Numerical Methods in Geotechnical Engineering Benz & Nordal (eds) 2010 Taylor & Francis Group, London, ISBN 978-0-415-59239-0

Modeling liquefaction behavior of sands by means of hypoplastic model

A.B. TsegayeDelft University of Technology, Delft, The NetherlandsPlaxis B.V., Delft, Netherlands

F. MolenkampDelft University of Technology, Delft, The Netherlands

R.B.J. BrinkgreveDelft University of Technology, Delft, The NetherlandsPlaxis B.V., Delft, The Netherlands

P.G. BonnierPlaxis B.V., Delft, The Netherlands

R. de JagerDelft University of Technology, Delft, The Netherlands

V. GalaviPlaxis B.V., Delft, The Netherlands

ABSTRACT: In this paper the hypoplastic model by Wolffersdorff with the Intergranular Strain extension byNiemunis and Herle has been used for modeling the undrained behavior of sand during static and cyclic loads.The paper presents the hypoplastic equations and the Intergranular Strain concept in brief. Numerical simulationsof undrained triaxial compression and cyclic simple shear tests are performed.

1 INTRODUCTION

Hypoplasticity is an incrementally non-linear pathdependent constitutive model. The basic function inhypoplasticity is of a general form:

Where is the Jaumanns objective stress rate, is the

current stress state and is the current strain rate.There are a number of hypoplastic models (Kolym-

bas, 1977, Wu, 1992, Gudehus 1996, Niemunis &Herle, 1997, Wolffersdorff, 1996). In this study Wolf-fersdorffs version of hypoplasticity (Wolffersdorf,1996) with the so called Intergranular Strain extension(Niemunis and Herle, 1997) has been used to simulatethe undrained behavior of sand during static and cyclicloading.

2 GENERAL FORMULATION OFWOLFFERSDORFFS HYPOPLASTICMODEL

Elaborating the tensor valued isotropic function inequation (1) using the representation theorem and

imposing assumptions of stress homogeneity, rateindependence and incremental non-linearity, the gen-eral form of the hypoplastic equation (e.g. Kolymbas,2000, Lanier, et al., 2004, Gudehus, 1996) is written as:

Where L and N are literally the linear and thenon-linear parts of the hypoplastic stiffness matrixrespectively. The L and N matrices of Wolffers-dorffs hypoplastic model, which uses a predefinedMatsuoka-Nakai yield criterion, are written as:

F and a define the Matsuoka-Nakais yield surface.

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Where c = the critical state friction angle, tr is thetrace of a matrix obtained by summation of the diag-onal terms, is a second order identity tensor, and fsand fd are scalar factors included in the early 1990s toaccount for the effect of density and pressure.

fs is a generalized function that contains the influ-ence of the void ratio on the incremental stiffness andthe influence of barotropy (pressure) and is given as:

The influence of density (pycnotropy) is controlled viathe scalar, fd given as:

Where , , ei0, ec0, ed0, hs and n are model parameters.All but and are determined from the evolutionof the critical state, the upper and the lower boundvoid ratio in e-logp plane following Bauers isotropiccompression law (Bauer, 1996) which is given as:

Where ei, ec and ed are the upper bound, critical andthe lower bound void ratios at mean normal pressurep, respectively; ei0, ec0 and ed0 are the correspondingvalues at zero mean pressure; hs = granulate hardness,n = exponent to take pressure sensitivity into account.

3 STIFNESS AT SMALL STRAINS ANDDURING CHANGE OF LOADINGDIRECTION

The hypoplastic model by Wolffersdorff could predictthe mechanical behavior of state dependent granularmaterial during monotonic deformation. The modelcould also differentiate unloading and reloading paths.The model however, is said to accumulate excessiveof plastic strains at small strain and during changeof loading direction leading to excessive pore pres-sure accumulation (Niemunis and Herle, 1997). Hencethey proposed the so called Intergranular Strain toaccount for stiff behavior of soils at small strains andduring change of loading direction. A tangential stiff-ness tenor, M, which considers increased magnitude

Table 1. Boundary conditions for linear interpolation over.

