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Implementation of an elasto-plastic constitutive model for cement stabilized clay in a non-linear finite element analysis N.N.S. Yapage and D.S. Liyanapathirana School of Engineering, University of Western Sydney, Penrith, Australia Abstract Purpose – Several constitutive models are available in the literature to describe the mechanical behaviour of cement stabilized soils. However, difficulties in implementing such models within commercial finite element programs have hindered their application to solve related boundary value problems. Therefore, the aim of this study is to implement a constitutive model, which has the capability to simulate cement stabilized soil behaviour, into the finite element program ABAQUS through the user material subroutine UMAT. Design/methodology/approach – After a detailed review of existing constitutive models for cement stabilized soils, a model based on the elasto-plastic theory and the extended critical state concept with an associated flow rule is selected for the finite element implementation. A semi-implicit integration method (cutting plane algorithm) with a continuum elasto-plastic modulus and path dependent stress prediction strategy has been used in the implementation. The performance of the new finite element formulation of the constitutive model is verified by simulating triaxial test data using the finite element program with the new implementation and predictions from constitutive equations as well as experimental data. Findings – The paper provides the implementation procedure of the constitutive model into ABAQUS but this method is useful for the implementation of any other constitutive model into ABAQUS or any other finite element program. Simulated results for the volumetric deformation of cement stabilized soils show that the cement stabilized soils do not obey the associated flow rule at high confining pressures. The parametric study shows that the influence of cementation increases the brittle nature and the bearing capacity of treated clay. In addition the results show that proposed finite element implementation has the ability to illustrate key features of the cement stabilized clay. Originality/value – This paper presents an implementation of an elasto-plastic constitutive model, based on the extended critical state concept, for cement stabilized soils into a finite element programme, which has been identified as an important and challenging topic in computational geomechanics. This implementation is useful in solving boundary value problems in geomechanics involving cement stabilized soils, incorporating key characteristics of these soils. Keywords Cement stabilized soils, Constitutive modelling, Finite element method, Cutting plane algorithm, Triaxial tests, Modelling, Soils Paper type Research paper 1. Introduction Soft ground improvement using cement stabilization has been widely used in coastal and low lying regions since 1970s. Although there is a wide range of ground The current issue and full text archive of this journal is available at www.emeraldinsight.com/0264-4401.htm The authors would like to acknowledge the financial support for this research provided by the Australian Research Council and Coffey Geotechnics Pty Ltd under the Linkage project LP0990581. EC 30,1 74 Received 9 September 2011 Revised 6 February 2012 9 February 2012 Accepted 15 February 2012 Engineering Computations: International Journal for Computer-Aided Engineering and Software Vol. 30 No. 1, 2013 pp. 74-96 q Emerald Group Publishing Limited 0264-4401 DOI 10.1108/02644401311286017

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  • Implementation of anelasto-plastic constitutive modelfor cement stabilized clay in a

    non-linear finite element analysisN.N.S. Yapage and D.S. Liyanapathirana

    School of Engineering, University of Western Sydney, Penrith, Australia

    Abstract

    Purpose Several constitutive models are available in the literature to describe the mechanicalbehaviour of cement stabilized soils. However, difficulties in implementing such models withincommercial finite element programs have hindered their application to solve related boundary valueproblems. Therefore, the aim of this study is to implement a constitutive model, which has thecapability to simulate cement stabilized soil behaviour, into the finite element program ABAQUSthrough the user material subroutine UMAT.

    Design/methodology/approach After a detailed review of existing constitutive models forcement stabilized soils, a model based on the elasto-plastic theory and the extended critical stateconcept with an associated flow rule is selected for the finite element implementation. A semi-implicitintegration method (cutting plane algorithm) with a continuum elasto-plastic modulus and pathdependent stress prediction strategy has been used in the implementation. The performance of the newfinite element formulation of the constitutive model is verified by simulating triaxial test data usingthe finite element program with the new implementation and predictions from constitutive equationsas well as experimental data.

    Findings The paper provides the implementation procedure of the constitutive model intoABAQUS but this method is useful for the implementation of any other constitutive model intoABAQUS or any other finite element program. Simulated results for the volumetric deformation ofcement stabilized soils show that the cement stabilized soils do not obey the associated flow rule athigh confining pressures. The parametric study shows that the influence of cementation increases thebrittle nature and the bearing capacity of treated clay. In addition the results show that proposed finiteelement implementation has the ability to illustrate key features of the cement stabilized clay.

    Originality/value This paper presents an implementation of an elasto-plastic constitutive model,based on the extended critical state concept, for cement stabilized soils into a finite elementprogramme, which has been identified as an important and challenging topic in computationalgeomechanics. This implementation is useful in solving boundary value problems in geomechanicsinvolving cement stabilized soils, incorporating key characteristics of these soils.

    Keywords Cement stabilized soils, Constitutive modelling, Finite element method,Cutting plane algorithm, Triaxial tests, Modelling, Soils

    Paper type Research paper

    1. IntroductionSoft ground improvement using cement stabilization has been widely used in coastaland low lying regions since 1970s. Although there is a wide range of ground

    The current issue and full text archive of this journal is available at

    www.emeraldinsight.com/0264-4401.htm

    The authors would like to acknowledge the financial support for this research provided by theAustralian Research Council and Coffey Geotechnics Pty Ltd under the Linkage projectLP0990581.

    EC30,1

    74

    Received 9 September 2011Revised 6 February 20129 February 2012Accepted 15 February 2012

    Engineering Computations:International Journal forComputer-Aided Engineering andSoftwareVol. 30 No. 1, 2013pp. 74-96q Emerald Group Publishing Limited0264-4401DOI 10.1108/02644401311286017

  • improvement methods available, cement stabilization has become an attractivesolution for soft ground related problems due to advantages such as:

    . cost-effectiveness;

    . better stiffness, strength and deformation properties of the treated ground thanthe surrounding soil; and

    . suitability for fast-track construction.

    When soils are mixed with cement, soil behaviour is totally different to natural orreconstituted states as soil and cement form a new material. Then these soils are calledartificially structured soils. Therefore, classical constitutive models such as ModifiedCam Clay are inadequate to predict the behaviour of this new material. Hence newconstitutive models are required to model the induced soil-cement structure.

    Constitutive models describe the mechanical behaviour of engineering materialsin the form of a relationship between stresses and strains. Constitutive models areessential for all numerical calculations of routine boundary value problems ingeotechnical engineering. Basically constitutive models consist of a number ofmathematical equations and controlling parameters depending on the complexity ofthe model and the ability of the model in describing the different facets of the soilbehaviour. Majority of the models are developed considering the behaviour of naturallystructured soils (Baudet and Stallebrass, 2004; Kavvadas and Amorosi, 2000; Rouainiaand Muir Wood, 2000; Taiebat et al., 2009; Wheeler et al., 2003). In artificially cementedsoils, size of the yield surface is predominantly related to the bond strength betweensoil particles in addition to the void ratio. Hence cement stabilized soils possess atensile strength. Recently few constitutive models have been developed especially forartificially cemented soils (Lee et al., 2004; Horpibulsuk et al., 2010; Kasama et al., 2000;Suebsuk et al., 2010; Vatsala et al., 2001) within the framework of Modified Cam Claymodel. Accuracy and efficiency of these models are mostly verified simulating thebehaviour of a single gauss integration point of a soil element within the context of thefinite element method. Their implementation into finite element programs and thenapplication to boundary value problems are rather limited.

