplastic strainrate and the final viscoplastic strain

(5) The existence of rheology will significantly slow downthe dissipation of the excess pore pressure Consid-ering the rheological effect the total displacement

is larger and the displacement in the early stage ismainly caused by consolidation while in the late stageit is mainly caused by rheology

(6) For consolidation problems of high compressibilitysoft clay foundation great error will be caused bysmall strain assumption so finite strain effect shouldbe considered When finite strain effect is consideredthe consolidation process is developed more rapidlyand the computed consolidation displacement will beless than that of small strain

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

(Jacobian matrix) and the equation to update the stress andstrain should be defined in the UAMT The Jacobian matrixD is defined as follows

D =

120597Δ120590

120597Δ120576

(15)

where Δ120590 and Δ120576 denote the stress increment matrix andstrain increment matrix respectively

For the proposedmodel (4)sim(6) can be written as followsin 2D or 3D problems

120576119890

119899=

120583

1198640

A120590119899 (16a)

V119890119899=

1

1205781

A120590119899minus

1198661

1205781

120576V119890119899 (16b)

V119901119899=

1

1205782

⟨120601 (119865)⟩

120597119865

120597120590

(16c)

In (16a) (16b) and (16c) 120590119899 120576119899 120576V119890119899 and 120576V119901

119899denote the

Voigt matrix form of stress and strain component tensors Adenotes the flexibilitymatrix for plane strain problem and 3Dproblems and it can be written as follows

A =

[[[

[

1 minus120583 minus120583 0

minus120583 1 minus120583 0

minus120583 minus120583 1 0

0 0 0 2 (1 + 120583)

]]]

]

(17a)

A =

[[[[[[[

[

1 minus120583 minus120583 0 0 0

minus120583 1 minus120583 0 0 0

minus120583 minus120583 1 0 0 0

0 0 0 2 (1 + 120583) 0 0

0 0 0 0 2 (1 + 120583) 0

0 0 0 0 0 2 (1 + 120583)

]]]]]]]

]

(17b)

Suppose that the time step of the 119899th increment is Δ119905119899 so

the viscoelastic strain increment in the increment step can bewritten as follows

Δ120576V119890119899= Δ119905119899[(1 minus Θ)

V119890119899+ Θ

V119890119899+1] (18)

where Θ is differential coefficient 0 le Θ le 1 Suppose that119861119899= 1(1 + ΘΔ119905

11989911986411205781) C119899= 119861119899ΘΔ119905119899A1205781 (18) can be

written as follows

Δ120576V119890119899= 119861119899V119890119899Δ119905119899+ C119899Δ120590119899 (19)

The viscoplastic strain increment in the increment stepcan be written as follows

Δ120576V119901119899= Δ119905119899[(1 minus Θ)

V119901119899+ Θ

V119901119899+1] (20)

In order to calculate the viscoplastic strain rate at theend of the 119899th increment apply the Taylor series expansionand ignore the second order trace then we can obtain thefollowing equation

V119901119899+1

= V119901119899+

120597V119901119899

120597120590

120597120590

120597119905

Δ119905119899 (21)

Suppose thatH119899= 120597

V119901119899120597120590 (21) can be written as

V119901119899+1

= V119901119899+H119899Δ120590119899 (22)

Substituting (22) into (21) we can obtain the followingequation

Δ120576V119901119899= Δ119905119899V119901119899+ ΘΔ119905

119899H119899Δ120590119899 (23)

From (16c)H119899can be written as

H119899=

1

1205782

[⟨120601 (119865)⟩

120597a119879

120597120590

+

120597 ⟨120601 (119865)⟩

120597119865

aa119879] (24)

where a = 120597119865120597120590 According to (9) a can be written asfollows

a = 120597119865

120597120590

= 1198621a1+ 1198622a2 (25)

where 1198621= 2119902119872

2 1198622= 2119901 minus 119901

0 a1= 120597119902120597120590 a

2= 120597119901120597120590

From (25) the following equation can be derived

120597a119879

120597120590

=

1205971198621

120597119902

a1a1198791+ 1198621

120597a1198791

120597120590

+

1205971198622

120597119901

a2a1198792 (26)

where (120597a1198791120597120590) = (32119902)M

1minus(94119902

3

)M2 a1a1198791= (94119902

2

)M2

a2a1198792= (19)M

3 M1 M2 and M

3are matrices written as

follows

M1=

[[[[[[[

[

23 minus13 minus13 0 0 0

minus13 23 minus13 0 0 0

minus13 minus13 23 0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

]]]]]]]

]

M2=

[[[[[[

[

1198782

111198781111987822

1198781111987833

21198781111987812

21198781111987823

21198781111987813

1198781111987822

1198782

221198782211987833

21198782211987812

21198782211987823

21198782211987813

1198781111987833

1198782211987833

1198782

3321198783311987812

21198783311987823

21198783311987813

21198781111987812

21198782211987812

21198783311987812

41198782

1241198781211987823

41198781211987813

21198781111987823

21198782211987823

21198783311987823

41198781211987823

41198782

2341198782311987813

21198781111987813

21198782211987813

21198783311987813

41198781211987813

41198782311987813

41198782

13

]]]]]]

