J. Cent. South Univ. (2013) 20: 3305−3313 DOI: 10.1007/s11771-013-1854-7

Numerical and theoretical verification of modified cam-clay model and discussion on its problems

DAI Zi-hang(戴自航)1, 2, QIN Zhi-zhong(秦志忠)1

1. Institute of Geotechnical Engineering, Fuzhou University, Fuzhou 350108, China;

2. School of Civil, Mining and Enviromental Engineering, University of Wollongong, Wollongong 2522, Australia

© Central South University Press and Springer-Verlag Berlin Heidelberg 2013

Abstract: Isotropic consolidation test and consolidated-undrained triaxial test were first undertaken to obtain the parameters of the modified cam-clay (MCC) model and the behavior of natural clayey soil. Then, for the first time, numerical simulation of the two tests was performed by three-dimensional finite element method (FEM) using ABAQUS program. The consolidated-drained triaxial test was also simulated by FEM and compared with theoretical results of MCC model. Especially, the behaviors of MCC model during unloading and reloading were analyzed in detail by FEM. The analysis and comparison indicate that the MCC model is able to accurately describe many features of the mechanical behavior of the soil in isotropic consolidation test and consolidated-drained triaxial test. And the MCC model can well describe the variation of excess pore water pressure with the development of axial strain in consolidated-undrained triaxial test, but its ability to predict the relationship between axial strain and shear stress is relatively poor. The comparison also shows that FEM solutions of the MCC model are basically identical to the theoretical ones. In addition, Mandel-Cryer effect unable to be discovered by the conventional triaxial test in laboratories was disclosed by FEM. The analysis of unloading-reloading by FEM demonstrates that the MCC model disobeys the law of energy conservation under the cyclic loading condition if the elastic shear modulus is linearly pressure-dependent. Key words: modified cam-clay (MCC) model; isotropic consolidation test; consolidated-undrained triaxial test; consolidated-drained triaxial test; Mandel-Cryer effect; energy conversation

1 Introduction

The cam-clay model proposed by ROSCOE et al [1−2] is the first critical-state model for describing mechanical behavior of soft soils. It is originally developed for reconstituted clayey soils under triaxial loading conditions. ROSCOE and BURLAND [3] amended the shape of the yield curve and obtained the modified cam-clay (MCC) model. The MCC model is based on few parameters which can be obtained from conventional laboratory tests [4]. It is good for representing normal to moderately overconsolidated soils. Recently, many researchers have improved the MCC model to make it compatible for overconsolidated soils [5−8]. Moreover, it can predict the pressure-dependent soil strength as well as the compression and dilatancy caused by shearing [9]. All these advantages make the MCC model popularly used in geotechnical problems [10−12]. However, the properties of natural soil are more complicated than reconstituted soil in aspects such as anisotropy, structure and viscosity [13−15], of which the MCC model could not consider. On the other hand, the MCC model is based on the results of isotropic

compression test, but clay is usually K0-consolidated in practice. PARRY and NADARAJAH [16] and TAVENAS and LEROUEIL [17] have evidenced that the yield surface is centered on K0 consolidation line in stress space. For simplicity, the original critical-state was formulated without considering the elastic shear strain, i.e., assuming an infinite shear modulus. However, it is in contrast with the experimental response of soil [18]. There are two commonly used forms of shear modulus. One is to take a constant shear modulus, but it also does not agree well with the experimental results and may lead to a negative Poisson ratio [19]. Another is to adopt a variable shear modulus which is linearly pressure- dependent and related through a constant Poisson ratio. However, it leads to a non-conservative stress−strain relation [20].