    Numerical modelling using the finite element method has been carried out by manyresearchers to study the behaviour of cement stabilized soft ground in the form ofdeep cement mixed (DCM) columns in different geotechnical applications such asembankments for railways and roads (Filz, 2007; Han and Gabr, 2002; Huang et al.,2006), bearing capacity and settlement of treated soil (Dong et al., 2004) and seismicmitigation applications (Namikawa et al., 2007). To model the cement stabilized soil,Mohr-Coulomb model (Filz, 2007; Huang et al., 2006; Dong et al., 2004) andDrucker-Prager model (Han and Gabr, 2002) have been used. However, neitherMohr-Coulomb model nor Drucker-Prager model has the ability to describe the keycharacteristics such as the strain softening behaviour of cement stabilized soil and maylead to unsafe and uneconomical designs. The main reason for using these simplemodels to simulate the cement stabilized soil behaviour is the lack of availability ofadvanced soil models suitable for the simulation of cement treated soil in manycommercially available finite element programs.

    When implementing new constitutive models into finite element programs,algorithms are used to numerically integrate the rate constitutive equations andthen update the stress state. Existing approaches can be mainly classified as explicit

    An elasto-plasticconstitutive

    model

    75

  • integration methods and implicit integration methods. Explicit schemes are simple andthe implementation is straightforward. However, the calculated stresses may notsatisfy the yield criterion automatically. When stress state changes from elastic toplastic, intersection of the stress state with the yield surface should be computed.Accuracy of these schemes can be improved by introducing error control andsub-stepping in the finite element formulation (Sloan, 1987, 2001).

    Conversely, the finite element formulation of the implicit scheme is complicated.The derivation of the consistent tangent modulus involves solution of a system ofequations but the advantage is the resulting stresses will automatically satisfy theyield criterion to a specified tolerance (Sloan et al., 2001). However, the majordrawbacks of the implicit scheme when used for advanced constitutive models are:

    . there is a possibility for divergence; and

    . the difficulty of deriving the consistent tangent modulus.

    To overcome these complexities of the implicit schemes, Oritz and Popov (1985) andSimo and Taylor (1986) have introduced a semi-implicit return mapping approachcalled the cutting plane algorithm (CPA), which bypasses the calculation of consistenttangent modulus.

    The aim of the present study is to implement the constitutive model developed byLee et al. (2004), which has the capability to simulate cement stabilized soil behaviour,into the finite element program ABAQUS through the user material subroutine UMAT(ABAQUS Inc., 2010). A semi-implicit algorithm based on CPA with a continuumtangent modulus has been used. Although the implementation presented in this paperis focused on ABAQUS, the steps involved in the integration process are applicable toany other finite element program.

    General steps used for integrating the stress-strain relationship with an isotropichardening law have been discussed. The implemented material subroutine is verifiedby simulating the behaviour of cement stabilized soft clay presented by Lee et al. (2004)based on the constitutive equations and drained triaxial test data. Also the paperpresents finite element simulations carried out for drained triaxial tests for cementstabilized Ariake clay presented by Suebsuk et al. (2010).

    2. Constitutive model for cement stabilized soilThe constitutive model selected for the finite element implementation is the modeldeveloped by Lee et al. (2004) for artificially cemented clays. The model is formulatedwithin the framework of the critical state theory. The concept behind this model is that thestrength of bonds between soil particles increases by adding cement. When there is noadded cement, there is no added bond strength component in the model, and hence themodel behaviour is similar to the MCC model. With the increase in plastic deviatoric strain,soil will reach the critical state due to breakage of cementation bonds between soil particles.

    This model has assumed that the failure state line of cemented clay is parallel to thatof the reconstituted clay and the intercept of the failure state line exhibits the tensilestrength due to the effects of cementation similar to Horpibulsuk et al. (2010) andSuebsuk et al. (2010). Therefore, cement stabilized clay has an elliptical yield surface inthe p 0 2 q stress space similar to that of the reconstituted clay, where p 0 is the meaneffective stress and q is the deviatoric stress. This concept has widely been employed by

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  • many researchers (Horpibulsuk et al., 2010; Kasama et al., 2000; Suebsuk et al., 2010;Liu et al., 2006; Muir Wood, 1990) and graphically shown in Figure 1.

    It is assumed that the plastic potential, G, and the yield loci, F, change with theplastic strain. The model follows associated flow rule obeying normality condition.Considering cementation effect, modified effective stress concept is employed to get theyield function by rearranging the yield function of the MCC model, as shown below:

    G F q2 M 2cs p 0 p0t

    p 0 p 0c

    1where p

    0c is the size of the initial yield surface, p

    0t is the cementation parameter, which

    takes into account the effect of bonding between soil particles. p0t controls the size of the

    yield surface in conjunction with the plastic deviatoric strain. p0t is given by the

    following equation:

    p0t p

    0t;o e2kp1

    pq 2

    where 1 pq is the cumulative plastic deviatoric strain, kp is the bonding control parameter,

    Mcs is the critical stress ratio. p0t;o is the initial value of the cementation parameter.

    According to equation (2), when plastic deviatoric strain increases, p0t decreases and it

    becomes zero after complete elimination of bonds due to shearing. This stress state isequivalent to the reconstituted state of the soil. Cumulative plastic deviatoric straincomponent is calculated by getting the summation of all incremental plastic deviatoricstrain terms at each step. Incremental plastic deviatoric strain is calculated bysubtracting the total elastic deviatoric strain component from the total deviatoric straincomponent. Equations (3)-(5) give the total plastic deviatoric strain, total deviatoricstrain and incremental elastic deviatoric strain components, respectively:

    1pq 1q 2Xni1

    d1eq

    i

    3

    Figure 1.Graphical representationof the constitutive model

    An elasto-plasticconstitutive

    model

    77

  • 1q 2

    p3

    1rr 2 1zz2 1zz 2 1uu2 1uu 2 1rr2 3212rz

    1=24

    d1eq

    i 21 vk

    912 2v1 ep 0 dqi 5

    where v is the Poissons ratio of the soil, k is the swelling index of the soil, p 0 is the meaneffective stress, dqi is the increment in deviatoric stress during the increment i, e is thevoid ratio, 1rr, 1zz and 1uu are, respectively, the radial, vertical and tangentialcomponents of the total direct strains and 1rz is the total shear strain component for anaxisymmetric situation. The number of strain increments is denoted by i and varies from1 to n.

