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Article info.
1.  Introduction
2.  Evidence of variation with void ratio
3.  Limitation of current approaches
4.  Proposed modified creep index
5.  Elastic viscoplastic (EVP) model using a modified creep index
6.  Model verification
7.  Conclusions
8. Reference List
Open peer comments

Journal of Zhejiang University SCIENCE A 2014 Vol.15 No.4 P.272-281


A modified creep index and its application to viscoplastic modelling of soft clays*

Author(s):  Qi-yin Zhu1, Ze-xiang Wu1, Yan-ling Li1, Chang-jie Xu2,3, Jian-hua Wang1, Xiao-he Xia1

Affiliation(s):  1. Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China; more

Corresponding email(s):   qiyin.zhu@gmail.com

Key Words:  Clays, Creep, Consolidation test, Embankment, Finite element method, Viscoplasticity

Qi-yin Zhu, Ze-xiang Wu, Yan-ling Li, Chang-jie Xu, Jian-hua Wang, Xiao-he Xia. A modified creep index and its application to viscoplastic modelling of soft clays[J]. Journal of Zhejiang University Science A, 2014, 15(4): 272-281.

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author="Qi-yin Zhu, Ze-xiang Wu, Yan-ling Li, Chang-jie Xu, Jian-hua Wang, Xiao-he Xia",
journal="Journal of Zhejiang University Science A",
publisher="Zhejiang University Press & Springer",

%0 Journal Article
%T A modified creep index and its application to viscoplastic modelling of soft clays
%A Qi-yin Zhu
%A Ze-xiang Wu
%A Yan-ling Li
%A Chang-jie Xu
%A Jian-hua Wang
%A Xiao-he Xia
%J Journal of Zhejiang University SCIENCE A
%V 15
%N 4
%P 272-281
%@ 1673-565X
%D 2014
%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.A1300331

T1 - A modified creep index and its application to viscoplastic modelling of soft clays
A1 - Qi-yin Zhu
A1 - Ze-xiang Wu
A1 - Yan-ling Li
A1 - Chang-jie Xu
A1 - Jian-hua Wang
A1 - Xiao-he Xia
J0 - Journal of Zhejiang University Science A
VL - 15
IS - 4
SP - 272
EP - 281
%@ 1673-565X
Y1 - 2014
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/jzus.A1300331

Conventional consolidation tests on reconstituted specimens of numerous natural soft clays show a decreasing of creep index C αe with increasing soil density. Based on all selected and conducted experimental results, a modified creep index C αe * defined in double logarithmic plane lge-lgt, was plotted for various clays, from which C αe * can be assumed as a constant for different soil densities. Then, the modified creep index was applied to a newly developed elastic viscoplastic model. In this way, the modified creep index C αe * can naturally take into account the nonlinear C αe revealing the influence of soil density in the soil assemblies without additional parameters. Finally, the enhanced model was incorporated into the finite element code ABAQUS and used to simulate a consolidation test and a test embankment. The improvement of simulations by the modified creep index was highlighted by comparing simulations using the conventional creep index C αe.




Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Article Content

1.  Introduction

 From real construction projects, it is well known that soft clays exhibit time-dependent behaviors due to their viscosity (Mesri and Godlewski, 1977; Leroueil et al., 1985; Yin et al., 2002; Augustesen et al., 2004; Yin and Hicher, 2008; Yin and Wang, 2012). The creep index C αe, conventionally expressed by C αee/Δlgt (where e is the void ratio and t is the time) based on conventional consolidation test (1D) (Fig. 1), is widely used to describe the time-dependent behavior and develop viscoplastic models (Kutter and Sathialingam 1992; Leoni et al. 2008; Yin et al., 2010; 2011a; 2011b; Zhu et al., 2013). In these models, C αe is usually suggested as a constant value. However, this method has a flaw that when the time up to infinite, the void ratio will become negative. Therefore, a precise and reasonable description of C αe is practically important for viscoplastic modelling of soft clays and geotechnical simulations.

Definition of creep index

 In this study, the evidence of C αe variation with void ratio (reflecting soil density) is first investigated. The limitation of current approaches in describing the C αe variation is then discussed. A modified creep index is proposed based on this C αe variation of soft clays. Applying this modified creep index to an elastic viscoplastic model, an enhanced model accounting for the influence of soil density in soils is developed. The model is implemented into a finite element code and used to simulate a 1D consolidation test and a test embankment for the validation.

