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Technical Brief: Technical Briefs

# A Simple Effective Stress Approach for Modeling Rate-Dependent Strength of Soft ClayOPEN ACCESS

[+] Author and Article Information
Jiang Tao Yi

School of Civil Engineering,
Chongqing University,
No.83 Shabei Street,
Chongqing 400045, China
e-mail: yijt@foxmail.com

Yu Ping Li

Key Laboratory of Geomechanics
and Embankment Engineering,
Ministry of Education,
Hohai University;
Geotechnical Research Institute of Hohai University,
Hohai University,
Nanjing 210098, China
e-mail: juliya-li@hotmail.com

Shan Bai

School of Civil Engineering,
Chongqing University,
No.83 Shabei Street,
Chongqing 400045, China
e-mail: baishan.cqu@foxmail.com

Yong Fu

Department of Civil and Environmental Engineering,
National University of Singapore,
Blk E1A #07-03 1 Engineering Drive 2,
10 Kent Ridge Crescent,
Singapore 117576
e-mail: fuyong@u.nus.edu

Fook Hou Lee

Department of Civil and Environmental Engineering,
National University of Singapore,
Blk E1A #07-03 1 Engineering Drive 2,
10 Kent Ridge Crescent,
Singapore 117576
e-mail: leefookhou@nus.edu.sg

Xi Ying Zhang

Mem. ASME
American Bureau of Shipping,
ABS Plaza,
16855 Northchase Drive,
Houston, TX 77060
e-mail: xyzhang@eagle.org

1Corresponding author.

Contributed by the Ocean, Offshore, and Arctic Engineering Division of ASME for publication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received June 30, 2016; final manuscript received November 3, 2017; published online January 9, 2018. Assoc. Editor: Lizhong Wang.

J. Offshore Mech. Arct. Eng 140(4), 044501 (Jan 09, 2018) (5 pages) Paper No: OMAE-16-1074; doi: 10.1115/1.4038732 History: Received June 30, 2016; Revised November 03, 2017

## Abstract

This paper proposes a simple effective stress method for modeling the strain rate-dependent strength behavior that is experienced by many fine-grained soils in offshore events when subjected to rapid, large strain, undrained shearing. The approach is based on correlating the size of the modified Cam-Clay yield locus with strain rate, i.e., yield locus enlarging or diminishing dependent on the strain rate. A viscometer-based method for evaluating the needed parameters for this approach is provided. The viscometer measurements showed that strain rate parameters are largely independent of water content and agree closely with data from a previous study. Numerical analysis of the annular simple shear situation induced by the viscometer shows remarkable agreement with the experimental data provided the remolding-induced strength degradation effect is accounted for. The proposed method allows offshore foundation installation processes such as dynamically installed offshore anchors, free-falling penetrometer, and submarine landslides to be more realistically analyzed through effective stress calculations.

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## Introduction

Offshore situations such as installation of dynamically installed anchors, free-falling penetrometer, and submarine landslides often involve rapid undrained soil flow, which involves large strain at high strain rates [14]. Many fine-grained soils exhibit increasing shear strength when subjected to high shear strain rate [1,5,6]. Such problems were often modeled using a total stress approach wherein the elastic-perfectly plastic model such as Tresca model was extended to reflect the strain rate effect [2,68]. However, the total stress analysis does not allow excess pore pressure, and therefore long-term performance of rapidly installed foundation elements, to be determined. To do this would require an effective stress analysis with an appropriate strain rate model. Existing visco-plastic effective stress models (e.g., see Refs. [912]), generally based on the overstress theory of Perzyna [13], are mainly applicable for consideration of the creep and secondary compression of soils, rather than the strain rate-dependent strength behavior of soil when subjected to rapid, large strain, undrained shearing.

This paper proposes an effective stress approach to model the strain rate-dependent strength behavior of fine-grained soils when subjected to rapid undrained shearing. The approach is based on correlating the size of the modified Cam-Clay yield locus with strain rate. The general formulation is first derived, followed by the procedures for evaluating the strain rate parameters. Finally, the method is validated by using it to analyze a ring shear problem. While the deformation behavior such as compressibility of fine-grained soils is also affected by the strain rate, the present paper is only concerned with the strain rate-dependent strength at large strain. This kind of loading scenario is of particular interest in the aforementioned offshore situations, which are accompanied by a change in strength (or mobilized shear stress) of several order of magnitude and strain rate far higher than those strain rates encountered in usual geotechnical problems [1].