= 0 = 90 = 180

0 mRL mRL mRL Mo M90 M1801 L+Nh mT L mRL

Table 2. Boundary conditions for linear interpolation overf()

0 (0, 90) 90 (0, 90) 180

f () d hh:d 0 hh:d dM M0 M M90 M M180

at the intergranular strain and during change of load-ing direction, is calculated from the basic hypoplasticstiffness tensors L and N.

The various assumptions are depicted in Figure 1a.The recent deformation history is stored in Intergranu-lar Strain tensor with a generalized objective evolutionrule (Niemunis and Herle, 1997) given as:

Where h = h/||h|| is the direction of the intergranularstrain, R is a material constant and r is a parameterthat controls the Intergranular Stain evolution rate.

The tangent stiffness is assumed to degrade linearlyover as shown in Figure 1b, where is modelparameter for non-linearity of the tangent stiffnessdegradation with .

Linear interpolation over of the set of constraintsgiven in Table 1 gives:

A second interpolation follows based on the direc-tion of the current strain rate, d = /||||, relative tothe recent strain rate direction, h. This interpolation isalso linear with the direction parameter, f() = hh:d,between M180 and M90 and between M90 and M0.

Linear interpolation over f() following the con-straints in Table 2 gives:

Where denotes the dyadic product, and the colon:denotes the scalar product between two tensors.

To the authors knowledge, this interpolation func-tion has not so far been compared to experimental

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Figure 1. a) The Intergranular Stiffness dial (for N = 0) b)tangent stiffness degradation with .

results. The law of inertia can be utilized to reasonout the validity of the assumptions (Tsegaye, 2009).

As a consequence of modeling the small strain andcyclic behavior, the formulation of the IntergranularStrain levies the hypoplastic model with five moreparameters- mR, mT , r R and . The procedures todetermine these parameters have been presented byNiemunis (Niemunis, 2003). Nevertheless, the testsrequired are cumbersome and some of the parame-ters r and may be abstract to the user. As such,in hypoplastic simulations involving the IntergranularStrain default values have often been used.

Using various empirical relations summarized byBenz (2006), the parameter mR may be estimated froma semi empirical relation given as:

Wherein A is a correlation constant f (e) is functionof the void ratio, OCR is the over consolidation ratio(which can fairly be left out for sand), vur and E

refur

are the unload-reload Poissons ratio and the unloadreload reference elastic stiffness respectively at a ref-erence mean normal pressure (usually considered atatmospheric pressure).

Proper determination of the parameter mT requiresa test with a 90 load reversal. In this study mT = 0.4mRhas been used. The parameter R can be obtained fromcyclic shear test. To observe the effect of the otherparameters r and we shall consider a one dimen-sional monotonic simple shearing, h > 0, whereequation (12) can be reduced to the form:

Up on integration of both sides of equation (17)(Tsegaye, 2009) we obtain:

Figure 2. Effect of the parameters and r on the tangentstiffness degradation curve (the reference curves (in bold) arefor mR = 5, = 6, r = 0.5).

Figure 3. The first set of curves show the small strain stiff-ness degradation for various values of the parameter mR(varying mR from 2 up to 10 and holding = 6 and r = 0.5)while the second set show the effect of the parameter andr on the normalized limit strain, lim/R (the normalized limitstrain should be read for various values of and the valuesr at the right axis).

Using linear interpolation between the maximumsmall strain stiffness GR and the residual shear stiffnessG in terms of , the following relation can be derived:

Considering equations (18) and (19) the purposesand effects ofr and can be observed as shown in Fig-ure 2. Very low values of r may not be desirable fromnumerical convergence point of view and higher val-ues increase the rate of stiffness degradation. Highervalues of tend to add to the constant (very smallstrain) regime.

As shown in Figure 3, the normalized small strainstiffness is asymptotic to G/GR = 1/mR. This markswhere the small strain stiffness is completely forgottenand the hypoplastic model takes full charge. The cor-responding strain level may be obtained from Figure 3.

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Niemunis (2003) defined this strain as swept out ofmemory strain, som in which the additional stiffnessis swept out (decayed out by more than a 90%).