]

M3=

[[[[[[[

[

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]]]]]]]

]

(27)

Combining the above equations the value of H119899can

be derived Substituting H119899into (22) the viscoplastic strain

increment Δ120576V119901119899

is obtainedFor current increment Δ120590

119899is calculated by (28) as

follows

Δ120590119899= D119890Δ120576119890

119899 (28)

Substituting (19) and (23) into (28) we can obtain thefollowing equation

Δ120590119899= D (Δ120576

119899minus 119861Δ119905

119899V119890119899minus Δ119905119899V119901119899) (29)

10 Mathematical Problems in Engineering

Table 7 Material parameters of verification model

1198640(MPa) 119864

1(MPa) 120578

1(GPasdots) 120578

2(GPasdots) 119875

0(kPa) 119890

0120583 119872 120582 120581

10 20 1000 1000 50 108 035 0898 0102 0015

12059021205902

1205903

1205903

1205901

1205901

Figure 8 Sketch of verification model

Equation (29) is the relationship between stress incre-ment and strain increment in which matrix D is defined asfollows

D = [(D119890)minus1 + C119899+ ΘΔ119905

119899H119899]

minus1

(30)

Based on the above equations a user material subroutinewas coded for the proposed constitutive model on theplatform of ABAQUS

42 Verification of UMAT In order to verify the validity ofthe UMAT we study a soil column of triaxial compressionsuch as Figure 8 the size of the column is 01m times 01m times

02m Supposing that 1205901= 100 kPa 120590

2= 1205903= 50 kPa

and the model parameters are given in Table 7 The columnwas cut into one element the element type is C3D8 and theconsolidation effect was not considered here Embedding theUMAT coded in this paper on the platform of ABAQUS wecalculated the triaxial compression property of the columnthe results were summarized in Figure 9simFigure 10

From Figure 9(a) we can see the deformation of thecolumn exhibits apparent rheology the rheological deforma-tion takes about 37 of the whole deformation Figure 9(b)shows that the static yield surface gradually expanded withthe time until it reached steady state when it coincided withthe dynamic yield surface From Figures 9(c) and 9(d) wecan see that the viscoelastic strain and the viscoplastic strainboth gradually increased with time until they reached theirsteady state accordingly the viscoelastic strain rate and theviscoplastic strain rate were gradually deduced to nearly zeroThe above results fit the property of Nishihara model andthe hardening law of modified Cambridge model very wellwhich proved the correctness of the UMAT

In the proposed model 1205781 1205782 and 119901

0are the most

important parameters Figure 10 showed the sensitivity ofthese parameters From Figures 10(a) and 10(b) we can seethe final viscous deformation is a constant regardless of thedifferent values of 120578

1 1205782 but the time when it reaches steady

varies greatly so the accuracy of 1205781and 1205782is directly related

to the accuracy of computed displacement in the rheologicalprocess Figure 10(c) shows that the value of 119901

0has a great

influence on the viscoplastic strain the smaller 1199010is the

more rapidly the viscoplastic strain develops and the largerthe final viscoplastic strain occurs and so the accuracy of1199010is not only directly related to the accuracy of computed

displacement in the rheological process but also affected thefinal displacement which needs to be paid more attention

5 Analysis on Rheological Consolidation ofSoft Clay

51 Computed Model Consolidation problems can be classi-fied as small strain theory and finite strain theory accordingto the difference of geometric assumption Small straintheory assumed that the strain is little but the consolida-tion deformation is usually very large for deep soft clayfoundation Take dredger fill for example the deformationmay exceed 80 for which the small strain theory is nolonger suitable [14 15] As the pioneer researchers studiedon finite strain consolidation theory Mikasa [16] and Gib-son et al [17] derived their one-dimensional finite strainconsolidation equations respectively in 1960s and manyscholars developed further research on the topic in thefollowing decades Because of strong nonlinearity most ofexisting researches on finite strain consolidation adoptedsome simplified assumptionswithout considering rheologicalcharacteristics By means of ABAQUS this paper studied therheological consolidation of soft clay considering finite straineffect

The computed model is plane strain problem as shown inFigure 11 The soil foundation is homogeneous anisotropicsaturated and normally consolidated soft clay with a thick-ness of 100m and only the top boundary is permeable Inaddition the semilogarithmic empirical formula proposed byTavenas et al [18] was adopted to consider the variation ofpermeability in the process of consolidation as follows

119896V = 119896V0 sdot 10(119890minus1198900)119862119896 (31)

where 119896V0 is the initial permeability coefficients and 119896V is thepermeability coefficient in the consolidation process 119890

0is the

initial void ratio and 119890 is the void ratio in the consolidationprocess 119862