## Methodology

The strategy used herein is to extend an effective stress constitutive model, or more precisely the modified Cam-Clay model, to capture the strain rate effect. The constitutive model after extension can thus be implemented into the effective stress finite element calculation to account for the strain rate-dependent strength behavior of fine-grained soils.

The basic postulate adopted herein is that shear strain rate effect can be described by allowing the yield locus of modified Cam-Clay to expand beyond its static size by an amount, which depends upon the strain rate (Fig. 1). This can be achieved through a modification to the modified Cam-Clay yield function F so that Display Formula

(1)$F=M2p′2+q2−M2p′pcr′(ε˙ŝ)$
in which $pcr′(ε˙ŝ)$ a strain rate-dependent isotropic precompression pressure, which is postulated to have the form Display Formula
(2)$pcr′=pc′(1+ξε˙ŝβ)m$
in which $pc′$ is the static precompression pressure; $ξ$ and $β$ are strain rate parameters; m = λ/(λ − κ) and λ and κ are the isotropic compression and swelling indices. $ε˙ŝ$ is a generalized shear strain rate term, which can either be the deviatoric shear strain rate $ε˙s$ or its normalized form $ε˙s$/$ε˙s,ref$ where $ε˙s,ref$ is the reference shear strain rate. The former is more aligned with the fluid dynamics framework while the latter, a dimensionless quantity, is more in line with the geotechnical framework [1,14].

Following Wroth [15], the resulting rate-dependent undrained shear strength $sur$ can be shown to be given by Display Formula

(3)$sur=M2p′1−Λpcr′2Λ$

in which $Λ$ = (λ − κ)/λ. Combining Eqs. (2) and (3) gives Display Formula

(4)$sur=su(1+ξε˙ŝβ)$

where $su$ is the static undrained shear strength and $su=(M/2)p′1−Λ(pc′/2)Λ$ according to Wroth [15]. Equation (4) resembles the widely used additive power-law formulation (Eq. (5)) of Zhu and Randolph [14] except that deviatoric shear rate $ε˙s$ is used in Eq. (4) instead of the engineering shear strain rate $γ˙$ in Eq. (5) as the former is an invariant and has wider generality in three-dimensional problems. Display Formula

(5)$sur=su(1+ηγ˙γ˙refβ)$

In brief, the above methodology is essentially to correlate the yield locus of the modified Cam-Clay with the strain rate so as to reflect the rate-dependent strength behavior of soft clay within the Cam-Clay framework. The strain rate parameters $ξ$ and $β$ are the two needed parameters in addition to the standard Cam-Clay parameters for its implementation. As will be described, $ξ$ and $β$ are obtainable from viscometer test (or other alternatives such as T-bar penetrometer test [1]).

## Determination of Parameters

###### Test Procedures.

The strain rate parameters $ξ$ and $β$ were here experimentally determined by a HAAKE RotoVisco 1 rotational viscometer (Fig. 2). Locat and Demers' [16] viscometer tests on soils have shown that yield stress from viscometer and shear strength from the Swedish fall cone are fairly close. Compared to penetrometer or vane shear tests, the viscometer allows the shear rate to be explicitly determined so that the measured stress does not have to be associated with an “operative” shear strain rate, e.g., Ref. [1]. As will be explained later, the strain rate parameters have to be evaluated iteratively from viscometer test results by sequentially treating soil as a Newtonian and non-Newtonian fluid material. The deviatoric shear strain rate $ε˙s$ rather than its normalized form will thereby be used, as customarily practiced in fluid mechanics.

The soil specimen in the present viscometer tests was prepared from reconstituted Malaysian kaolin clay with its basic properties summarized in Table 1. In order to prepare soil specimens with various water contents, the dry clay powder was first thoroughly mixed with water at a water content of 120%. The clay slurry was then transferred to a large cylindrical container measuring 580 mm in diameter and 400 mm in height and subjected to 50g gravity at geotechnical centrifuge. This could then produce a clay bed with different water contents at different depths. Soil specimens were subsequently extracted from different depths of the clay bed.