The limit strain plotted in Figure 3 for various valuesof and r is by using equations (18) and (19). Afair estimation can be obtained by using the followingcorrelations.

Where lim is the limit shear strain level the Intergran-ular Strain remains active and is the normalized limitshear strain. A and n may be estimated according to:

The parameter R is the strain range where the modelbehaves linear elastic with a shear modulus of GR.Experimental results show this region is of a lim-ited range to a shear strain level 106 or less (verysmall strain range). However, in this formulation Ralso determines the maximum shear strain level theIntergranular Strain is active:

For example, for = 6 and r = 0.5, we obtain 8.5.If we set lim = 103 , we require R 1.1 104 orrather a choice of R = 1 106 will give a lim ofapproximately 105 which is very small. The factthat the Intergranular Strain decays relatively fasterrequires higher value of R (than observed in experi-ments) to stay in the game. In fact this range can alsobe controlled by the choice of the parameter r (forsmaller magnitude of r , we can obtain higher valueof ). However, significantly lower values of r maylead to numerical non convergence yet it increases theinfluence zone in a similar fashion as using highervalue of R).

4 TRIAXIAL COMPRESSION STRESS STATE

From now on we can leave the Intergranular straincomplication aside as it will not affect the elaborationsqualitatively.

Elaboration of the general hypoplastic equation fortriaxial compression stress state (e.g. Niemunis, 2003,Tsegaye, 2009) gives

Where, = q/p, p = (1 + 23)/

3, q = 1 3,v = 1 + 23 and q = 2(1 + 23)

/3 are considered

The coupling between shear stress and volumestrain is inherent to the hypoplastic equation. Hencethe dilatancy behavior of the model can be easily inves-tigated from equation (24) by considering dq / dp = 3for drained triaxial compression condition.

For undrained condition, the volumetric strain ratecan fairly be assumed zero. Hence, the ratio of thedeviatoric stress rate to the isotropic stress rate can bewritten as:

Considering equation (25) and instability condition atdq / dp = 0,

Solving the quadratic equation (26), the slope of theundrained instability line, IL, for triaxial compressionis obtained as:

Equation (27) holds true for contractive soils wherefd,IL 1. The equation further shows the most impor-tant parameters that govern the slope of the instabilityline during undrained triaxial compression simulationare fd,IL and a. The function fd presented in equation(10), can also be written as:

Where = e ec is the state parameter as defined byBeen & Jefferies (1985) and e and ec are the currentmaterial void ratio and the corresponding critical statevoid ratio at the current confining pressure respec-tively. The function fd depends not only on the stateparameter but also on ec and ed which in turn aredependent on the mean normal pressure. Figure 3illustrates the effect of fd,IL and IL on the slope ofthe instability line during undrained triaxial compres-sion simulation. However, instead of equation (28),fd = (1 + a) has been used, where a is considered amaterial constant which is held 1 in the figure dis-regarding the pressure dependence. Similar curveshave been experimentally investigated (e.g. Chu, et al.,2003, Wanatowuski, 2007).

5 APPLICATION: MODELLING OF TRIAXIALCOMPRESSION AND CYCLIC SIMPLESHEAR

Castro (Castro, 1969) in his PhD thesis investigated theliquefaction behavior of the so called Castro Sand B(Been and Jefferies, 2004) during monotonic triaxial

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Table 3. Model parameters for Castro Sand B.

Basic parameters Intergranular Strain Parameters*

Symbol unit values symbol unit values

ed0 0.5 R 1E-4ec0 0.8 mR 5ei0 0.97 mT 2c [0] 30.5 r 0.5hs [MPa] 1107 6n 0.26 0.2 2

*The usual Intergranular Strain parameters

Figure 4. Slope of the Instability line for a triaxial com-pression test in hypoplastic constitutive model, M = 6sin c/(3-sin c). Parameters and functions with the subscript-IL areat the point of instability.

compression test. The hypoplastic parameters deter-mined for this sand are shown in Table 3 (Tsegaye,2009). Due to absence of cyclic shear data, the Inter-granular Strain parameters used are which we foundcommon in literatures.