    The bonding effect of cement stabilized clay increases the size of the yield surfaceexhibiting tensile strength. Therefore, yield surface crosses the q axis and has anegative value for p 0 in the p 0 2 q stress space, when q is zero. With the elimination ofbonds, yield surface shifts to the origin of the p 0 2 q stress space. In the modelproposed by Lee et al. (2004), it is assumed that l changes with the p 0 and thecorresponding void ratio, e.

    Since the model uses associated plasticity, the vector of plastic strain increment d1p

    is in the direction of the outward normal to the yield surface. Therefore, the flow rulecan be defined as:

    d1pvd1pq

    F=Pt

    F=q M

    2cs 2p

    0 2 p 0c p 0t

    2q6

    This model uses an isotropic hardening rule to describe the enlargement and shifting ofthe yield loci and it mainly depends on the plastic volumetric strain increment asshown below:

    dp0c

    1 el2 k p

    0c d1

    pv 7

    This model is developed assuming that the cement stabilized soils behave in a drainedmanner. This assumption is reasonable for cement stabilized soils created by thedry mixing method, where dry cement is in situ mixed with soil. At the same timethe model has been developed assuming that the cement stabilized soils behave inan isotropic manner. Haung and Airey (1998) and Rotta et al. (2003) showed that thevariation in mechanical properties in cement stabilized soil is fundamentally isotropic.This model is simple and it requires only eight model parameters

    MCS; p 0t;0 ; l; k; k1P; v; eo; p0c

    , which can be derived using standard laboratory tests.

    3. Stress integration algorithmThe updating scheme for the path dependent materials like elasto-plastic materialsrequires a numerical algorithm to integrate the plastic rate constitutive equations. It isimportant to mention that the accuracy of the overall non-linear finite element schemedepends on the accuracy of the numerical algorithm adopted to formulate the state updateprocedures and this is currently an active research area (Clausen et al., 2006; Simo andTaylor, 1986, 1985; Nazem et al., 2006; Simo and Hughes, 1998; Wang and Atluri, 1994).

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  • When the stress integration method is robust, accurate and fast, then the performanceof the finite element program significantly improves, especially with advancedelasto-plastic constitutive models. Stability, consistency and simplicity are the otherfactors which contribute to the superiority of integration schemes.

    The CPA used in this study is a special return mapping algorithm developed tobypass the calculation of second derivative of the plastic potential as it uses acontinuum tangent modulus. Thus, it makes the implementation easier when dealingwith complicated constitutive models. However, Simo and Hughes (1998) showed thatthe CPA method limits its use in practical finite element implementations withNewton-Rhapson solution strategy due to lack of consistent linearization. Nevertheless,Huang and Griffiths (2009) showed that this can be overcome using a path dependentstrategy and with that strategy, CPA can be made more stable and efficient. Therefore,in this study, the CPA with the continuum elasto-plastic modulus in conjunction with apath dependent strategy is implemented.

    This algorithm consists of two major steps. They are the elastic predictor andplastic corrector steps. Next sections describe the development of the stress integrationalgorithm to update the new stress state using the strain increments computed withinthe main finite element program and transferred to the UMAT.

    3.1 Derivation of the continuum elasto-plastic tangent modulusIn this section the derivation of continuumum tangent modulus relating incrementalstrains and stresses is discussed. Since the aim is to implement this model into a finiteelement program, all derivatives, strains and stresses are given in the matrix form.

    The yield function, F, for the model is given by equation (1). When the material isplastic, the stress state should be on the yield surface and hence it should satisfy thecondition, F 0. Further loading will cause plastic deformations and changes thestress state directing outwards the yield surface. If the hardening parameter remainsunchanged, this will make F . 0. However, plastic definitions are accompanied withincreasing hardening and the yield surface expands in such a way that the yieldsurface passes through the current stress state of the material. Therefore, loading andhardening together satisfy the condition that the material remains on the yield surface.Hence dF 0 and using the chain rule of differentiation, the incremental change of theyield function can be written as:

    dF Fs 0

    T{ds 0} F

    1pvd1 pv

    F

    1 pqd1 pq 0 8

    where d1pv is the plastic volumetric strain increment, d1pq is the plastic deviatoric strain

    increment and {ds 0} is the corresponding stress increment.This implementation is explained for an axisymmetric condition. Therefore, {ds 0}

    is given by:

    {ds 0}

    ds0r

    ds0z

    ds0u

    dt0rz

    8>>>>>>>>>:

    9>>>>>=>>>>>;

    9

    An elasto-plasticconstitutive

    model

    79

  • where ds0r , ds

    0z and ds

    0u are, respectively, the radial, vertical and tangential

    components of the effective stress increment and dt0rz is the shear stress increment.

    The incremental plastic strains d1pv and d1pq are related to the plastic potential

    function, G. Since the material model follows associated plasticity, G F.Consequently, the plastic strains are given by:

    {d1p} l Fs 0

    10a

    d1pv lF

    p 0

    10b

    d1pq lF

    q

    10c

    l is the scalar multiplier, which represents the magnitude of the plastic flow.Constitutive behaviour can be written using the incremental stress {ds 0} and the

    incremental elastic strain {d1 e} assuming only the component of the elastic strain cangenerate stresses through the elastic constitutive matrix as shown below:

    {ds 0} D e{d1 e} 11where [D e] is the elastic constitutive matrix. For an axisymmetric situation, [D e] is a4 4 matrix given by:

    De EMBED equation3 12where K and m are material parameters defined as:

    K 1 ep0

    k13

    m 312 2v21 v K 14

    where m is the elastic shear modulus.For the case of a small strain increment, the total strain increment {d1} consists of

    elastic, {d1 e} and plastic, {d1 p} strain increments. Hence {d1 e} can be written as:

    {d1 e} {d1}2 {d1 p} 15Therefore, the following constitutive relationship can be derived using equations(10a)-(10c), (11) and (15):

    {ds 0} D e{d1}2 {d1 p} D e {d1}2 l Fs 0

    16

    Using equations (8), (10), (11), (15) and (16) following equation for l can be derived:

    l {F=s0}TD e{d1}

    {F=s 0}TD e{F=s 0}2 F=1pv F=p 02 F=1pq F=q 17

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  • Finally the continuum tangent modulus relating incremental stresses and strains isgiven by:

    {ds 0} hD e2

    D e{F=s 0} 0{F=s 0}D e={F=s 0} 0TD e{F=s 0}

    2 F= 1pv

    F=p 02 F= 1pq

    F=qi

    {d1}

    18Equation (18) is the elasto-plastic constitutive equation in the form of{ds 0} D ep{d1}, where [D ep] is the continuum elasto-plastic tangent stiffnessmatrix (continuum Jacobian).