In order to examine the modified cam-clay constitutive relation, triaxial tests were undertaken on natural clayey soil and the numerical method as well as theoretical method was used for comparison in this work. Therein, the three-dimensional (3D) numerical analysis was conducted using ABAQUS. In particular, the dissipation of excess pore water pressures in the process of consolidation and the behavior of soil under cyclic

Foundation item: Project(2011J01308) supported by the Natural Science Foundation of Fujian Province, China Received date: 2012−07−02; Accepted date: 2012−11−20 Corresponding author: DAI Zi-hang, PhD, Professor; Tel: +86−13625098017; E-mail: Mr.dai@163.com

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loading were also analyzed by finite element method. Some of existing problems of MCC model were also discussed in this work. 2 Brief introduction of modified cam-clay

model

As shown in Fig. 1, in the q−p′ plane, the yield surface of modified cam-clay is given by

2 2 2

c c1 1

2 2

qp p p

M

(1)

where p′ is the mean effective stress, and q is the deviator stress (shear stress). In conventional axisymmetric triaxial stress condition, p′ and q can be calculated as

1 31

( 2 )3

p (2)

1 3q (3)

In Eq. (1), cp is used to specify the size of the yield

surface; it controls the hardening behavior as a function of the volumetric plastic strain

pv .

pcv

c

d 1d

p e

p

(4)

where λ is the slope of the normal consolidation line (NCL) and κ is the slope of the unloading−reloading line in the e−ln p′ plane. M is the slope of the critial-state line (CSL) and is defined as

f

f

qM

p

(5)

where fp is the mean effective stress at failure, and qf is the shear stress at failure (i.e., the shear strength). Equation (5) therefore provides the failure criterion.

The associated flow rule is adopted and the plastic volumetric strain increment can be calculated as

pv 2 2

d 2 dd

1

p

e p M

(6)

where /q p is the stress ratio. The plastic shear strain increment is calculated as

ps 2 2 2 2

2 d 2 dd

1

p

e pM M

(7)

In the modified cam-clay model, the elastic

behavior of soil is nonlinear and pressure-dependent. The bulk modulus K is related to p′ and e by

(1 )e p

K

(8)

Therefore,

Fig. 1 Illustration of modified cam-clay model

ev

d dd

1

p p

K e p

(9)

where

evd is the elastic volumetric strain increment. The

shear modulus G is given by

3(1 2 ) (1 )

2(1 )

e pG

(10)

where is Poisson ratio that keeps constant. Note that G is not a constant. Therefore,

es

d 2(1 ) dd

3 9(1 2 )(1 )

q q

G e p

(11)

where

esd is the elastic shear strain increment.

There are three material parameters of λ, κ and M besides the elastic moduli in the MCC model. λ and κ can be determined from the isotropic consolidation test, and M can be obtained from the results of consolidated- drained (CD) or consolidated-undrained (CU) triaxial compression tests. 3 Materials and test

The soil used in the present work is soft marine clay taken at depth of 6.8 m from the Putian-Yongding Expressway in Fujian Province, China. The cylindrical soil samples in all tests are 80 mm in height and 39.1 mm in diameter. Basic physical properties of the soil are listed as follows: unit weight γ=17.4 kN/m3, water content w=52%, void ratio e=1.54, liquid limit wL=41%, plastic limit wP=20%, and plastic index IP=21%.

Two types of tests were undertaken in the present work: isotropic consolidation test and consolidated- undrained (CU) triaxial test. All tests were conducted on

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the Model SLB-1 Stress and Strain Controlled Triaxial Shear Permeation Test Apparatus (Fig. 2). A back pressure of 300 kPa was applied to ensure saturation of the specimens.

Fig. 2 Model SLB-1 Stress & Strain Controlled Triaxial Shear

Permeation Test Apparatus

An isotropic consolidation test was performed to

determine the material parameters of λ and κ. The test contains three procedures, i.e., loading, unloading and reloading. During loading, the soil specimens were consolidated by steps at different effective confining stresses of 100, 150, 230, 350 and 600 kPa. During unloading, the confining stress decreases by steps according to 600, 350, 150 and 100 kPa. During reloading, the confining stress increases by steps according to 100, 150, 350, 600 and 900 kPa. The relationship between void ratio e and ln p′ (natural logarithm of mean effective stress) is shown in Fig. 3. As predicted, the normal consolidation line (λ-line) can be represented by a straight line. On the other hand, another straight line which is called unloading−reloading line (κ-line) is taken to describe the unloading−reloading behavior.