119896is the permeability index According to the

above work themodel parameters are summarized in Table 8approximately

52 Consolidation Property of Soft Clay From Figure 12 wecan see the differences of the consolidation process betweenfinite strain and small strain Figure 12(a) shows that for anylocation the excess pore pressure of finite strain is less thanthat of small strain at the same time and the difference isincreasing with the increase of time in beginning period

Mathematical Problems in Engineering 11

Table 8 Material parameters of computed model

1198640(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots) 120574sat (kNsdotm

minus3) 1198961199070(msdotsminus1) 119875

0(kPa) 119872 120582 120581 119890

0120583 119862

119896

25 115 150 24000 177 45119864 minus 9 50 0898 0102 0015 108 035 05

100000225

00200

00175

00150

00125

00100

Vert

ical

def

orm

atio

n (m

)

Time (s)10000 100000 1000000 1E7

(a)

40

50

60

70

80

90

100

110

120

1000Time (s)

10000 100000 1000000 1E7

Pcs

(kPa

)(b)

05

00

10

15

20

25

30

35

Viscoelastic strain

Visc

ous s

trai

n (

)

Viscoplastic strain

1000

Time (s)

10000 100000 1000000 1E71E7

(c)

Viscoelastic strain rateViscoplastic strain rate

0

2

4

6

8

10

12

14

16

1000Time (s)

10000 100000 1000000 1E71E7

Visc

ous s

trai

n ra

te (s

minus1)

(d)

Figure 9 Triaxial compression property of soil column

Figure 12(b) shows that the displacement of finite strain isless than that of small strain and the difference increasedsignificantly if the load is increasing For example when 119902 =200 kPa the differencemay take about 20while it is nearly 0when 119902= 50 kPa So it is essential to consider the error causedby small strain if the load is large and the compressibility ofthe soil is large

From Figure 13 we can see the influence of rheologyon the consolidation process Figure 13(a) shows that thedissipation of excess pore pressure is significantly slower ifthe rheological effect is considered for example when 119905 =1000 d the excess pore pressure is nearly 0 if the rheologicaleffect is not considered but it is still larger than 20 kPa ifthe rheological effect is considered Figure 13(a) shows that

the displacement is influenced by rheological effect too forexample in the early stage (119905 le 100 d) the displacement isnearly the same for two cases but the final displacementmay have a difference of more than 35 which indicatesthat the displacement in the early stage is mainly causedby consolidation while the displacement in the late stage ismainly caused by rheology

6 Conclusions

(1) An elastic-viscoplastic model developed to describethe time-dependent behavior of normally consoli-dated soft clay was presented in the paper on the

12 Mathematical Problems in Engineering

00

05

10

15

20

25

30Vi

scoe

lasti

c str

ain

()

100010 100Time (s)

10000 100000 1000000 1E7

1205781 = 10GPamiddots1205781 = 100GPamiddots1205781 = 1000GPamiddots

(a)

00

02

04

06

08

10

12

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

1205782 = 10GPamiddots1205782 = 100GPamiddots1205782 = 1000GPamiddots

(b)

00

04

08

12

16

20

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

P0 = 30kPaP0 = 40kPa

P0 = 50kPaP0 = 60kPa

(c)

Figure 10 Sensitivity of model parameters

O

Permeable

B = 5 mx

z

H = 10m

q = 100 kPa

Figure 11 Calculation model and mesh generation

framework of modified cam-clay elastic plastic modeland Perzynarsquos overstress viscoplastic theory

(2) A series of laboratory triaxial rheological tests ofNingbo soft clay with different confining pressure anddeviatoric stress are performed the results show thatthe presented constitutive model was suitable for therheological characteristic of Ningbo soft clay and it isachievable to determine the parameters of presentedmodel by curve fitting

(3) The analysis on the model parameters shows that thevalue of all parameters is related to confining pressureand deviatoric stress It may need further study on therelationship between parameters and the stress levelso as to use the model conveniently

Mathematical Problems in Engineering 13

0 20 40 60 80 100u (kPa)

0

2

4

6

8

10H

(m)

FS t = 100dSS t = 100dFS t = 500d

SS t = 500dFS t = 1000 dSS t = 1000 d

(a)

12

10

08

06

04

02

00

Def

orm

atio

n (m

)

Time (d)1000101 100 10000 100000

SS q= 100 kPaFS q= 50 kPaSS q= 50 kPa

SS q= 200 kPaFS q= 200 kPa

FS q= 100 kPa

(b)

Figure 12 Results comparison of large strain and small strain (SS small train FS finite strain)

00

2

4

6

8

10

H(m

)

20 40 60 80 100u (kPa)

t = 100d NRt = 100d Rt = 500d NR

t = 500d Rt = 1000 d NRt = 1000 d R

(a)

06

05

04

03

02

01

00

Def

orm

atio

n (m

)

Time (d)

NRR

1000101 100 10000 100000

(b)

Figure 13 Effects of rheology on calculated results (NR rheological effect not considered R rheological effect considered)