2.  Evidence of C αe variation with void ratio

 Conventional consolidation tests on six clays in various studies and two clays in the present study with their physical properties (Table 1) were selected. All of the selected experiments were conducted on reconstituted specimens to eliminate the effect of destructuration. The plasticity chart of these soils is shown in Fig. 2 plotted by liquid limit w L and plasticity index I P. According to this chart, all selected clays cover low plastic and high plastic inorganic clays.

Table 1

Summary of creep indexes and physical properties of typical reconstituted clays under normally consolidated state
Clay C αe Applied stress corresponding to C αe (kPa) C αe * e 0 w L I P Reference
 Nanjing clay 0.007–0.011 25–1600 0.0056 44 21 Zeng et al. (2012)
 Lianyungang clay 0.023–0.039 25–800 0.0096 86 55 Zeng et al. (2012)
 Suurpelto clay 0.0117–0.0585 80–640 0.0122 2.66 80 57 Stapelfeldt et al. (2007)
 Murro clay 0.0184–0.0375 11–600 0.0082 2.02 88 54 Karstunen and Yin (2010)
 HKMC* 0.0163–0.054 100–3200 0.0053 1.50 60 32 Yin et al. (2002)
 Shanghai clay 0.0072–0.0086 100–800 0.0043 1.27 51 24.6 Li et al. (2012)
 Ningbo clay 0.006–0.0095 200–1600 0.0061 0.90 39.5 16.5 This study
 Wenzhou clay 0.0078–0.0132 50–1600 0.0054 1.38 63 35 This study

  • *HKMC: Hong Kong Marine clay

  • Fig.2
    Classification of soils by plasticity chart

     Since the void ratio can reflect the physical conditions and represent the deformation potential of soils, the evolution of C αe for all selected clays was plotted against the void ratio (Fig. 3). Although the initial void ratio for all selected clays is varied, the creep behavior of each clay at a normally consolidated range is almost the same from Fig. 3a to Fig. 3h, i.e., C αe is decreasing with a decreasing of the void ratio for all clays.

    Conventional consolidation tests on reconstituted clays (a)–(h): creep index C αe and modified creep index C αe * versus void ratio
    *Present study

    3.  Limitation of current approaches

     To describe the evolution of C αe during creep, based on the experiments on Hong Kong deposited clay, Yin et al. (2002) proposed a formulation according to the nonlinear behavior of C αe: , where is the rate of viscoplastic volumetric strain; V=1+e 0 is the initial specific volume and e 0 is the initial void ratio; C αe0 is the initial value of C αe for each load stage; τ is a time-dependent parameter, and τ=1 d corresponding to the 1D consolidation test; ε vm is the volumetric strain relating to the current mean effective stress p m; is the reference volumetric strain corresponding to p m0; is equal to e 0/V. Eq. (1) describe that C αe decreases with time under constant stress levels without extra model parameters. However, when the applied stress is changed, the strain-rate is always accounted from the initial value C αe0 and there will be a jump of C αe between two different stress levels. As a result, this formulation cannot reproduce the consecutively decreasing C αe with different stress levels or soil densities. Later on, Karim et al. (2010) proposed a function describing the nonlinear behavior of C αe by the distance between the current and initial stress level. However, this method has the same deficiencies as Eq. (1).

     Based on the linear relationship between (1+e) and t in the double logarithmic plane, Zeng et al. (2012) gave a definition of C αL as .

     Eq. (2) describes that C αe will consecutively decrease with decreasing void ratio. However, due to its mathematical structure, Eq. (2) will deduce a negative void ratio when the creep is infinite.

     In conclusion, it is desirable that the creep formulation at least should have the following attributes: (1) C αe should consecutively decrease with the applied stress level or soil density; (2) the void ratio should always be positive during creep.

    4.  Proposed modified creep index

     In this study, we proposed a modified creep index C αe * defined as the slope of the lge-lgt curve of 1D consolidation test, which can be written as . Comparing Eq. (3) with Eq. (2), the only difference is that 1+e is substituted by e. The definition method for C αe * is similar to the concept adopted by Hong and Onitsuka (1998) and Hong (2007) in the studies of compression curves in double logarithm planes for natural clays.

     With Eq. (3), we calculated C αe * of all the selected clays in Fig. 3 at different void ratios. Then, the evolution of C αe * with void ratios was also plotted in Fig. 3. It can be found that for all clays C αe * slightly changes with the void ratio, and can be assumed as a constant. Then the average C αe * for clays was calculated and plotted in Fig. 3 with the horizontal dash line. Furthermore, the standard deviation (SD, , x represents random variable values and E(x) is their average value) and the ratio of the SD to the averaged C αe * (X/C αe *) were calculated and listed on each figure. As shown in Fig. 3, X/C αe * varies from 0.034 to 0.121. Generally speaking, errors are within the acceptable range for geomaterials. Therefore, it is reasonable to consider C αe * as a constant. Therefore, based on Eq. (3), the value of the creep index decreasing with a decreasing of void ratio is obtained by . Actually, Eq. (4) can capture all the essential features listed above for a creep formulation. First, Eq. (4) can be easily incorporated into elastic viscoplastic models without any additional parameters. Second, C αe is related to the void ratio; therefore, C αe will decrease consecutively with the void ratio during creep or loading. Third, benefiting from the mathematical structure of the logarithmic scale, the void ratio can always be positive when creeping.