Three CU triaxial tests (CU1, CU2 and CU3) were carried out to determine the parameter M. The initial

Fig. 3 λ-line and κ-line determined from isotropic consolidation

test results

mean effective confining stresses for CU1, CU2 and CU3 before shearing are 200, 300 and 400 kPa, respectively.

All specimens were sheared at the same axial compression rate of 0.1 mm/min and the shearing would stop when the total axial strain reached 20%. The at-failure stresses in the q−p′ plane of all the three CU triaxial tests are plotted in Fig. 4. The data are best fitted with a straight line (i.e., critical-state line). The line passes through the origin and its slope is equal to M= 1.344.

Fig. 4 Critical-state line determined from CU triaxial test

results

4 Numerical simulation

The three-dimensional model of the cylindrical soil specimen with d=0.039 1 m and h=0.08 m is shown in Fig. 5. The top surface of the specimen is open and permeable. The loading plate is assumed to be perfectly smooth. The elements chosen are reduced-integration, pore fluid/stress twenty-node brick elements with quadratic displacement and linear pore pressure

Fig. 5 Three-dimensional model of cylindrical soil specimen:

(a) Before shearing; (b) After shearing

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(C3D20RP). The material is assumed elastoplasitc, obeying the modified cam-clay model. The parameters used are listed in Table 1 and Table 2. The initial conditions of the soil strata (initial geostatic stresses, initial pore water pressures and initial void ratio) should be considered herein for all analysis. Table 1 Cam-clay model parameters

Parameter Value Parameter Value

k/(m·s−1) 5×10−9 M 1.344

γ/(kN·m−3) 17.4 ν 0.3

κ 0.032 2 β 1

λ 0.129 7 Κ 1

Table 2 Parameters used in cam-clay model

Confining pressure, pc/kPa e0 a0/kPa

100 1.348 50

200 1.258 100

300 1.205 150

400 1.168 200

4.1 Simulation of isotropic consolidation test

In an isotropic consolidation test, the soil specimen was allowed to consolidate under different confining pressures. In order to get the reasonable results, it is very important to choose the initial time step in the soil consolidation. According to VERMEER and VERRUIJT [21], the criterion of the initial time step can be calculated as

2w ( )6

t hEk

(12)

where ∆h is the distance between nodes of the finite element mesh near the boundary condition change (0.005 m in this case), and E is the elastic modulus of the soil skeleton, which herein is calculated as E=p′/κ. Another parameter UTOL (allowable pore pressure stress change parameter) which controls the accuracy of the time integration during consolidation is chosen as 5 kPa for each consolidation step.

The results of the isotropic consolidation tests obtained from finite element analysis are shown in Figs. 6 and 7. In particular, Fig. 6 shows the consolidation curve in the void ratio versus mean effective stress (natural logarithm of p′) plane. It is noted that the numerical solution agrees very well with the experimental results. Because the unloading line and reloading line coincide with each other, the numerical solution could not truly reflect the hysteretic loop and its effects. The void ratio keeps the same after an unloading−reloading cycle by numerical solution, but it slight decreases compared to the measured one in test,

which results in slightly difference between the test data and the numerical results in the further analysis. It can be assumed that this difference would be widened after more unloading−reloading cycles.