(4) In the proposed model the viscoelastic viscositycoefficient 120578

1 the viscoplastic viscosity coefficient 120578

2

and the initial yield surface 1199010are the most important

parameters 1205781and 120578

2have important influence on

the time of viscous strain developed and 1199010have

important influence on both the viscoplastic strainrate and the final viscoplastic strain

(5) The existence of rheology will significantly slow downthe dissipation of the excess pore pressure Consid-ering the rheological effect the total displacement

is larger and the displacement in the early stage ismainly caused by consolidation while in the late stageit is mainly caused by rheology

(6) For consolidation problems of high compressibilitysoft clay foundation great error will be caused bysmall strain assumption so finite strain effect shouldbe considered When finite strain effect is consideredthe consolidation process is developed more rapidlyand the computed consolidation displacement will beless than that of small strain

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 7 Material parameters of verification model

1198640(MPa) 119864

1(MPa) 120578

1(GPasdots) 120578

2(GPasdots) 119875

0(kPa) 119890

0120583 119872 120582 120581

10 20 1000 1000 50 108 035 0898 0102 0015

12059021205902

1205903

1205903

1205901

1205901

Figure 8 Sketch of verification model

Equation (29) is the relationship between stress incre-ment and strain increment in which matrix D is defined asfollows

D = [(D119890)minus1 + C119899+ ΘΔ119905

119899H119899]

minus1

(30)

Based on the above equations a user material subroutinewas coded for the proposed constitutive model on theplatform of ABAQUS

42 Verification of UMAT In order to verify the validity ofthe UMAT we study a soil column of triaxial compressionsuch as Figure 8 the size of the column is 01m times 01m times

02m Supposing that 1205901= 100 kPa 120590

2= 1205903= 50 kPa

and the model parameters are given in Table 7 The columnwas cut into one element the element type is C3D8 and theconsolidation effect was not considered here Embedding theUMAT coded in this paper on the platform of ABAQUS wecalculated the triaxial compression property of the columnthe results were summarized in Figure 9simFigure 10

From Figure 9(a) we can see the deformation of thecolumn exhibits apparent rheology the rheological deforma-tion takes about 37 of the whole deformation Figure 9(b)shows that the static yield surface gradually expanded withthe time until it reached steady state when it coincided withthe dynamic yield surface From Figures 9(c) and 9(d) wecan see that the viscoelastic strain and the viscoplastic strainboth gradually increased with time until they reached theirsteady state accordingly the viscoelastic strain rate and theviscoplastic strain rate were gradually deduced to nearly zeroThe above results fit the property of Nishihara model andthe hardening law of modified Cambridge model very wellwhich proved the correctness of the UMAT

In the proposed model 1205781 1205782 and 119901

0are the most

important parameters Figure 10 showed the sensitivity ofthese parameters From Figures 10(a) and 10(b) we can seethe final viscous deformation is a constant regardless of thedifferent values of 120578

1 1205782 but the time when it reaches steady

varies greatly so the accuracy of 1205781and 1205782is directly related

to the accuracy of computed displacement in the rheologicalprocess Figure 10(c) shows that the value of 119901

0has a great

influence on the viscoplastic strain the smaller 1199010is the

more rapidly the viscoplastic strain develops and the largerthe final viscoplastic strain occurs and so the accuracy of1199010is not only directly related to the accuracy of computed

displacement in the rheological process but also affected thefinal displacement which needs to be paid more attention

5 Analysis on Rheological Consolidation ofSoft Clay

51 Computed Model Consolidation problems can be classi-fied as small strain theory and finite strain theory accordingto the difference of geometric assumption Small straintheory assumed that the strain is little but the consolida-tion deformation is usually very large for deep soft clayfoundation Take dredger fill for example the deformationmay exceed 80 for which the small strain theory is nolonger suitable [14 15] As the pioneer researchers studiedon finite strain consolidation theory Mikasa [16] and Gib-son et al [17] derived their one-dimensional finite strainconsolidation equations respectively in 1960s and manyscholars developed further research on the topic in thefollowing decades Because of strong nonlinearity most ofexisting researches on finite strain consolidation adoptedsome simplified assumptionswithout considering rheologicalcharacteristics By means of ABAQUS this paper studied therheological consolidation of soft clay considering finite straineffect

The computed model is plane strain problem as shown inFigure 11 The soil foundation is homogeneous anisotropicsaturated and normally consolidated soft clay with a thick-ness of 100m and only the top boundary is permeable Inaddition the semilogarithmic empirical formula proposed byTavenas et al [18] was adopted to consider the variation ofpermeability in the process of consolidation as follows

119896V = 119896V0 sdot 10(119890minus1198900)119862119896 (31)

where 119896V0 is the initial permeability coefficients and 119896V is thepermeability coefficient in the consolidation process 119890

0is the

initial void ratio and 119890 is the void ratio in the consolidationprocess 119862