    When deriving the tangent stiffness matrix, the stress gradients of the yieldfunction are calculated as follows:

    F

    s 0

    F

    p 0p 0

    s 0r Fq

    q

    s 0r

    F

    p 0p 0

    s 0z Fq

    q

    s 0z

    F

    p 0p 0

    s0u

    Fq

    q

    s0u

    F

    q

    q

    trz

    8>>>>>>>>>>>>>>>>>>>>>>>>>>>:

    9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

    19

    where, for an axisymmetric problem, p 0 and q are given by:

    p 0 s0r s 0z s 0u

    3

    20

    q 12

    p s 0r 2 s0z

    2 s 0z 2 s 0u 2 s 0u 2 s 0r

    26t 2rz 1=2

    21

    Derivatives of the yield function with respect to p 0and q are given below:

    F

    p 0 M 2cs 2p 0 p

    0t 2 p

    0c

    h i22

    F

    q 2q 23

    Gradient of the stress invariants p 0 and q can be computed as follows:

    p 0

    s 0r p

    0

    s0u

    p0

    s 0z 1

    324

    q

    s 0r 2s

    0r 2 s

    0z 2 s

    0u

    2q;q

    s 0z 2s

    0u 2 s

    0r 2 s

    0z

    2q;q

    s0u

    2s0z 2 s

    0u 2 s

    0r

    2q25

    An elasto-plasticconstitutive

    model

    81

  • Finally, {F=s 0} can be evaluated as follows:

    F

    s 0

    2s0r 2 s

    0z 2 s

    0u

    M 2cs3 2p 0 p 0t 2 p 0c 2s

    0u 2 s

    0r 2 s

    0z

    M 2cs3 2p 0 p 0t 2 p 0c 2s

    0z 2 s

    0u 2 s

    0r

    M 2cs3 2p 0 p 0t 2 p 0c 6trz

    8>>>>>>>>>>>>>:

    9>>>>>>>=>>>>>>>;

    26

    A change in the yield surface updates d1pv and d1pq. Hence, F=1

    pv and F=1

    pq in the

    continuum tangent modulus can be evaluated using the chain rule as shown below:

    F

    1pv Fp

    0c

    p0c

    1pv27

    F=p 0c and p 0c=1pv can be evaluated using:F

    p0c

    2M 2cs p 0 p0t

    ;p

    0c

    1pv 1 el2 k p

    0c 28

    F

    1pq Fp

    0t

    p0t

    1pq29

    F=p 0t and p 0t=1pq can be evaluated using:F

    p0t

    M 2cs p 0 2 p0c

    ;p

    0t

    1pq 2kpp 0t;oexp 2kp1pq

    30

    3.2 Cutting plane algorithm (CPA)In this algorithm, the calculation of plastic strain increment and the occurrence ofconvergence at the time tn1 are completed at the end of the step and therefore theintegration scheme can be described as below.

    At the beginning of each stress update step (n 1), the total strain increment{d1n1} as well as total strain {1n}, total plastic strain {1pn}, total effective stress {s

    0n}

    and state variables (void ratio, e and preconsoildation pressure, p 0c) saved at the end ofprevious stress update, n, are given as input parameters. In the elastic predictor step,assuming total strain increment is completely elastic, a trial stress state is calculated asshown below:

    0trialn1

    n o s 0nn o

    D e{d1n1} 31

    Then the yield condition is checked. If the yield function meets the conditionFs 0trialn1 # 0, then the trial stress state is inside the yield surface and the materialbehaviour is elastic. Therefore, s 0trialn1 is the actual stress state. Otherwise the stressstate is outside the yield surface and hence the stress state is plastic. Then the plasticcorrector step is employed. The Newton-Raphson method is used to iteratively return

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  • the trial stress state to the updated yield surface by correcting the plastic strainincrement using the plasticity parameter l The derivation of plasticity parameter isdescribed in Section 3.2.

    In the CPA algorithm, the yield function is linearized around the current stress state,{s kn1}. Hence, the yield function at the iteration k 1 and k are related as shownbelow:

    F k1 F k akn1

    sk1n1

    2 skn1 F

    1pvd1pv

    F

    1pqd1pq 32

    By combining equations (10a)-(10c), (17) and (32), and F k1 0, the plasticityparameter can be written shown below:

    lkn1 F k

    akn1n oTD e akn1

    n o2 F=1pv F=p 02 F=1pq F=q

    33

    The process with respect to CPA is graphically shown in Figure 2.The plastic corrector, {s

    0correctorn1 } is calculated as:

    s0correctorn1

    n o ln1D e{an1}; {a} F

    s 0

    34

    Therefore, the corrected stress state is calculated as:

    s0n1

    n o s 0trialn1n o

    2 ln1D e{an1} 35The iteration loop can be summarised as shown below.

    Initialization and elastic predictor step: initialize the input parameters at thebeginning of the iteration loop (k is the iteration number), and calculate the trial stress:

    k 0 : 1p0n1n o

    1pn

    ; l0n1 0 36a

    Figure 2.Graphical interpretation

    of semi-implicitalgorithm (CPA)(a) Hardening (b) Softening

    An elasto-plasticconstitutive

    model

    83

  • ds00n1

    n o D e {d1n1}2 d1p0n1

    n o 36b

    s00n1

    n o s 0nn o

    ds 00n1n o

    36cCheck the yield condition and convergence at kth iteration:

    F k F skn1n o

    37

    If Fk , TOL, convergence is ok. Otherwise, proceed to the plastic corrector step.Plastic corrector step: compute the plastic corrector (plasticity parameter), lkn1

    given in equation (33).Obtain the stress correction:

    {ds k} 2lkn1D e akn1n o

    38Update the plastic and elastic strains as well as the stress state:

    1pk1n1n o

    1pkn1n o

    {d1 pk} 1pkn1n o

    ln1{an1} 39a

    1ek1n1n o

    1ekn1n o

    {d1pk} 1ekn1n o

    ln1{an1} 39b

    s0k1n1

    n o skn1n o

    2 {ds0k} 39c

    k k 1, go to equation (37).This iteration loop continues until the convergence is achieved.

    3.3 Development of subroutine, UMATThe numerical procedure described in the previous section can be used to implementany constitutive model into a finite element program. In this paper, the numericalprocedure has been used to implement a constitutive model for cement stabilized soils(Lee et al., 2004) and incorporated into the finite element software, ABAQUS/Standardthrough the user material subroutine (UMAT). Implementation of the material model,stress updates and tangential modulus computation (material Jacobian matrix) arecarried out within the UMAT subroutine.

    UMAT performs two main functions necessary for an analysis:

    (1) Computes incremental stresses based on the strain increments passed intoUMAT from the main program.

    (2) Computes the updated tangential stiffness modulus based on the constitutivemodel used for the material modelling, for the global Newton-Raphson iterationemployed by main program to solve non-linear problems.