Figure 4 shows the evolution of the critical stateand the bounding void ratios.The evolution parameters(hs and n) have been determined based on the grada-tion curve following the empirical relations given byGudehus & Herle (Herle, I. & Gudehus, G. 1999).

Where Cu = coefficient of uniformity, d50 = meangrain diameter and d0 = 1 mm

Results shown in Figure 5 are drained simulationsof triaxial compression test on Castro sand samplesunder different initial state. The numerical resultsshow remarkably close trend to experimental results.Drained softening is well predicted for dense samples.The dilatancy behavior has been captured. The modelhowever seems to accumulate more volumetric strainthan seen in the experimental results.

Model runs for undrained triaxial compressiontest, as shown in Figure 6, could show liquefactionbehavior. While contractive samples (whose initial

Figure 5. Evolution of the critical state, and the maximumand the minimum void ratios (following Bauers exponentialisotropic compression rule) and various Castro sand drainedtriaxial compression test results.

void ratio lie above the critical state line in e-logpplane) show liquefaction (stress path directing tozero effective stress), dense samples (with an initialvoid ratio below the critical state void ratio) couldshow increase in undrained strength climbing up afterthe phase transformation line. Moreover, undrainedcyclic simple shear simulations show cyclic mobil-ity and liquefaction. However, the hypoplastic modelwithout application of the Intergranular Strain accu-mulates excessive pore pressure underestimating theundrained shear strength of the samples. Applica-tion of the Intergranular Strain helped to reduce thisexcessive accumulation of pore pressure around thehydrostatic axis in the undrained triaxial compressionsimulations. During undrained cyclic shear simula-tions, the number of cycles leading to liquefaction isvery much underestimated if the Intergranular Strainis not considered.

6 CONCLUSION

In modeling the mechanical behavior of granular mate-rials, soil mechanics offers two strong theoreticalconcepts: the theory of presence of a critical state andthe stress dilatancy theory. The attempt of modeling ofdeformation behavior of granular soil under the criticalstate theory involves at a minimum the initial state andan experimentally well defined critical state. Interme-diate states can be considered as interpolation betweenthese known boundaries, similar to boundary valueproblem (Tsegaye, 2009). The stress dilatancy theoryoffers a mathematical tool that captures the experi-mentally proved shear volume coupling. In modelingthe liquefaction behavior both frameworks are essen-tial. The reference hypoplastic model has a stronggrip on both frameworks which makes it an inter-esting tool for modeling the deformation behavior ingeneral and liquefaction behavior in particular of gran-ular soils. While the model appears appealing due itsfirm theoretical and experimental base, the questionof uncertainty and fuzziness of the initial state and the

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Figure 6. Drained triaxial compression test and simulationon various Castro sand samples at different initial states.

critical state poses a challenge on the predicted results.The measurement of the initial state is prone to distur-bance. Determination of the initial state is also liable tothe assumption of homogeneity. The determination ofthe critical state requires performing a number of testsat various confining pressures. Various samples arelikely to show scatter in reaching the critical state. Forrelatively dense samples, reaching the critical state isdifficult because of stress localization. The limitation

Figure 7. Undrained triaxial compression and undrainedcyclic shear simulations on various samples of CastroSand (Exp. = Experiment, W =With Intergranular Strain,Wo = with out Intergranular Strain).

also lays on the test apparatuses. Reaching the criti-cal state requires apparently a very large deformationwhich can be beyond the apparatuses allow.

The stress dilatancy formulation quantifies the vol-ume change due to shearing (contractive or dilative).This coupling is captured by the hypoplastic modelused in this study. However, it gives stronger con-tractive sense than shown by experiments. This leadsto unrealistic accumulation of pore pressure duringundrained monotonic and cyclic loading. As shown invarious undrained simulations, application of the Inter-granular Strain reduces the strong contractive sense ofthe hypoplastic model during monotonic loading andthe excessive ratcheting behavior during cyclic load-ing leading to better prediction of the pore pressure

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generation during undrained simulations. In spite ofits importance, the Intergranular Strain formulationsuffers from parameters that require complicated testprocedures and perhaps some parameters which aretoo abstract. In this regard, we feel the need for exper-imental investigation and appropriation of the variousparameters.

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