The plate–plate sensor system of HAAKE RotoVisco 1 was used in this study, which consists of two parallel 35 mm diameter circular plates (Fig. 2). The setting up consisted of first placing the 38 mm diameter and 3 mm thick soil specimen on the lower plate. The upper plate was then lowered and allowed to reach the designated 1 mm clearance. Excess soil was squeezed out, thereby ensuring complete filling of the gap. The sample was then trimmed with the excess soil removed. The upper plate was then rotated at the prescribed rate while the lower plate was held stationary. The torque, from which shear stress was deduced, was observed to require several revolutions before reaching a stable value. It is very likely that the soil would have reached a fully remolded state [18] by then and torque measurement was thereby taken.

###### Experimental Results.

The engineering shear strain rate $γ˙$ in the plate–plate viscometer is given by Display Formula

(6)$γ˙r=Ωhr$
in which $γ˙r$ is the shear strain rate at the radial distance $r$,$Ω$ the angular velocity of the rotating disk and h the separation between the two parallel plates. As the viscometer test imposes simple shearing only in the circumferential direction, the generalized shear strain $ε˙sr$ can be readily evaluated from $γ˙r$Display Formula
(7)$ε˙s(r)=3γ˙(r)/3$

The shear stress is not measured directly, but is instead deduced from the torque. Since the specimen is continuously sheared, we can assume that the undrained shear strength has been reached at every point. The measured torque $T$ is thus given by Display Formula

(8)$T=∫0R2πr2surrdr$

As Eqs. (6) and (7) suggest, the shear strain rate on the soil specimen varies with the radial distance. So does the rate-dependent undrained shear strength $sur(r)$. The relationship between shear strength at the edge of plate $surR$ and torque $T$ can be proven (see Appendix) to have the following form: Display Formula

(9)$surR=3+β2πR3T−β3su$

Since $β$ and $su$ are unknown a priori, they have to be evaluated iteratively. The soil is first treated as a Newtonian material, which enables $surR$ to be directly calculated from the torque $T$ via Eq. (10) (see Appendix) Display Formula

(10)$surR=2πR3T$

The first estimates of $β,ξ$, and $su$ can then be obtained from the plot of$surR$ versus $ε˙s(R)$, which is then fitted by Eq. (4) where $ε˙ŝ$ is replaced with $ε˙s$. Substituting these estimates of $β$ and $su$ into Eq. (9) allows the values of $surR$ to be refined. The second estimates of $β,ξ,andsu$ can be subsequently obtained from the replot of the $surR$ versus $ε˙s(R)$, which is refitted by Eq. (4). This process is repeated until the values of $β,ξ,andsu$ stabilize.

Figures 36 plot the $surR$ versus $ε˙s(R)$ by the end of iteration with stabilized values of $β,ξ,andsu$ shown in the embedded equations. As can be seen, the strain rate dependency of the soft soil is well fitted by Eq. (4) for various water contents ranging from 72% to 87%. Moreover, as Table 2 shows, strain rate parameters $β$ and $ξ$ are nearly unchanged over the range of water contents studied ($β$ = 0.36–0.37 and $ξ$ = 0.30–0.32). Attempts to test soil samples with lower than 72% were unsuccessful as the motor was unable to provide sufficient torque required to shear these stiffer soil samples. Nonetheless, this is not expected to significantly affect the main finding since $β$ and $ξ$ appear nearly insensitive to water content. Similar characteristics were noted by Boukpeti et al. [1] and Jeong et al. [19]. Moreover, as shown in Fig. 7, normalizing the present data yielded good agreement with those reported by Jeong et al. [19] using Mediterranean Sea soil.

In addition to viscometer test, laboratory vane shear test was carried out herein for soil with water content of approximately 83% prior to the extraction of such soil specimen from the clay bed formed by centrifuge high-g (50g) consolidation as described earlier. Following the vane shear test procedure specified in BS 1377 (part 7) [20], both the intact and remolded strengths were measured and included in Fig. 5. As the standard vane tests conducted at very slow rate (0.1°s−1) (equivalent to shear rate less than 0.05 s−1), its measurement of strength is deemed to be the static shear strengths. As can be seen from the graph, the “static” shear strength from the viscometer tests is much lower than the intact strength from vane shear, but agrees well with the remolded strength. This is consistent with preceding notion that soil in the viscometer is very likely to be in a remolded state.