    The total stress, strain and the user-defined solution-dependent state variables(STATEV) from the last converged equilibrium state and the total strain increment forthe current step, total analysis time, time increment and other predefined field

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  • variables are passed into the UMAT subroutine. From the equilibrium state at time tn,ABAQUS computes the total strain increment d1totaln based on the applied incrementalloading at the time increment Dtn to find the updated stress state according to theconstitutive law used for the material behaviour. After that the convergence is checkedusing the updated stresses and this procedure is performed iteratively until theconvergence is reached. The maximum number of iterations and convergence toleranceused in the subroutine are 50 and 1023, respectively.

    Within the UMAT subroutine, increments of total stress, plastic strain, statevariables and tangential stiffness matrix, which is called the material Jacobian matrix(DDSDDE), are computed. The steps involved within the ABAQUS user definedmaterial subroutine, UMAT, from time tn to tn1 can be illustrated using the flow chartshown in Figure 3.

    Figure 3.Flow chart for the UMAT

    in ABAQUS/Standard

    Initialization of history variables to the UMAT at t = tn

    Total strain total

    Time increment tn

    n

    Total stress total

    Obtain stress increments dnUpdate plastic strain n+1

    Update total stress n+1 using semi implicit integration method

    Total strain increment dtotaln

    n

    Call UMAT

    Specify new t in UMAT

    Convergence

    Calculate the material tangent modulus

    Update state variables

    Compute the global stiffness matrixGenerates new strain increment

    ABAQUS main programme updates strain

    No

    Yes

    p

    total

    total = total + dtotaln+1 n n

    n+1 dtotal at tn+1

    An elasto-plasticconstitutive

    model

    85

  • After the solution is converged to the correct stress state, the degree of cementation, p0t ,

    and the size of the yield surface, p0c , are updated at the end of each time increment using

    equations (2) and (40) (Borja and Lee, 1990), respectively:

    p0cn1 p

    0cn

    exp1 en1ln1 2 k

    d1pv;n1

    40

    Then the continuum Jacobian is calculated using equation (18). The Jacobian should beaccurate to achieve the fast convergence and the accuracy of the overall solution as themain finite element program needs this constitutive matrix to compute the globaltangent stiffness matrix, which is used to estimate the strain increment for the next step.

    Performance of the UMAT and the constitutive model are investigated in thefollowing sections using the drained triaxial test data reported by Lee et al. (2004) andSuebsuk et al. (2010).

    4. Verification of the formulationThe performance of the new finite element formulation of the constitutive model isverified using an Excel spreadsheet program developed by the authors to compute thestress paths and variation of volumetric strain with the axial strain at a single Gaussintegration point. This is the only possible method we can use to verify the formulationbecause results given by Lee et al. (2004) are incorrect due to some errors in the equations(derivatives) in Formulation of the Constitutive Model section of their paper.

    Due to the symmetry of the triaxial specimen, finite element mesh is developed onlyfor a quarter of the cylindrical specimen of 5 cm in diameter and 10 cm in height. Theelement type used for this analysis is an eight-node axisymmetric continuum elementand reduced integration (four Gauss integration points). Table I shows the materialproperties used for the finite element analysis carried out using ABAQUS/Standard.Figure 4 shows the variation of deviatoric stress with axial strain and Figure 5 showsthe variation of volumetric strain with the axial strain for a triaxial test with confiningpressures of 20 and 160 kPa obtained from the finite element analysis and the Excelspreadsheet program based on the constitutive equations for a single Gauss integrationpoint. Finite element simulation results overlap with the Excel spreadsheet resultscalculated for a single Gauss point. These results confirm that the numericalimplementation presented in the paper is in agreement with the cement stabilized soilbehaviour described by the constitutive model.

    5. Effect of confining pressure on the cement stabilized soil behaviourDrained triaxial tests are simulated using the constitutive model implemented inABAQUS/Standard by varying the confining pressure from 50 to 400 kPa. The modelparameters used for this simulation are given in Table II.

    Figure 6 shows the variation of deviatoric stress with axial strain when theconfining pressure increases. According to this figure, the deviatoric stress developed

    Model parameters for soil type used by Lee et al. (2004)v k MCS kp p

    0t(kPa) Confining pressure (kPa)

    0.25 0.003 1.9 10, 45 250 20, 160

    Table I.Model parameters for thedrained triaxial test

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  • within the soil sample increases with the increasing confining pressure. However, therate of destructuring decreases with the increasing confining pressure. Therefore, thecement stabilized soil shows highly brittle behaviour under low confining pressuresand the ductile behaviour under high confining pressures. Furthermore, at higherconfining pressures, the soil can undergo large strains before failure.

    Model parameters for soil type used by Lee et al. (2004)v k MCS kp p

    0t(kPa) Confining pressure (kPa)

    0.25 0.01 1.4 20, 40, 45, 50 200 50, 100, 200, 400

    Table II.Model parameters used to

    investigate the effect ofconfining pressure

    Figure 4.Variation of deviator

    stress with axial strain

    1,200

    1,000

    Simulation-160 kPa

    Simulation-20 kPa

    Experimental-160 kPa

    Experimental-20 kPa

    Single Gauss-160 kPa

    Single Gauss-20 kPa

    800

    600

    400

    200

    00 0.05 0.1 0.15

    Axial strain (ea)

    Dev

    iato

    ric

    stre

    ss (

    kPa)

    0.2 0.25

    Source: Experimental data by Lee et al. (2004)

    Figure 5.Variation of volumetricstrain with axial strain

    0.01

    0.005

    Simulation-160 kPa

    Simulation-20 kPa

    Experimental-160 kPa

    Experimental-20 kPa

    Single Gauss-160 kPa

    Single Gauss-20 kPa

    0

    0.005

    0.01

    0.015

    0.02

    0.03

    0.025

    0 0.05 0.1 0.15

    Axial strain (ea)

    Vo

    lum

    etri

    c st

    rain

    (e n

    )

    0.2 0.25

    Source: Experimental data by Lee et al. (2004)

    An elasto-plasticconstitutive

    model

    87

  • Figure 7 shows the variation of volumetric strain with axial strain when the confiningpressure is increasing. This figure shows that the soil samples subjected to lowconfining pressures behave more dilatantly than the samples subjected to highconfining pressures. This is due to the higher bonding degradation at low confiningpressures and lower bonding degradation of samples at higher confining pressures.Furthermore, the volume change characteristics during drained shear are largelyinfluenced by the confining pressure. These results explain that the cement stabilizedsoil behaviour is similar to the behaviour of highly overconsolidated clays or soft rocks.