119896is the permeability index According to the

above work themodel parameters are summarized in Table 8approximately

52 Consolidation Property of Soft Clay From Figure 12 wecan see the differences of the consolidation process betweenfinite strain and small strain Figure 12(a) shows that for anylocation the excess pore pressure of finite strain is less thanthat of small strain at the same time and the difference isincreasing with the increase of time in beginning period

Mathematical Problems in Engineering 11

Table 8 Material parameters of computed model

1198640(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots) 120574sat (kNsdotm

minus3) 1198961199070(msdotsminus1) 119875

0(kPa) 119872 120582 120581 119890

0120583 119862

119896

25 115 150 24000 177 45119864 minus 9 50 0898 0102 0015 108 035 05

100000225

00200

00175

00150

00125

00100

Vert

ical

def

orm

atio

n (m

)

Time (s)10000 100000 1000000 1E7

(a)

40

50

60

70

80

90

100

110

120

1000Time (s)

10000 100000 1000000 1E7

Pcs

(kPa

)(b)

05

00

10

15

20

25

30

35

Viscoelastic strain

Visc

ous s

trai

n (

)

Viscoplastic strain

1000

Time (s)

10000 100000 1000000 1E71E7

(c)

Viscoelastic strain rateViscoplastic strain rate

0

2

4

6

8

10

12

14

16

1000Time (s)

10000 100000 1000000 1E71E7

Visc

ous s

trai

n ra

te (s

minus1)

(d)

Figure 9 Triaxial compression property of soil column

Figure 12(b) shows that the displacement of finite strain isless than that of small strain and the difference increasedsignificantly if the load is increasing For example when 119902 =200 kPa the differencemay take about 20while it is nearly 0when 119902= 50 kPa So it is essential to consider the error causedby small strain if the load is large and the compressibility ofthe soil is large

From Figure 13 we can see the influence of rheologyon the consolidation process Figure 13(a) shows that thedissipation of excess pore pressure is significantly slower ifthe rheological effect is considered for example when 119905 =1000 d the excess pore pressure is nearly 0 if the rheologicaleffect is not considered but it is still larger than 20 kPa ifthe rheological effect is considered Figure 13(a) shows that

the displacement is influenced by rheological effect too forexample in the early stage (119905 le 100 d) the displacement isnearly the same for two cases but the final displacementmay have a difference of more than 35 which indicatesthat the displacement in the early stage is mainly causedby consolidation while the displacement in the late stage ismainly caused by rheology

6 Conclusions

(1) An elastic-viscoplastic model developed to describethe time-dependent behavior of normally consoli-dated soft clay was presented in the paper on the

12 Mathematical Problems in Engineering

00

05

10

15

20

25

30Vi

scoe

lasti

c str

ain

()

100010 100Time (s)

10000 100000 1000000 1E7

1205781 = 10GPamiddots1205781 = 100GPamiddots1205781 = 1000GPamiddots

(a)

00

02

04

06

08

10

12

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

1205782 = 10GPamiddots1205782 = 100GPamiddots1205782 = 1000GPamiddots

(b)

00

04

08

12

16

20

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

P0 = 30kPaP0 = 40kPa

P0 = 50kPaP0 = 60kPa

(c)

Figure 10 Sensitivity of model parameters

O

Permeable

B = 5 mx

z

H = 10m

q = 100 kPa

Figure 11 Calculation model and mesh generation

framework of modified cam-clay elastic plastic modeland Perzynarsquos overstress viscoplastic theory

(2) A series of laboratory triaxial rheological tests ofNingbo soft clay with different confining pressure anddeviatoric stress are performed the results show thatthe presented constitutive model was suitable for therheological characteristic of Ningbo soft clay and it isachievable to determine the parameters of presentedmodel by curve fitting

(3) The analysis on the model parameters shows that thevalue of all parameters is related to confining pressureand deviatoric stress It may need further study on therelationship between parameters and the stress levelso as to use the model conveniently

Mathematical Problems in Engineering 13

0 20 40 60 80 100u (kPa)

0

2

4

6

8

10H

(m)

FS t = 100dSS t = 100dFS t = 500d

SS t = 500dFS t = 1000 dSS t = 1000 d

(a)

12

10

08

06

04

02

00

Def

orm

atio

n (m

)

Time (d)1000101 100 10000 100000

SS q= 100 kPaFS q= 50 kPaSS q= 50 kPa

SS q= 200 kPaFS q= 200 kPa

FS q= 100 kPa

(b)

Figure 12 Results comparison of large strain and small strain (SS small train FS finite strain)

00

2

4

6

8

10

H(m

)

20 40 60 80 100u (kPa)

t = 100d NRt = 100d Rt = 500d NR

t = 500d Rt = 1000 d NRt = 1000 d R

(a)

06

05

04

03

02

01

00

Def

orm

atio

n (m

)

Time (d)

NRR

1000101 100 10000 100000

(b)

Figure 13 Effects of rheology on calculated results (NR rheological effect not considered R rheological effect considered)