## Implementation and Verification

With the assumption of associated flow rule, the preceding constitutive model (Eq. (1)) was subsequently implemented into ABAQUS/Explicit via user-subroutine VUMAT for model verification as well as evaluation of its predictive ability. Similar strategy of performing model verification exercise via finite element implementation was practiced and reported in the literature [2123]. Finite element analysis modeled a ring shear problem, which is essentially an annulus of the viscometer specimen. A finite element model (FEM) was established to model simple shear of a ring that has a 10.1 m outer radius, a 10 m inner radius and a 0.1 m thickness. The use of a large diameter-to-thickness ratio of ring allowed a uniform stress state to be reached throughout the element and a large strain rate to be achieved with a small angular velocity (with reference to Eq. (6)). The soil properties used were those in Table 1, together with rate parameters $β$ = 0.37 and $ξ$ = 0.30. The soil was initially subjected to an isotropic stress of 2.3 kPa which is consistent with a water content of 83% under normally consolidated condition. Shearing was imposed by rotating the top surface of ring at prescribed angular velocities while maintaining its bottom surface stationary, in the same manner as viscometer.

As Fig. 8 shows, the trend of strength changes with shear rate was well replicated by finite element computation (“FEM” in the graph). However, the computed strengths are consistently larger than those experimental values from viscometer. This can be attributed to the remolding-induced strength degradation effect, i.e., the gradual reduction in strength with the cumulative shear strain [6], which was not considered by this model. As stated above, soil in the viscometer had probably been fully remolded. For such a fully remolded soil, a simple approach to factor the strain softening effect into calculation is to multiply those results by the fully remolded strength ratio δrem [6]. Based on vane shear rests, the remolded strength ratio δrem in this case is 0.52. Multiplying those computed strengths by this leaded to remarkable agreement with the measured strengths.

## Conclusion

This study proposes a simple effective stress approach for modeling the strain rate-dependent strength behavior. The yield locus of modified Cam-Clay was expanded beyond its static size by an amount, which depends upon the strain rate. The resulting strain rate-dependent undrained shear strength has been shown to have a consistent form with the widely used additive power-law model. Viscometer tests can be undertaken to determine needed strain-rate parameters. The validation calculation of modeling annular simple shear situation shows that, in situations involving rapid undrained shearing of soil to failure, the proposed simple approach can adequately reflect the strain rate-dependent strength behavior if the remolding-induced strength degradation is accounted for. The practical implication of the proposed method lies mainly in the incorporation of shear strain rate effect into the Cam-Clay soil model, which allows offshore foundation installation processes such as dynamically installed offshore anchors, free-falling penetrometer, and offshore submarine slides (the run-out of offshore submarine slides as well as its influence on infrastructure and oil pipelines) to be more realistically analyzed. So far, these analyses have been conducted largely in a total stress framework. The method proposed herein will allow similar analyses to be conducted in an effective stress framework.

## Funding Data

• National Natural Science Foundation of China (Grant Nos. 51509025 and 51778091).

• China Postdoctoral Science Foundation (Grant No. 2015M581988).

• Maritime and Port Authority of Singapore (Grant No. R-302-501-021-490).

• Ministry of Education of the People's Republic of China (Grant No. 2015CDJXZ).

## Nomenclature

• h =

separation between rotating and stationary disk plate

• M =

friction coefficient

• p′, q =

mean effective and deviator stresses

• $pc′$ =

static isotropic precompression pressure

• $pcr′$ =

strain rate dependent isotropic precompression pressure

• $r$ =

• R =

• $su$ =

static undrained shear strength (i.e., $sur$ at zero strain rate)