    6. Cement stabilized soil vs same soil at the reconstituted stateIn order to investigate the influence of additional structure due to cementation bondson the soil behaviour, another set of simulations were performed using p

    0t 0, which

    depicts the reconstituted state of the cement stabilized soil (i.e. when the cementationbonds are completely removed from the cement stabilized soil). The model parametersused for this study are given in Table III. According to Figure 8(a), after breaking all

    Model parameters for soil type used by Lee et al. (2004)v k MCS kp p

    0t(kPa) Confining pressure (kPa)

    0.25 0.01 1.85 50 0, 200 100

    Table III.Model parameters usedfor cemented andremoulded clay

    Figure 6.Effect of confiningpressure on deviatoricstress

    50 kPa100 kPa200 kPa400 kPa

    Confining Pressure

    0 0.05

    1,000

    800

    600

    400

    200

    00.1 0.15

    Axial strain (ea)

    Dev

    iato

    ric

    stra

    in, q

    (kP

    a)

    0.2 0.25

    Figure 7.Effect of confiningpressure on volumetricbehaviour

    50 kPa

    100 kPa

    200 kPa

    400 kPa

    Confining Pressure

    00

    0.01

    0.02

    0.03

    0.04

    0.05

    0.05 0.1 0.15Axial strain (ea)

    Vo

    lum

    etri

    c st

    rain

    (e n

    )

    0.2 0.25

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    88

  • cementation bonds, cemented sample reaches a state, which is exactly similar to thefailure state of the reconstituted soil. However, the cement stabilized soil can sustainhigher loads than the reconstituted soil. Figure 8(b) shows the higher peak values forthe volumetric deformation than that of the reconstituted clay because the cementstabilized soil contains larger void ratio within the same volume compared to the samesoil at the reconstituted state. After breakage of bonds created by the added cement,additional void ratio sustained by the cement stabilized soil diminishes. Hence only thecement stabilized sample shows dilatancy behaviour.

    7. Simulation of the experimentally observed behaviour of cementstabilized clayIn this section, the experimentally observed behaviour of cement stabilized soil issimulated using the finite element formulation presented in Section 3 to investigate the

    Figure 8.Comparison of model

    simulations for cementtreated and untreated clay

    Cement treatedclay

    Uncementedclay, OCR = 1

    00

    0.005

    0.01

    0.015

    0.025

    0.02

    0.03

    700

    600

    500

    400

    300

    200

    100

    0

    0.05 0.1 0.15

    Axial strain (ea)0.2 0.25

    0 0.05 0.1 0.15

    Axial strain (ea)0.2 0.25

    Dev

    iato

    ric

    stre

    ss, q

    (kP

    a)

    Cement treatedclay

    Uncementedclay, OCR = 1

    Vo

    lum

    etri

    c st

    rain

    (e n

    )

    (a)

    (b)Notes: (a) Deviatoric stress vs axial strain; (b) volumetric strain vs axial strain

    An elasto-plasticconstitutive

    model

    89

  • applicability of the constitutive model as well as the numerical implementation for thesimulation of cement stabilized soil behaviour. The experimental results given byLee et al. (2004), Horpibulsuk et al. (2010) and Suebsuk et al. (2010) for drained triaxialtests carried out for cement stabilized soils are used in this section.

    Figures 4 and 5 show the drained triaxial test data given by Lee et al. (2004). The soiltype used to prepare the cement stabilized soil or the cement content used to prepare thesoil samples are not given in their paper. These figures clearly show the capability of thefinite element implementation as well as the constitutive model in describing the keycharacteristics of the cement stabilized soil. It is clear that after the deviatoric stressreaches the peak value, it starts to decrease due to the crushing of the soil-cementstructure. After that soil reaches a residual value at the critical state conditionwhile the axial strain further increases. This is because the cement stabilized soil reachesthe reconstituted state due to the complete elimination of its structure. Figure 5 showsthe dilation behaviour of the cement stabilized soil after the peak volumetric strain.

    Another simulation was carried out to simulate the behaviour of Ariake clay withcement content of 9 per cent given by Suebsuk et al. (2010). The cement content Aw isdefined as the ratio of cement to clay based on the dry mass. In the constitutive model,l is not a constant. It varies with the mean effective stress and the void ratio. Isotropicconsolidation curves in the e2 ln p0 space for different soils are different. Theconstitutive model uses a third-order polynomial equation to get the void ratio atdifferent mean effective stresses as shown in Figure 9. Therefore, equation (41) isderived to fit the experimental data given by Horpibulsuk et al. (2004) for Ariake claywith cement content of 9 per cent:

    e 4:3552 2 1026p02 2 9 10211p 03 41Therefore, l0 is given by the following equation:

    l0 dedln p0

    de

    dp0 p0 24 1026p0 2 27 10211p02 p0 42

    where p0 is in kPa and the equation (41) is derived using p0 in the range of 0-1430 kPa.Comparisons of simulated model results with experimental data for Ariake clay

    with Aw 9 per cent are shown in Figures 10(a) and (b). According to Figure 10(a),

    Figure 9.Regression method for thevariable l 0 for isotropiccompression behaviour ofthe cemented Ariake withAw 9 per cent

    100 1,000 10,0001

    4.5

    4Regression line

    Aw = 9%

    Aw = 0%3.5

    3

    2.5

    2

    1.510

    Mean effective stress, p' (kPa)

    Vo

    id r

    atio

    , e

    Source: Experimental data by Horpibulsuk et al. (2004)

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    90

  • simulations have given the same peak deviatoric stresses as in the experimental resultsfor the range of confining pressures considered for the analysis. For high confiningpressures, the experimental results show high strain softening behaviour but thesimulations do not show any softening. At low confining pressures, model simulationsagree well with the experimental results demonstrating the softening behaviour of the

    Figure 10.Comparison of simulated

    model resultswith experimental

    data for Ariake clay withAw 9 per cent

    900

    800

    700

    600

    500

    Dev

    iato

    ric

    stre

    ss, q

    (kP

    a)

    Deviatoric strain, ed (%)

    400

    300

    200

    100

    00 5 10 15 20 25 30

    Deviatoric strain, ed (%)0

    0

    2

    4

    6

    8

    10

    12

    14

    16

    18

    5 10 15 20 25 30

    50 kPa

    100 kPa

    200 kPa

    Simulations

    Confining Pressure

    50 kPa

    100 kPa

    200 kPa

    Simulations

    Confining Pressure

    Vo

    lum

    etri

    c st

    rain

    , en

    (%)

    (a)

    (b)Notes: (a) Deviatoric stress vs axial strain; (b) volumetricstrain vs axial strainSource: Experimental data by Suebsuk et al. (2010)

    An elasto-plasticconstitutive

    model

    91

  • cement stabilized soil. Although the experimental results given by Suebsuk et al. (2010)show strain softening at high confining pressures, the experimental results given byLee et al. (2004) does not show softening at high confining pressures. The contradictoryexperimental results given by Suebsuk et al. (2010) and Lee et al. (2004) needs to befurther investigated by carrying out drained triaxial tests for clay soils with differentpercentages of added cement by varying the confining pressures applied to the soilsamples.