(4) In the proposed model the viscoelastic viscositycoefficient 120578

1 the viscoplastic viscosity coefficient 120578

2

and the initial yield surface 1199010are the most important

parameters 1205781and 120578

2have important influence on

the time of viscous strain developed and 1199010have

important influence on both the viscoplastic strainrate and the final viscoplastic strain

(5) The existence of rheology will significantly slow downthe dissipation of the excess pore pressure Consid-ering the rheological effect the total displacement

is larger and the displacement in the early stage ismainly caused by consolidation while in the late stageit is mainly caused by rheology

(6) For consolidation problems of high compressibilitysoft clay foundation great error will be caused bysmall strain assumption so finite strain effect shouldbe considered When finite strain effect is consideredthe consolidation process is developed more rapidlyand the computed consolidation displacement will beless than that of small strain

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 8 Material parameters of computed model

1198640(MPa) 119864

1(MPa) 120578

1(MPasdots) 120578

2(GPasdots) 120574sat (kNsdotm

minus3) 1198961199070(msdotsminus1) 119875

0(kPa) 119872 120582 120581 119890

0120583 119862

119896

25 115 150 24000 177 45119864 minus 9 50 0898 0102 0015 108 035 05

100000225

00200

00175

00150

00125

00100

Vert

ical

def

orm

atio

n (m

)

Time (s)10000 100000 1000000 1E7

(a)

40

50

60

70

80

90

100

110

120

1000Time (s)

10000 100000 1000000 1E7

Pcs

(kPa

)(b)

05

00

10

15

20

25

30

35

Viscoelastic strain

Visc

ous s

trai

n (

)

Viscoplastic strain

1000

Time (s)

10000 100000 1000000 1E71E7

(c)

Viscoelastic strain rateViscoplastic strain rate

0

2

4

6

8

10

12

14

16

1000Time (s)

10000 100000 1000000 1E71E7

Visc

ous s

trai

n ra

te (s

minus1)

(d)

Figure 9 Triaxial compression property of soil column

Figure 12(b) shows that the displacement of finite strain isless than that of small strain and the difference increasedsignificantly if the load is increasing For example when 119902 =200 kPa the differencemay take about 20while it is nearly 0when 119902= 50 kPa So it is essential to consider the error causedby small strain if the load is large and the compressibility ofthe soil is large

From Figure 13 we can see the influence of rheologyon the consolidation process Figure 13(a) shows that thedissipation of excess pore pressure is significantly slower ifthe rheological effect is considered for example when 119905 =1000 d the excess pore pressure is nearly 0 if the rheologicaleffect is not considered but it is still larger than 20 kPa ifthe rheological effect is considered Figure 13(a) shows that

the displacement is influenced by rheological effect too forexample in the early stage (119905 le 100 d) the displacement isnearly the same for two cases but the final displacementmay have a difference of more than 35 which indicatesthat the displacement in the early stage is mainly causedby consolidation while the displacement in the late stage ismainly caused by rheology

6 Conclusions

(1) An elastic-viscoplastic model developed to describethe time-dependent behavior of normally consoli-dated soft clay was presented in the paper on the

12 Mathematical Problems in Engineering

00

05

10

15

20

25

30Vi

scoe

lasti

c str

ain

()

100010 100Time (s)

10000 100000 1000000 1E7

1205781 = 10GPamiddots1205781 = 100GPamiddots1205781 = 1000GPamiddots

(a)

00

02

04

06

08

10

12

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

1205782 = 10GPamiddots1205782 = 100GPamiddots1205782 = 1000GPamiddots

(b)

00

04

08

12

16

20

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

P0 = 30kPaP0 = 40kPa

P0 = 50kPaP0 = 60kPa

(c)

Figure 10 Sensitivity of model parameters

O

Permeable

B = 5 mx

z

H = 10m

q = 100 kPa

Figure 11 Calculation model and mesh generation

framework of modified cam-clay elastic plastic modeland Perzynarsquos overstress viscoplastic theory

(2) A series of laboratory triaxial rheological tests ofNingbo soft clay with different confining pressure anddeviatoric stress are performed the results show thatthe presented constitutive model was suitable for therheological characteristic of Ningbo soft clay and it isachievable to determine the parameters of presentedmodel by curve fitting

(3) The analysis on the model parameters shows that thevalue of all parameters is related to confining pressureand deviatoric stress It may need further study on therelationship between parameters and the stress levelso as to use the model conveniently

Mathematical Problems in Engineering 13

0 20 40 60 80 100u (kPa)

0

2

4

6

8

10H

(m)

FS t = 100dSS t = 100dFS t = 500d

SS t = 500dFS t = 1000 dSS t = 1000 d

(a)

12

10

08

06

04

02

00

Def

orm

atio

n (m

)

Time (d)1000101 100 10000 100000

SS q= 100 kPaFS q= 50 kPaSS q= 50 kPa

SS q= 200 kPaFS q= 200 kPa

FS q= 100 kPa

(b)

Figure 12 Results comparison of large strain and small strain (SS small train FS finite strain)

00

2

4

6

8

10

H(m

)