• $sur$ =

rate-dependent undrained shear strength

• T =

torque

• $v$ =

specific volume

• $γ˙$ =

shear strain rate

• $γ˙ref,ε˙s,ref$ =

reference shear strain rates

• δrem =

fully remolded strength ratio

• $ε˙s$ =

deviatoric shear strain rate

• $ε˙ŝ$ =

generalized shear strain rate

• κ =

isotropic swelling index

• λ =

isotropic compression index

• $ξ,β,η$ =

strain rate parameters

• $Ω$ =

angular velocity of rotation of moving disk

## Appendices

###### Appendix: Relationship Between Sur(R) and T

For a Newtonian material Display Formula

(A1)$surr=μγ˙r$

where $μ$ is viscosity of the Newtonian material

With reference to Eq. (6), Display Formula

(A2)$surr=ar$

where $a=μΩ/h$

Then Display Formula

(A3)$surR=aR$

Substituting Eq. (A2) into Eq. (8) leads to Display Formula

(A4)$T=2πa∫0Rr3dr=aπ2R4=πR32sur(R)$

Thus Display Formula

(A5)$surR=2πR3T$

For a non-Newtonian material obeying Eq. (4)

With reference to Eq. (4) and setting $ε˙ŝ$ = $ε˙s$Display Formula

(A6)$surr=su1+ξε˙sβ(r)=su1+ζrβ$

where $ζ=ξ(3Ω/3h)β$ in view of Eqs. (6) and (7)

Then Display Formula

(A7)$surR=su1+ζRβ$

Substituting Eq. (A6) into Eq. (8) leads to Display Formula

(A8)$T=2π∫0Rr2su1+ζrβdr=2πR33+β3+β3su+ζRβsu=2πR33+βsurR+β3su$

Hence Display Formula

(A9)$surR=3+β2πR3T−β3su$

## References

Boukpeti, N. , White, D. J. , Randolph, M. F. , and Low, H. E. , 2012, “ Strength of Fine-Grained Soils at the Solid-Fluid Transition,” Geotechnique, 62(3), pp. 213–226.
Nazem, M. , Carter, J. P. , Airey, D. W. , and Chow, S. H. , 2012, “ Dynamic Analysis of a Smooth Penetrometer Free-Falling Into Uniform Clay,” Geotechnique, 62(10), pp. 893–905.
Hossain, M. S. , Kim, Y. , and Wang, D. , 2013, “Physical and Numerical Modelling of Installation and Pull-out of Dynamically Penetrating Anchors in Clay and Silt,” ASME Paper No. OMAE2013-10322.
Zakeri, A. , 2009, “ Review of State-of-the-Art: Drag Forces on Submarine Pipelines and Piles Caused by Landslide or Debris Flow Impact,” ASME J. Offshore Mech. Arct. Eng., 131(1), p. 014001.
Dayal, U. , and Allen, J. H. , 1975, “ The Effect of Penetration Rate on the Strength of Remolded Clay and Sand Samples,” Can. Geotech. J., 336(3), pp. 336–348.
Einav, I. , and Randolph, M. , 2005, “ Combining Upper Bound and Strain Path Methods for Evaluating Penetration Resistance,” Int. J. Numer. Methods Eng., 63(14), pp. 1991–2016.
Zhou, H. , and Randolph, M. F. , 2007, “ Computational Techniques and Shear Band Development for Cylindrical and Spherical Penetrometers in Strain-Softening Clay,” Int. J. Geomech., 7(4), pp. 287–295.