Fig. 6 Relationship between e and ln p′

Fig. 7 Dissipation of excess pore pressure with time on center

line of specimen at different depths (top of specimen is taken as

0 m)

Figure 7 shows the excess pore-pressure at different

depths during consolidation with respect to time under confining pressure of 150 kPa. The initial confining pressure is 100 kPa. As can be seen, the pore water pressures at depths of 0.015 m and 0.025 m increase rather than decrease in the initial stage of the consolidation process. This phenomenon is just the well-known Mandel-Cryer effect [22−23]. On the other hand, there is no significant Mandel-Cryer effect when the depth is larger than 0.04 m. This suggests that the Mandel-Cryer effect is more pronounced for depths more adjacent to the pervious surface. With the increase of the depth, the peak of excess pore water pressure gradually diminishes, and it also takes longer time to reach the peak. The pore water pressure contours at different time can be shown in half-cutting model (Fig. 8). Note that the pore water pressure increases in the local area near

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the top surface of the soil specimen at the early stage of consolidation process and decreases with elapsed time. When the consolidation time is up to 112 s, the phenomenon becomes unobvious.

Fig. 8 Distribution of pore water pressure (POR) at different

elapsed time: (a) t=12 s; (b) t=24 s; (c) t=39 s; (d) t=112 s 4.2 Simulation of CU triaxial test

Similar to the real test, finite element analyses for CU1, CU2 and CU3 were carried out in two steps: a consolidation step and an undrained shearing step. During the consolidation step, a confining pressure was applied to the outer surface while drainage was permitted across the top surface. At the same time, the “geostatic” command was invoked to make sure that equilibrium was satisfied within the soil specimen. During the undrained shearing step, the top surface was made to be impervious and forced to displace downward a distance of 0.016 m at a rate of 0.001 67 mm/s. The process of shearing had a duration of 9 600 s. Automatic time stepping was used and the parameter UTOL was chosen as 2 kPa to control the accuracy of the time integration.

The numerical results for CU triaxial tests are shown in Fig. 9. The experimental results are also plotted for comparison. In particular, Fig. 9(a) shows the deviator stress versus axial strain and Fig. 9(b) shows the excess pore water pressure versus axial strain. The

calculated deviator stresses are very close to the measured data at the initial and final parts of the curves but obviously deviate from the measured data in the middle part of the curves. The calculated curves quickly approach to the peaks while the specimens only bear small axial strains. However, the test curves slow down clearly before reaching the maximum deviator stresses. In fact, the same problem can be seen in Ref. [24], but not be pointed out. By contrast, the predicted excess pore water pressures are closer to the test results and just slightly less than test results in the initial part of the curves.

Fig. 9 Comparison between CU triaxial test results and finite

element prediction: (a) Deviator stress versus axial stain;

(b) Excess pore water pressure versus axial stain

The mean effective stress, deviator stress and pore pressure at failure obtained by theoretical method, finite element method and test are listed in Table 3. As can be found, the finite element results are almost the same with the theoretical results and, in general, close to the test results.

Generally speaking, the comparison shows that the finite element results agree well with the test results. This implies that the implementation of modified cam- clay model in the finite element program used herein is done correctly.

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Table 3 Comparison of test, theory and FEM results at failure

Test Method fp /kPa qf/kPa uf/kPa

Test 122.3 162.8 132

CU1 Theory 118.8 15.6 134.4

FEM 118.6 159.4 134.5

Test 178.6 24.7 204

CU2 Theory 178.2 239.5 201.6

FEM 177.9 239.1 201.8

Test 230.8 308.4 272

CU3 Theory 237.6 319.3 268.8

FEM 237.2 318.8 269.1

4.3 Simulation of CD triaxial test Three CD triaxial tests (CD1, CD2 and CD3) were

performed in the finite element program ABAQUS. Finite element analysis for CD triaxial test was also carried out in two steps: a consolidation step and a drained shearing step. The initial confining pressures for CD1, CD2 and CD3 were 200, 300 and 400 kPa, respectively. Different from CU tests, the top surface kept pervious and forced to displace downward a distance of 0.032 m at a very slow rate of 3.2×10−8 mm/s during drained shearing step. This small rate of displacing is crucial to ensure a drained shearing condition. In addition, the initial time was set to be 8 000 s. This value is very small compared to time scale of shearing, but much greater than the characteristic time for pore pressure dissipation which is calculated as

2wH

TkE

(13)

where H is the height of the soil sample. Automatic time stepping was used and the parameter UTOL was chosen as 0.005 kPa.