    According to Figure 10(b), model simulations successfully describe the dilationbehaviour of cement stabilized Ariake clay. However, some discrepancy in volumetricdeformation can be seen especially at high confining pressures. This deviationinherently comes from the MCC model as all models belonging to MCC family followthe associated flow rule and according to Lee et al. (2004), volumetric strain at highconfining pressure does not exactly follow the associated flow rule. However,Suebsuk et al., 2010 has stated that this is because of the influence of anisotropy. Thishas become a controversial point and needs further development of constitutive modelsfor cement stabilized soils with non-associated flow rules, taking into account theinfluence of anisotropy on the volumetric behaviour.

    In order to understand the effect of cementation on the maximum shear stress,another simulation was carried out changing the cement content of Ariake clay. Therelationships between void ratio and mean effective stress of the cement mixed Ariakeclay at different cement contents are shown in Figure 11 and the model parameters forAriake clay with different cement contents are given in Table IV.

    The following equations describe the isotropic consolidation regression lines forcement stabilized Ariake clay with cement contents of 6 and 18 per cent, respectively:

    Model parameters for Ariake clay Suebsuk et al. (2010)Aw v k MCS Kp p

    0t (kPa) Confining pressure (kPa)

    6% 0.25 0.06 1.6 30 50 509% 0.25 0.024 1.45 30, 45, 50 100 50, 100, 200

    18% 0.25 0.001 1.35 30 650 50

    Table IV.Model parameters forcemented Ariake claywith different cementcontents

    Figure 11.Isotropic compressionbehaviour of Ariake claywith different cementcontents

    10 100 1,000 10,0001

    4.5

    4

    Aw = 0%

    Aw = 6%

    Aw = 18%

    3.5

    3

    2.5

    2

    1.5

    Mean effective stress, p' (kPa)

    Vo

    id r

    atio

    , e

    Source: Horpibulsuk et al. (2004)

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    92

  • e 4:3362 7 1023p0 9 1026p02 2 3 1029p03 43

    e 3:7602 3 1025p0 2 1 1028p02 2 3 10212p03 44

    where p0 is in kPa and for equation (43), p0 varies from 0 to 760 kPa and for equation (44),p0 varies from 0 to1500 kPa.

    For the Ariake clay with cement content of 9 per cent, regression line is shown inFigure 9 and the relationship between the void ratio and the mean effective stress isgiven by equation (41).

    The parameter describing the bond strength, p0t , increases with the increasing

    cement content. Figures 12(a) and (b) show the variation of deviatoric stress andvolumetric strain with deviatoric strain. According to Figure 12, with increasingcement content, an increase in the peak deviator stress as well as dilation behaviour arepredicted for the same confining pressure. However, according to Figure 11, theadditional void ratio sustained by the clay-cementation structure of the Ariake claywith cement content of 18 per cent is much lower than that of 9 per cent cement content.Therefore, it can be seen from Figure 12(b) that the maximum volumetric strain forAw 18 per cent is lower than that of Aw 9 per cent.

    With the increasing degree of cementation, the compression index should increasedue to the sudden breakdown of clay-cementation structure (Horpibulsuk et al.,2004a, b; Lorenzo and Bergado, 2004). Nevertheless, the experimental curve of cementcontent of 18 per cent shown in Figure 11 shows some discrepancy based onthis concept. However, this parametric study shows that with the increasing cementcontent, maximum shear stress that can be applied to the clay sample before failureincreases. Furthermore, at low cement percentages, post-peak stress decreasesgradually with strain showing ductile behaviour. Conversely, post-peak stresssuddenly drops with increasing strain, demonstrating the brittle nature of treated clayat higher cement percentages.

    8. ConclusionsThis paper presented a finite element implementation based on the CPA for theconstitutive model developed by Lee et al. (2004) for cement stabilized clays. Variationof volumetric strain and deviatoric stress with the axial strain predicted by the finiteelement implementation agrees well with the model predictions simulated for a singleGauss integration point using constitutive equations.

    Some discrepancy in volumetric deformation simulated by the model and theexperimental data could be seen, especially at high confining pressures, due to the factthat the cement stabilized soils do not obey the associated flow rule under highconfining pressures as assumed in the model development and the model does not takeinto account the anisotropy of the cemented clay. The parametric study showed thatthe influence of cementation increases the brittle nature and the bearing capacity oftreated clay.

    The applicability of the model to predict behaviour of the cement stabilized soilsare investigated by simulating drained triaxial test data published by Lee et al. (2004)and Suebsuk et al. (2010). Although deviatoric stress vs axial strain agrees wellwith the data provided by Lee et al. (2004), at high confining pressures, deviatoricstress vs axial strain deviates from the data for Ariake clay presented

    An elasto-plasticconstitutive

    model

    93

  • by Suebsuk et al. (2010). It was observed that the experimental data provided byLee et al. (2004) and Suebsuk et al. (2010) also contradict each other at high confiningpressures. Hence this needs to be further investigated using experimental data.

    Finally, the results presented in the paper showed that the proposed finite elementimplementation has the ability to illustrate key features of the cement treated clay suchas softening with destructuring and the dilation behaviour after the peak stress state.Therefore, this numerical implementation is a valuable tool for geotechnical engineersto solve boundary value problems related to the structures founded on groundimproved with dry soil mixing.

    Figure 12.Simulated consolidateddrained test resultsof cemented Ariakeclay under confiningpressure, 50 kPawith cement contentsof Aw 6-18 per cent

    10 15 20 25 3000

    5

    10 15 20 25 3000

    1

    2

    3

    4

    5

    6

    7

    8

    9

    5

    1,400

    1,200

    1,000

    800

    600

    400

    200

    Aw = 6%

    Aw = 9%

    Aw = 18%

    Aw = 6%

    Aw = 9%

    Aw = 18%

    Dev

    iato

    ric

    stre

    ss, q

    (kP

    a)

    Deviatoric strain, ed (%)

    Deviatoric strain, ed (%)

    Vo

    lum

    etri

    c st

    rain

    , q (

    kPa)

    (a)

    (b)Notes: (a) Deviatoric stress vs deviatoric strain; (b) volumetric strain vsdeviatoric strain

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  • References

    ABAQUS Inc. (2010), ABAQUS Version 6.10 Users Manual, ABAQUS Inc., Providence.

    Baudet, B. and Stallebrass, S. (2004), A constitutive model for structured clays, Geotechnique,Vol. 54 No. 4, pp. 269-78.

    Borja, R.I. and Lee, S.R. (1990), Cam-Clay plasticity, part I: implicit integration of elasto-plasticconstitutive relations, Computer Methods in Applied Mechanics and Engieering, Vol. 78,pp. 49-72.

    Clausen, J., Damkilde, L. and Anderson, L. (2006), Effcient return algorithms for associatedplasticity with multiple yield planes, International Journal for Numerical Methods inEngineering, Vol. 66 No. 6, pp. 1036-59.