Zhou, H. , and Randolph, M. F. , 2009, “ Resistance of Full-Flow Penetrometers in Rate-Dependent and Strain-Softening Clay,” Geotechnique, 59(2), pp. 79–86.
Adachi, T. , and Oka, F. , 1982, “ Constitutive Equations for Normally Consolidated Clay Based on Elasto-Viscoplasticity,” Soils Found., 22(4), pp. 57–70.
Yin, J. H. , Zhu, J. G. , and Graham, J. , 2002, “ A New Elastic Viscoplastic Model for Time-Dependent Behaviour of Normally and Overconsolidated Clays: Theory and Verification,” Can. Geotech. J., 39(1), pp. 157–173.
Kelln, C. , Sharma, J. , Hughes, D. , and Graham, J. , 2008, “ An Improved Elastic–Viscoplastic Soil Model,” Can. Geotech. J., 45(10), pp. 1356–1376.
Yin, Z. , and Wang, J. , 2012, “ A One-Dimensional Strain-Rate Based Model for Soft Structured Clays,” Sci. China. Technol. Sci., 55(1), pp. 90–100.
Perzyna, P. , 1963, “ The Constitutive Equations for Rate Sensitive Plastic Materials,” Q. Appl. Math., 20, pp. 321–332.
Zhu, H. , and Randolph, M. F. , 2011, “ Numerical Analysis of a Cylinder Moving Through Rate-Dependent Undrained Soil,” Ocean Eng., 38(7), pp. 943–953.
Wroth, C. P. , 1984, “ The Interpretation of In Situ Soil Tests,” Geotechnique, 34(4), pp. 449–489.
Locat, J. , and Demers, D. , 1988, “ Viscosity, Yield Stress, Remolded Strength, and Liquidity Index Relationships for Sensitive Clays,” Can. Geotech. J., 25(4), pp. 799–806.
Goh, T. L. , 2003, “Stabilisation of an Excavation by an Embedded Improved Soil Layer,” Ph.D. thesis, National University of Singapore, Singapore.
Hong, Y. , He, M. B. , Wang, L. , Wang, Z. , Ng, C. W. W. , and Mašín, D. , 2017, “ Cyclic Lateral Response and Failure Mechanisms of Semi-Rigid Pile in Soft Clay: Centrifuge Tests and Numerical Modelling,” Can. Geotech. J., 54(6), pp. 806–824.
Jeong, S. W. , Leroueil, S. , and Locat, J. , 2009, “ Applicability of Power Law for Describing the Rheology of Soils of Different Origins and Characteristics,” Can. Geotech. J., 46(9), pp. 1011–1023.
BSI, 1990, “Methods of Test for Soils for Civil Engineering Purposes,” British Standards Institution, London, Standard No. BS1377.
Karstunen, M. , and Yin, Z. Y. , 2010, “ Modelling Time-Dependent Behaviour of Murro Test Embankment,” Geotechnique, 60(10), pp. 735–749.
Zhu, Q. , Wu, Z. , Li, Y. , Xu, C. , Wang, J. , and Xia, X. , 2014, “ A Modified Creep Index and Its Application to Viscoplastic Modelling of Soft Clays,” J. Zhejiang. Univ., Sci. A, 15(4), pp. 272–281.
Liu, Y. , Hu, J. , Wei, H. , and Saw, A. L. , 2017, “ A Direct Simulation Algorithm for a Class of Beta Random Fields in Modelling Material Properties,” Comput. Methods Appl. Mech. Eng., 326(1), pp. 642–655.
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## References