     In addition, based on the available experimental tests, we investigated the correlations between C αe * and the Atterberg limits. Figs. 4a and 4b show the variation of C αe * with w L and I P, respectively. Fig. 4c shows that the accuracy of the correlation of C αe * with w L and I P is increasing by using a simple linear formula based on the method of least squares with R 2=0.8696. Therefore, the correlation of C αe * with the Atterberg limits is proposed as follows: .

    Relationship between modified creep index C αe * and Atterburg limits

    5.  Elastic viscoplastic (EVP) model using a modified creep index

     In this section, the proposed modified creep index will be incorporated into a recently developed elastic viscoplastic (EVP) model to consider the density effect of soft clays.

    5.1.  Brief introduction of adopted EVP model

     Recently, based on the strain rate-dependency of the preconsolidation pressure, Yin et al. (2010; 2011a; 2011b) developed an EVP model which can consider the characteristics of anisotropy, destructuration and viscosity of natural clays. The key constitutive relationships are: , , , . where σ d and α d are the deviatoric stress and fabric tensor, respectively; p′ is the mean effective stress; equals size of reference yield surface; represents the size of the dynamic loading surface relating to the current stress state; the soil constants ξ and ξ d control the effect of viscoplastic volumetric () and deviatoric () strains in destroying the bonds; f d is the dynamic yield surface; G is the shear modulus; is the K 0 consolidation critical stress ratio. Model parameters with their definitions are summarized in Table 2. Details of the model can be found in (Yin et al., 2010; 2011a; 2011b). As determined by Kutter and Sathialingam (1992), the constitutive relationship Eq. (6) follows the hypothesis of Bjerrum (1967) that there is no “instant compression” in elastic strains, which does not mean the creep does not occur before the end of the primary consolidation (EOP).

    Table 2

    Model parameters and definitions
    Group Parameter Definition
    Standard model parameter Initial reference preconsolidation pressure
    e 0 Initial void ratio
    ν Poisson’s ratio
    κ Swelling index
    λ i Intrinsic compression index
    M Slope of the critical state line

    Anisotropy parameter α 0 Initial anisotropy
    ω Absolute rate of yield surface rotation
    ω d Relative rate of yield surface rotation

    Destructuration parameter χ 0 Initial amount of bonding
    ξ Absolute rate of bond degradation
    ξ d Relative rate of bond degradation

    Viscosity parameter C αe * Creep index

    Hydraulic parameter k v0, k h0 Initial permeability
    c k Permeability coefficient

    5.2.  Incorporation of modified creep index

     Substituting Eq. (4) into Eq. (6), the constitutive equation in the form of total strain-rate can be written as . Then, the EVP model was enhanced by considering the density effect of soft clays. According to Eq. (10), no additional parameter is required in comparing with the older equation versions of Yin et al. (2011a; 2011b). The value of C αe * can be easily measured based on the conventional consolidation test or by the Atterberg limit based on Eq. (5).

    5.3.  Finite element implementation

     Based on a numerical solution proposed by Katona (1984), the original and enhanced elastic viscoplastic models were numerically implemented into the finite element code ABAQUS (version 6.10) as user-material soil models.

     For a soil-water coupled analysis based on Biot’s consolidation theory, the load increment is given by applying the principle of virtual work to the equilibrium equation. The coupled analyses of the finite element method are well conducted by various studies (Oka et al., 1986; Hinchberger and Rowe, 1998; Karstunen and Yin, 2010). According to experimental data, the evolution of the permeability with the void ratio can be described as , where the initial permeability k 0 is corresponding to the initial void ratio; permeability coefficient c k can be easily measured from the 1D consolidation experimental results by plotting e-lgk. The sensitive study of parameters in the model have been discussed in details by Yin et al. (2010; 2011a; 2011b). Consequently, the emphasis of the following section will only focus on the effectiveness of the proposed creep index.

    6.  Model verification

     To evaluate the predictive ability of the enhanced model considering the soil density effect, a 1D consolidation test on a reconstituted specimen of Wenzhou clay and a test embankment were simulated.