The finite element results compared with theoretical results for three CD triaxial tests are shown in Fig. 10. For simplicity, the critical-state theory assumes that no recoverable energy is associated with shear distortion (i.e., e

sd 0) and the theoretical method given in Ref. [9] ignores the elastic shear strain

esd . As noted, with the

precondition of esd 0, the curves obtained from the

finite element method (FEM2) and theoretical method both in the aq plane (Fig. 10(a)) and the v a

plane (Fig. 10(b)) are almost overlapped. This suggests that finite element method has good uniformity with theoretical method. On the other hand, it does not make much difference whether or not to consider elastic shear strain. In addition, the curves of q versus εa vary with confining pressure, but the curves of εv versus εa seem not to be affected by the confining pressure. The excess pore water pressure u at the center of the soil specimen as a function of axial εa is plotted in Fig. 11. It can be

found that no significant pore pressure should ever arise in the analysis. This means that the above commands are set correctly. Moreover, the curves are independent on the confining pressures, but related to shearing rate. Lower rate results in lower excess pore water pressure.

Fig. 10 Comparison of theoretical solution and finite element

prediction in consolidated drained shear (FEM1—Considering

elastic shear strain; FEM2— Ignoring elastic shear strain):

(a) Deviator stress versus axial stain; (b) Volumetric stain

versus axial stain

Fig. 11 Variation of excess pore pressure during shearing

(FEM—Shearing rate of 3.2×10−8 mm/s; FEM3—Shearing rate

of 1.6×10−8 mm/s)

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For normally consolidated soils, the theoretically predicted results of aq and v a by using MCC model coincide well with CD triaxial tests [8]. 4.4 Unloading−reloading behavior of soil for CU

triaxial test In the simulation, a case where the specimen was

subjected to a 200 kPa confining pressure was performed.

The predicted effective stress path is shown in Fig. 12(a). Initially, the specimen was normally consolidated under a 200 kPa confining pressure and located on point 1, which corresponded to the pre-consolidation pressure. When shearing started, the soil specimen traveled along the curves 1 and 2 and the state of stress reached Point 2, which corresponded to q=120 kPa. At the same time, the yield surface expanded and the hardening behavior occurred. The specimen was then unloaded to Point 3 by reducing the deviator stress q from 120 kPa to 0, and then increasing q to 120 kPa that brought the state of stress to Point 4. We can see from Fig. 12 that the unloading and reloading effective stress paths (lines 2 and 3 and lines 3 and 4) are two overlapped vertical lines. This means the mean effective stress p′ in the elastic region within the yield surface is a constant during

undrained shearing. If p′ was not a constant, there would be volumetric strains resulted from changes in the mean effective stress p′. This apparently conflicts with the undrained conditions. Deviated from Point 4, yield surface expands again until the effective stress path reaches the critical line at Point 5. Note that Points 1, 2, 3, 4 and 5 are all on a horizontal line between NCL and CSL in the e−p′ plane as shown in Fig. 12(b). Because there is no volumetric change in an undrained conditions, the void ratio will keep constant with the value of e=e0=1.258.

The consolidated−undrained stress−strain behavior is shown in Fig. 12(c). Note that in a triaxial stress state, εa is equal to εv/3+εs, and under an undrained condition, εv=0, therefore, εs=εa. According to the calculated results, the total shear strain, elastic shear strain and plastic shear strain are εs=1.0%,

es 0.785% and

ps 0.215%,

respectively, before unloading. Although the shear stress reaches about 75% of the shear strength before unloading, the axial strain is small and the specimen mainly bears elastic strain.

After unloading, the total shear strain reduces to

s 0.104%. The value is smaller than the plastic shear strain which should keep on a value of

ps 0.215%.

This means that the elastic shear strain es is negative.