Boukpeti, N. , White, D. J. , Randolph, M. F. , and Low, H. E. , 2012, “ Strength of Fine-Grained Soils at the Solid-Fluid Transition,” Geotechnique, 62(3), pp. 213–226.
Nazem, M. , Carter, J. P. , Airey, D. W. , and Chow, S. H. , 2012, “ Dynamic Analysis of a Smooth Penetrometer Free-Falling Into Uniform Clay,” Geotechnique, 62(10), pp. 893–905.
Hossain, M. S. , Kim, Y. , and Wang, D. , 2013, “Physical and Numerical Modelling of Installation and Pull-out of Dynamically Penetrating Anchors in Clay and Silt,” ASME Paper No. OMAE2013-10322.
Zakeri, A. , 2009, “ Review of State-of-the-Art: Drag Forces on Submarine Pipelines and Piles Caused by Landslide or Debris Flow Impact,” ASME J. Offshore Mech. Arct. Eng., 131(1), p. 014001.
Dayal, U. , and Allen, J. H. , 1975, “ The Effect of Penetration Rate on the Strength of Remolded Clay and Sand Samples,” Can. Geotech. J., 336(3), pp. 336–348.
Einav, I. , and Randolph, M. , 2005, “ Combining Upper Bound and Strain Path Methods for Evaluating Penetration Resistance,” Int. J. Numer. Methods Eng., 63(14), pp. 1991–2016.
Zhou, H. , and Randolph, M. F. , 2007, “ Computational Techniques and Shear Band Development for Cylindrical and Spherical Penetrometers in Strain-Softening Clay,” Int. J. Geomech., 7(4), pp. 287–295.
Zhou, H. , and Randolph, M. F. , 2009, “ Resistance of Full-Flow Penetrometers in Rate-Dependent and Strain-Softening Clay,” Geotechnique, 59(2), pp. 79–86.
Adachi, T. , and Oka, F. , 1982, “ Constitutive Equations for Normally Consolidated Clay Based on Elasto-Viscoplasticity,” Soils Found., 22(4), pp. 57–70.
Yin, J. H. , Zhu, J. G. , and Graham, J. , 2002, “ A New Elastic Viscoplastic Model for Time-Dependent Behaviour of Normally and Overconsolidated Clays: Theory and Verification,” Can. Geotech. J., 39(1), pp. 157–173.
Kelln, C. , Sharma, J. , Hughes, D. , and Graham, J. , 2008, “ An Improved Elastic–Viscoplastic Soil Model,” Can. Geotech. J., 45(10), pp. 1356–1376.
Yin, Z. , and Wang, J. , 2012, “ A One-Dimensional Strain-Rate Based Model for Soft Structured Clays,” Sci. China. Technol. Sci., 55(1), pp. 90–100.
Perzyna, P. , 1963, “ The Constitutive Equations for Rate Sensitive Plastic Materials,” Q. Appl. Math., 20, pp. 321–332.
Zhu, H. , and Randolph, M. F. , 2011, “ Numerical Analysis of a Cylinder Moving Through Rate-Dependent Undrained Soil,” Ocean Eng., 38(7), pp. 943–953.
Wroth, C. P. , 1984, “ The Interpretation of In Situ Soil Tests,” Geotechnique, 34(4), pp. 449–489.
Locat, J. , and Demers, D. , 1988, “ Viscosity, Yield Stress, Remolded Strength, and Liquidity Index Relationships for Sensitive Clays,” Can. Geotech. J., 25(4), pp. 799–806.
Goh, T. L. , 2003, “Stabilisation of an Excavation by an Embedded Improved Soil Layer,” Ph.D. thesis, National University of Singapore, Singapore.
Hong, Y. , He, M. B. , Wang, L. , Wang, Z. , Ng, C. W. W. , and Mašín, D. , 2017, “ Cyclic Lateral Response and Failure Mechanisms of Semi-Rigid Pile in Soft Clay: Centrifuge Tests and Numerical Modelling,” Can. Geotech. J., 54(6), pp. 806–824.
Jeong, S. W. , Leroueil, S. , and Locat, J. , 2009, “ Applicability of Power Law for Describing the Rheology of Soils of Different Origins and Characteristics,” Can. Geotech. J., 46(9), pp. 1011–1023.
BSI, 1990, “Methods of Test for Soils for Civil Engineering Purposes,” British Standards Institution, London, Standard No. BS1377.
Karstunen, M. , and Yin, Z. Y. , 2010, “ Modelling Time-Dependent Behaviour of Murro Test Embankment,” Geotechnique, 60(10), pp. 735–749.
Zhu, Q. , Wu, Z. , Li, Y. , Xu, C. , Wang, J. , and Xia, X. , 2014, “ A Modified Creep Index and Its Application to Viscoplastic Modelling of Soft Clays,” J. Zhejiang. Univ., Sci. A, 15(4), pp. 272–281.
Liu, Y. , Hu, J. , Wei, H. , and Saw, A. L. , 2017, “ A Direct Simulation Algorithm for a Class of Beta Random Fields in Modelling Material Properties,” Comput. Methods Appl. Mech. Eng., 326(1), pp. 642–655.

## Figures

Fig. 1

Extending the modified Cam-Clay yield locus to reflect strain rate effect

Fig. 2

Viscometer (HAAKE RotoVisco 1): (a) its plate–plate sensor system

Fig. 3

Shear stress strength against shear strain rate for soil with 72% water content and 1.87 void ratio

Fig. 4

Shear stress strength against shear strain rate for soil with 74% water content and 1.92 void ratio

Fig. 5

Shear stress strength against shear strain rate for soil with 83% water content and 2.16 void ratio

Fig. 6

Shear stress strength against shear strain rate for soil with 87% water content and 2.26 void ratio

Fig. 7

Normalized shear stress versus shear strain rate datafor soil of various liquidity index (IL) with a reference shear strain rate γ˙ref of 1 s−1 (sur,ref is the shear strength at reference shear strain rate γ˙ref)

Fig. 8

The calculated and measured shear strength versus shear strain rate data for soil with 83% water content

## Tables

Table 1 Cam-Clay properties of Malaysian kaolin clay [17]
Table 2 Parameters inferred from viscometer test results of Malaysian kaolin clay

## Discussions

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