    6.1.  1D consolidation test

     A full load stage 1D consolidation test was conducted on Wenzhou clay with its creep indexes shown in Fig. 3h. The destructuration process of reconstituted clays are neglected. Therefore, the parameters related to soil structure are set to zero in the simulation. The values of other parameters measured from laboratory tests are summarized in Table 3.

    Table 3

    Parameters of EVP model for the consolidation test on Wenzhou clay
    ν λ κ e 0 M c
    0.3 0.185 0.035 1.38 1.23

    C αe * (kPa) τ k v0 (m/h) c k

    0.0054 40 24 3.6×10-4 0.6

     The modified creep index C αe *=0.0054 was used in the simulation as plotted in Fig. 3h. To highlight the improvement by the consideration of the soil density effect, two additional simulations were carried out with constant C αe=0.0138 (the maximum value shown in Fig. 3h) and 0.0087 (the minimum value shown in Fig. 3h).

     All simulated results and experimental data were plotted in terms of the void ratio versus time. Fig. 5 shows that the simulated results of C αe=0.0138 without density effect keep a high creep rate, and the simulated results of C αe=0.0087 without density effect keep a low creep rate during the test. By comparison, it can be seen that the simulation of C αe *=0.0054 with density effect can predict the consecutively decreasing creep rate with stresses and void ratios. Thus, the consideration of the soil density effect can greatly improve the predictive ability of the EVP model. Note that the identical predicated void ratio by models with and without accounting for the density effect at the end of each load stage with 1 d duration is assured following the principle of the model. Thus, the comparison of the creep rate between different models can be clearly observed.

    Simulations of 1D consolidation test on reconstituted Wenzhou clay

    6.2.  Murro test embankment

    6.2.1.  Element properties and boundary conditions

     The Murro test embankment was constructed on a soft clay deposit in Finland in 1993, which has been studied by Karstunen et al. (2005), Karstunen and Yin (2010) and Yin et al. (2011a; 2011b). In the finite element method analysis, the plain strain condition was assumed and the finite element mesh is shown in Fig. 6. The groundwater table was located at a depth of 0.8 m. The horizontal boundary was 36 m away from the embankment center-line and the vertical boundary is at a depth of 23 m. The horizontal displacements were fixed in the lateral boundaries, and the vertical and horizontal displacements were fixed at the bottom. The finite element model consists of 1680 elements resulting in 4562 nodes. The stress-strain behavior of the embankment fill was modelled by Mohr-Coulomb model and typical values of model parameters were summarised as: Young’s modulus E=40 000 kPa, Poisson’s ratio v=0.35, friction angle ϕ=40°, and unit weight γ=19.6 kN/m3. The construction of the embankment took 2 d and the same was simulated in the modelling.

    Finite element analysis mesh

    6.2.2.  Parameters of foundation clays

     1D consolidation tests on high quality natural clays were conducted and selected to determine the parameters for the enhanced model. The modified creep index C αe *=0.0082 determined from Murro clay at 7 m depth (Fig. 3d) was used for layer 4. Using the same method, we also measured C αe * for all the other layers and used it for the simulation, as listed in Table 3. Other parameters used in the modelling by Karstunen and Yin (2010) and Yin et al. (2011a; 2011b) were adopted (Table 4) and the details of the process of determination will not be repeated here. To study the influence of soil density, one extra test was simulated with a constant C αe as listed in Table 4. The constant C αe for each soil layer was the averaged value of C αe at the normally consolidated state.

    Table 4

    Enhanced EVP model parameters for subsoil of Murro embankment
    Depth (m) γ (kN/m3) e 0 M c K 0 κ λ i ν C αe *
    0.0–1.6 16.1 1.57 1.7 1.25 0.01 0.18 0.3 0.0051
    1.6–3.0 15.7 1.81 1.7 0.34 0.024 0.18 0.3 0.0061
    3.0–6.7 14.4 2.45 1.65 0.35 0.041 0.25 0.3 0.0081
    6.7–10.0 15.2 2.16 1.5 0.4 0.024 0.21 0.3 0.0082
    10.0–15.0 15.7 1.76 1.45 0.42 0.024 0.21 0.3 0.0061
    15.0–23.0 16.2 1.53 1.4 0.43 0.02 0.15 0.3 0.0053