20 40 60 80 100u (kPa)

t = 100d NRt = 100d Rt = 500d NR

t = 500d Rt = 1000 d NRt = 1000 d R

(a)

06

05

04

03

02

01

00

Def

orm

atio

n (m

)

Time (d)

NRR

1000101 100 10000 100000

(b)

Figure 13 Effects of rheology on calculated results (NR rheological effect not considered R rheological effect considered)

(4) In the proposed model the viscoelastic viscositycoefficient 120578

1 the viscoplastic viscosity coefficient 120578

2

and the initial yield surface 1199010are the most important

parameters 1205781and 120578

2have important influence on

the time of viscous strain developed and 1199010have

important influence on both the viscoplastic strainrate and the final viscoplastic strain

(5) The existence of rheology will significantly slow downthe dissipation of the excess pore pressure Consid-ering the rheological effect the total displacement

is larger and the displacement in the early stage ismainly caused by consolidation while in the late stageit is mainly caused by rheology

(6) For consolidation problems of high compressibilitysoft clay foundation great error will be caused bysmall strain assumption so finite strain effect shouldbe considered When finite strain effect is consideredthe consolidation process is developed more rapidlyand the computed consolidation displacement will beless than that of small strain

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

00

05

10

15

20

25

30Vi

scoe

lasti

c str

ain

()

100010 100Time (s)

10000 100000 1000000 1E7

1205781 = 10GPamiddots1205781 = 100GPamiddots1205781 = 1000GPamiddots

(a)

00

02

04

06

08

10

12

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

1205782 = 10GPamiddots1205782 = 100GPamiddots1205782 = 1000GPamiddots

(b)

00

04

08

12

16

20

Visc

opla

stic s

trai

n (

)

100010 100Time (s)

10000 100000 1000000 1E7

P0 = 30kPaP0 = 40kPa

P0 = 50kPaP0 = 60kPa

(c)

Figure 10 Sensitivity of model parameters

O

Permeable

B = 5 mx

z

H = 10m

q = 100 kPa

Figure 11 Calculation model and mesh generation

framework of modified cam-clay elastic plastic modeland Perzynarsquos overstress viscoplastic theory

(2) A series of laboratory triaxial rheological tests ofNingbo soft clay with different confining pressure anddeviatoric stress are performed the results show thatthe presented constitutive model was suitable for therheological characteristic of Ningbo soft clay and it isachievable to determine the parameters of presentedmodel by curve fitting

(3) The analysis on the model parameters shows that thevalue of all parameters is related to confining pressureand deviatoric stress It may need further study on therelationship between parameters and the stress levelso as to use the model conveniently

Mathematical Problems in Engineering 13

0 20 40 60 80 100u (kPa)

0

2

4

6

8

10H

(m)

FS t = 100dSS t = 100dFS t = 500d

SS t = 500dFS t = 1000 dSS t = 1000 d

(a)

12

10

08

06

04

02

00

Def

orm

atio

n (m

)

Time (d)1000101 100 10000 100000

SS q= 100 kPaFS q= 50 kPaSS q= 50 kPa

SS q= 200 kPaFS q= 200 kPa

FS q= 100 kPa

(b)

Figure 12 Results comparison of large strain and small strain (SS small train FS finite strain)

00

2

4

6

8

10

H(m

)

20 40 60 80 100u (kPa)

t = 100d NRt = 100d Rt = 500d NR

t = 500d Rt = 1000 d NRt = 1000 d R

(a)

06

05

04

03

02

01

00

Def

orm

atio

n (m

)

Time (d)

NRR

1000101 100 10000 100000

(b)

Figure 13 Effects of rheology on calculated results (NR rheological effect not considered R rheological effect considered)

(4) In the proposed model the viscoelastic viscositycoefficient 120578

1 the viscoplastic viscosity coefficient 120578

2

and the initial yield surface 1199010are the most important

parameters 1205781and 120578

2have important influence on

the time of viscous strain developed and 1199010have

important influence on both the viscoplastic strainrate and the final viscoplastic strain

(5) The existence of rheology will significantly slow downthe dissipation of the excess pore pressure Consid-ering the rheological effect the total displacement

is larger and the displacement in the early stage ismainly caused by consolidation while in the late stageit is mainly caused by rheology

(6) For consolidation problems of high compressibilitysoft clay foundation great error will be caused bysmall strain assumption so finite strain effect shouldbe considered When finite strain effect is consideredthe consolidation process is developed more rapidlyand the computed consolidation displacement will beless than that of small strain

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

0 20 40 60 80 100u (kPa)

0

2

4

6

8

10H

(m)

FS t = 100dSS t = 100dFS t = 500d

SS t = 500dFS t = 1000 dSS t = 1000 d

(a)

12

10

08

06

04

02

00

Def

orm

atio

n (m

)

Time (d)1000101 100 10000 100000

SS q= 100 kPaFS q= 50 kPaSS q= 50 kPa

SS q= 200 kPaFS q= 200 kPa

FS q= 100 kPa

(b)