    Dong, P., Qin, R. and Chen, Z. (2004), Bearing capacity and settlement of concrete cored DCMpile in soft ground, Geotechnical and Geological Engineering, Vol. 22 No. 1, pp. 105-19.

    Filz, G.M. (2007), Load Transfer, Settlement, and Stability of Embankments Founded on ColumnsInstalled by Deep Mixing Method, National Technical University of Athens School of CivilEngineering Geotechnical Department Foundation Engineering Laboratory, Athens.

    Han, J. and Gabr, M. (2002), Numerical analysis of geosynthetic-reinforced and pile-supportedearth platforms over soft soil, Journal of Geotechnical and Geoenvironmental Engineering,ASCE, Vol. 128 No. 1, pp. 44-53.

    Haung, J.T. and Airey, D.W. (1998), Properties of an artificially cemented carbonate sand, Journalof Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 124 No. 6, pp. 492-9.

    Horpibulsuk, S., Bergado, D.T. and Lorenzo, G.A. (2004a), Compressibility of cement admixedclays at high water content, Geotechnique, Vol. 54 No. 2, pp. 151-4.

    Horpibulsuk, S., Miura, N. and Bergado, D.T. (2004b), Undrained shear behaviour of cementadmixed clay at high water content, Journal of Geotechnical and GeoenvironmentalEngineering, ASCE, Vol. 130 No. 10, pp. 1096-105.

    Horpibulsuk, S., Liu, M., Liyanapathirana, D. and Suebsuk, J. (2010), Behaviour of cemented claysimulated via the theoretical framework of structured Cam Clay model, Computers andGeotechnique, Vol. 37 Nos 1/2, pp. 1-9.

    Huang, J. and Griffiths, D. (2009), Return mapping algorithms and stress predictors for failureanalysis in Geomechanics, Journal of Engineering Mechanics, Vol. 135 No. 4, pp. 276-84.

    Huang, J., Han, J. and Porbaha, A. (2006), Two and three-dimensional modelling of DM columns underembankments, in DeGroot, D.J., DeJong, J.T., Frost, J.D. and Baise, L.G. (Eds), GeoCongress2006: Geotechnical Engineering in the Information Technology Age, ASCE, Atlanta, GA.

    Kasama, K., Ochiai, H. and Yasufuku, N. (2000), On the stress-strain behaviour of lightlycemented clay based on an extended critical state concept, Soils and Foundations, Vol. 40No. 5, pp. 37-47.

    Kavvadas, M. and Amorosi, A. (2000), A constitutive model for structured soils, Geotechnique,Vol. 50 No. 3, pp. 263-73.

    Lee, K., Chan, D. and Lam, K. (2004), Constitutive model for cement treated clay in a critical stateframework, Soils and Foundations, Vol. 44 No. 3, pp. 69-77.

    Liu, M.D., Carter, J.P., Horpibulsuk, S. and Liyanapathirana, D.S. (2006), Modelling thebehaviour of cemented clay, Proceedings of Sessions of GeoShanghai 2006: GroundModification and Seismic Mitigation. ASCE (GSP 152), pp. 65-72.

    Lorenzo, G.A. and Bergado, D.T. (2004), Fundamental parameters of cement-admixed clay: newapproach, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 130No. 10, pp. 1042-50.

    An elasto-plasticconstitutive

    model

    95

  • Muir Wood, D. (1990), Soil Behaviour and Critical State Soil Mechanics, Cambridge UniversityPress, Cambridge.

    Namikawa, T., Koseki, J. and Suzuki, Y. (2007), Finite Element Analysis of Lattice-shaped GroundImprovement by Cement-mixing for Liquefaction Mitigation, Soils and Foundations,Japanese Geotechnical Society, Tokyo, June, pp. 559-76.

    Nazem, M., Sheng, D. and Carter, J.P. (2006), Stress integration and mesh refinement for largedeformation in geomechanics, International Journal for Numerical Methods inEngineering, Vol. 65, pp. 1002-27.

    Oritz, M. and Popov, E. (1985), Accuracy and stability of integration algorithms for elastoplasticconstitutive relations, International Journal for Numerical Methods in Engineering, Vol. 21No. 9, pp. 1561-76.

    Rotta, G.V., Consoli, N.C., Prietto, P.M.D., Coop, M.R. and Graham, J. (2003), Isotropic yielding inan artificially cemented soil cured under stress, Geotechnique, Vol. 53 No. 5, pp. 493-501.

    Rouainia, M. and Muir Wood, D.A. (2000), Kinematic hardening constitutive model for naturalclays with loss of structure, Geotechnique, Vol. 50 No. 2, pp. 153-64.

    Simo, J.C. and Hughes, T.J.R. (1998), Computational Inelasticity, Springer Verlag, London.

    Simo, J.C. and Taylor, R.L. (1985), Consistent tangent operators for rate independent plasticity,Computer Methods in Applied Mechanics and Engineering, Vol. 48 No. 1, pp. 101-18.

    Simo, J.C. and Taylor, R.L. (1986), A return mapping algorithm for plane stress elastoplasticity,International Journal for Numerical Methods in Engineering, Vol. 22 No. 3, pp. 649-70.

    Sloan, S. (1987), Sub-stepping schemes for the numerical integration of elastoplasticstress-strain relations, International Journal for Numerical and Analytical Methods inGeomechanics, Vol. 24 No. 5, pp. 893-911.

    Sloan, S., Abbo, A. and Sheng, D. (2001), Redefined explicit integration of elastoplastic modelswith error control, International Journal for Numerical and Analytical Methods inGeomechanics, Vol. 18 Nos 1/2, pp. 121-54.

    Suebsuk, J., Horpibulsuk, S. and Liu, M. (2010), Modified structured Cam Clay: a generalisedcritical state model for destructured, naturally structured and artificially structured clays,Computers and Geotechnique, Vol. 37 Nos 7/8, pp. 956-68.

    Taiebat, M., Dafalias, Y. and Peek, P. (2009), A destructuration theory and its application toSANICLAY model, International Journal for Numerical and Analytical Methods inGeomechanics, Vol. 34 No. 10, pp. 1009-40.

    Vatsala, A., Nova, R. and Sirinivasa Murthy, B.R. (2001), Elastoplastic model for cementedsoils, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 127 No. 8,pp. 678-87.

    Wang, L.H. and Atluri, S.N. (1994), An analysis of an explicit algorithm and the radial returnalgorithm, and proposed modification, in finite plasticity, Computational Mechanics,Vol. 13 No. 5, pp. 380-9.

    Wheeler, S.J., Naatanen, A., Karstunen, M. and Lojander, M. (2003), An anisotropic elasto plasticmodel for soft clays, Canadian Geotechnical Journal, Vol. 40 No. 2, pp. 403-18.

    Corresponding authorD.S. Liyanapathirana can be contacted at: [email protected]

    To purchase reprints of this article please e-mail: [email protected] visit our web site for further details: www.emeraldinsight.com/reprints

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