    Depth (m) (kPa) χ 0 ξ ξ d k v0 (m/h) k h0 (m/h) c k C αe

    0.0–1.6 100 2.5 5 0.2 6.5×10−4 8.3×10−4 0.43 0.0201
    1.6–3.0 22 6 12 0.2 2.0×10−5 2.6×10−5 0.65 0.0224
    3.0–6.7 22 6 9 0.2 1.6×10−5 2.2×10−5 0.69 0.0278
    6.7–10.0 22 6 10 0.2 1.0×10−5 1.4×10−5 0.49 0.0254
    10.0–15.0 35 4 5 0.2 5.4×10−6 7.2×10−6 0.44 0.0219
    15.0–23.0 40 6 8 0.2 2.2×10−6 2.9×10−6 0.45 0.0198

    6.2.3.  Settlements

     Figs. 7a and 7b present the simulated and observed surface settlements under and 5 m off the centreline of the embankment, respectively. Both creep and modified creep models can very well predict the surface settlements in the short term. The major difference can be found between the simulations with and without consideration of the soil density effect after 1000 d of construction. However, their differences become larger with time increasing. In short, the simulation according to density effect leads to a smaller settlement due to the fact that the creep rate is reduced with the decreasing void ratio. In conclusion, the comparison shows that the consideration of density effect only influences the long-term settlement behavior.

    Surface settlements under (a) and 5 m off (b) the centerline

    6.2.4.  Horizontal displacements

     Fig. 8 shows the simulated horizontal displacements of inclinometer I2 by both creep and modified creep models. At 754 d after construction, both models gave the same trend for the horizontal displacement. However, the simulation considering the density effect predicted very well the maximum values with the depth. At 3201 d after construction, the simulation considering density effect predicted a slightly smaller displacement, but also gave an exact depth of the maximum horizontal displacement. At 100 years after construction, the same trend was predicted for horizontal displacement as with the 3201 d and their differences are becoming more and more obvious. Fig. 8 also compared the influence of the soil density on the horizontal displacement and the results show that the influence of soil density is less important than that on the settlements. Note that the differences between simulations and measurements for the horizontal displacement below 13 m were also observed, and this unsatisfying result is probably due to the constant Poisson’s ratio which needs further studies.

    Horizontal displacement at I2: (a) 754 d; (b) 3201 d, and (c) 100 years after construction

    6.2.5.  Excess pore pressure

     Fig. 9 presents the model simulations with and without the soil density effect for excess pore pressure in the subsoil at 12 m below the ground surface, which before 300 d after construction. The model considering density effect predicted a slightly higher excess pore pressure. However, both simulations show the excess pore pressure gradually dissipating with time and the influence of considering the density effect on its evolution is negligible. Since the dissipation rate of excess pore pressure in the subsoil depends mainly on its permeability, thus, the results are reasonable.

    Excess pore pressure at point U7 (12 m depth below ground surface)

     Although the predicted horizontal displacements and excess pore pressure by using parameters measured from laboratory tests are generally a little unsatisfactory with the measurements, it does not overly influence the emphasis of this study by comparing the results with and without consideration of the soil density effect. Actually, due to the sample disturbance, the values of the parameters are difficult to be obtained precisely for soft sensitive clay. If the sample disturbance is accounted for by measuring parameters, then the prediction will be improved, which needs to be further investigated. However, it does not mean that the consideration of the soil density effect is not necessary.

    7.  Conclusions

     The evolution of creep index C αe was investigated throughout the conventional consolidation test on reconstituted specimens of eight natural soft clays. The consecutively decreasing results of C αe with the void ratio were studied. Then, a modified creep index C αe * was proposed in the lge-lgt plane. Experimental results show that C αe * can be assumed as a constant for soft clays.

     The proposed modified creep index was substituted into a recently developed EVP model to consider the influence of soil density. No extra parameters and no extra experimental costs are needed for the enhanced model. The model was then introduced into the finite element code ABAQUS for analyzing boundary value problems.

     The enhanced model was firstly verified by simulating a 1D consolidation test on a reconstituted sample of Wenzhou clay. Two simulations according to test for the presence of a density effect or not were made and compared to the observed results. The comparisons show that the enhanced model determines very well the consecutively decreasing property of C αe during the test.

     Furthermore, the enhanced model was used to simulate an embankment. The simulation was also compared to that without considering the density effect. Comparisons demonstrate that: (a) accordingly, the density effect for creep influences significantly the long-time settlement; (b) for horizontal displacement, the influence of the density effect consideration is less significant than that for the settlement; and (c) the density effect on the dissipation of excess pore pressure can be neglected.

    * Project supported by the National Natural Science Foundation of China (Nos. 41240024, 41372285, 41272317, 51278449, and 51238009), the Research Fund for the Doctoral Program of Higher Education of China (No. 20110073120012), and the European Project CREEP (PIAPP-GA-2011-286397)


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