Figure 12 Results comparison of large strain and small strain (SS small train FS finite strain)

00

2

4

6

8

10

H(m

)

20 40 60 80 100u (kPa)

t = 100d NRt = 100d Rt = 500d NR

t = 500d Rt = 1000 d NRt = 1000 d R

(a)

06

05

04

03

02

01

00

Def

orm

atio

n (m

)

Time (d)

NRR

1000101 100 10000 100000

(b)

Figure 13 Effects of rheology on calculated results (NR rheological effect not considered R rheological effect considered)

(4) In the proposed model the viscoelastic viscositycoefficient 120578

1 the viscoplastic viscosity coefficient 120578

2

and the initial yield surface 1199010are the most important

parameters 1205781and 120578

2have important influence on

the time of viscous strain developed and 1199010have

important influence on both the viscoplastic strainrate and the final viscoplastic strain

(5) The existence of rheology will significantly slow downthe dissipation of the excess pore pressure Consid-ering the rheological effect the total displacement

is larger and the displacement in the early stage ismainly caused by consolidation while in the late stageit is mainly caused by rheology

(6) For consolidation problems of high compressibilitysoft clay foundation great error will be caused bysmall strain assumption so finite strain effect shouldbe considered When finite strain effect is consideredthe consolidation process is developed more rapidlyand the computed consolidation displacement will beless than that of small strain

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

Conflict of Interests

Theauthors do not have any conflict of interests regarding thecontents of the paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (Grant no 51378469) and theNingbo Municipal Natural Science Foundation (Grant no2013A610196) The authors wish to express their gratitudeto the above for financial support (1) Project (51378469) issupported by the National Natural Science Foundation ofChina (2) Project (2013A610196) is supported by the NaturalScience Foundation of Ningbo City China

References

[1] L Zeevaert Consolidation of Mexico City Volcanic Clay Confer-ence on Soils For Engineering Purpose ASTM Philadelphia PaUSA 1958

[2] G A Leonards and B K Ramiah Time Effects in the Consoli-dation of Clays Symposium on Time Rate of Loading in TestingSoils ASTM Philadelphia Pa USA 1960

[3] A Singh and J KMitchell ldquoGeneral stress-strain-time functionfor clayrdquo Journal of the ClayMechanics and Foundation DivisionASCE vol 94 no 1 pp 21ndash46 1968

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

[5] GMesri E Febres-CorderoD R Shields andACastro ldquoShearstress strain time behaviour of claysrdquo Geotechnique vol 31 no4 pp 537ndash552 1981

[6] T Adachi and F Oka ldquoConstitutive equations for normallyconsolidation clay based on elastic-viscoplasticrdquo Soils andFoundations vol 22 no 4 pp 57ndash70 1982

[7] A Fodil W Aloulou and P Y Hicher ldquoViscoplastic behaviourof soft clayrdquo Geotechnique vol 47 no 3 pp 581ndash591 1997

[8] Z Y Yin P Y Hicher Y Riou and H W Huang ldquoAn elasto-viscoplastic model for soft clayrdquo in Proceedings of the Geoshang-hai Conference of Soil and Rock Behavior andModeling pp 312ndash319 Geotechnical Special Publication Shanghai China June2006

[9] Z Y Yin and P Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressuremeter testingrdquo International Journal for Numerical and Ana-lytical Methods in Geomechanics vol 32 no 12 pp 1515ndash15352008

[10] S D Hinchberger and R K Rowe ldquoEvaluation of the predictiveability of two elastic-viscoplastic constitutive modelsrdquo Cana-dian Geotechnical Journal vol 42 no 6 pp 1675ndash1694 2005

[11] S D Hinchberger and G Qu ldquoViscoplastic constitutive ap-proach for ratesensitive structured claysrdquo Canadian Geotechni-cal Journal vol 46 no 6 pp 609ndash626 2009

[12] P Perzyna ldquoFundamental problems in viscoplasticityrdquoAdvancesin Applied Mechanics vol 9 pp 243ndash377 1966

[13] L M Wei Double non-linear fluid-solid coupled simulatinganalysis and settlement prediction for soft soil subgrade [PhDthesis] Central South University Changsha China

[14] W G Weber ldquoPerformance of embankments constructed overpeatrdquo Journal of Soil Mechanics and Foundations ASCE vol 95no 1 pp 53ndash76 1969

[15] K W Cargill ldquoPrediction of consolidation of very soft soilrdquoJournal of Geotechnical Engineering ASCE vol 110 no 6 pp775ndash795 1984

[16] M Mikasa ldquoThe consolidation of soft claymdasha new consolida-tion theory and its applicationrdquo Civil Engineering in JapanaJSCE vol 1 no 1 pp 21ndash26 1965

[17] R E Gibson G L England and M J L Hussey ldquoThetheory of one-dimensional consolidation of saturated clays IIfinite nonlinear consolidation of thick homogeneous layersrdquoGeotechnique vol 17 no 2 pp 261ndash237 1967

[18] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of