Fig. 12 Results of CU triaxial test under unloading−reloading condition: (a) Effective stress path; (b) Path in e−p′ plane; (c) Deviator

stress versus shear strain; (d) Deviator stress versus elastic shear strain

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This phenomenon can be demonstrated clearly from the curves of the deviator stress q versus the elastic shear strain

es in Fig. 12(d). This is physically unacceptable.

The reason for this negative result is that the shear modulus G is linear pressure-dependent for constant Poisson ratio. The mean effective stress on the effective stress Paths 2 and 3 (see Fig. 13) is less than that on Paths 1 and 2. This results in a smaller shear modulus G in the unloading−reloading condition. Therefore, there will be larger elastic shear strain under the same shearing

Fig. 13 Results of CD triaxial test under unloading−reloading

condition: (a) Effective stress path; (b) Path in e−p′ plane;

(c) Deviator stress versus shear stain

stress increment during unloading. This means that the model is not conservative. This conclusion has also been proved by ZYTYNSKI et al [20]. 4.5 Unloading−reloading behavior of soil for CD

triaxial test The initial confining pressure was also 200 kPa for

this case. Figure 13 shows the calculated consolidated− drained stress−strain behavior for unloading−reloading. The three effective stress paths for loading (lines 1 and 2 and lines 4 and 5), unloading (lines 2 and 3) and reloading (lines 3 and 4) are all in a straight line with a slope of 3 (Fig. 13(a)). The curves of void ratio e versus the mean effective stress p′ for unloading and reloading are on the unloading−reloading line in the e−p′ plane as shown in the Fig. 13(b).

Figure 13(c) shows the relationship between deviator strength q and the shear strain εs. During unloading, the effective stress state is within the elastic region (Fig. 12). Because of the assumption of “pressure- dependent”, the value of G changes from 8 080 kPa to 5 400 kPa with the decrease of effective mean stress. Therefore, the relationship between q and εs during unloading is not a straight line but a curve. This is different from the elastic-perfectly plastic models such as Mohr-Coulomb model. Obviously, the recoverable elastic strain is small, and a majority of unrecoverable plastic strain is reserved. The fact coincides with the actual deformation of clayey soils. The MCC model is commonly accepted to be mainly suitable for loading situations of clayey soils, however, the authors believe that, under drained condition, the MCC model is also suitable for unloading situations. 5 Conclusions

1) The MCC model, based on the isotropic consolidation test and CU test of natural clay, can fit well the former test curve of e−ln p′ and the later test curve of u−εa; but it cannot predict accurately the later test curve of q−εa, and the difference is obvious especially in the middle part of the curve. This should be noticed in using the MCC model.

2) The MCC model can predict accurately the test curve of q−εa in CD test, and the FEM prediction results of MCC model are basically identical to theoretical solutions. This indicates that the MCC material model in ABAQUS program is very reliable.

3) According to the numerical solution of finite element method, Mandel-Cryer effect exists in the cylindrical soil specimen when only the top surface is drained in the process of consolidation. Moreover, the effect only occurs in the local region of the specimen and is more pronounced for depths more adjacent to the

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pervious surface. This effect should occur in conventional triaxial tests but cannot be detected by pore pressure sensors set on bottoms of soil samples.

4) Analysis results of finite element method demonstrate that the MCC model disobeys the law of energy conservation under cyclic loading and undrained condition if the elastic shear modulus is linear pressure-dependent. Therefore, further study is worth to be done on this problem.

5) The FEM simulations of loading, unloading and reloading processes in CU and CD triaxial tests are beneficial to understanding various stress paths. Meanwhile, this indicates that under drained condition, although the curves of q−εs for unloading−reloading are nonlinear, the MCC model is suitable to predict the deformations of clayey soils for unloading and reloading situations. References [1] ROSCOE K H, SCHOFIELD A N, WROTH C P. On yielding of soils

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(Edited by HE Yun-bin)