Geomechanics from Micro to Macro – Soga et al. (Eds)© 2015 Taylor & Francis Group, London, ISBN 978-1-138-02707-7

Constitutive modelling of fabric anisotropy in sand

Z.W. GaoSchool of Engineering, University of Glasgow, Scotland, UK

J.D. Zhao & X.S. LiDepartment of Civil and Environmental Engineering, Hong Kong University of Science and Technology, Hong KongSAR, China

Y.F. DafaliasDepartment of Civil and Environmental Engineering, University of California, Davis, USADepartment of Mechanics, School of Applied Mathematical and Physical Sciences, National Technical Universityof Athens, Greece

ABSTRACT: Fabric and its evolution have significant effect on the mechanical behaviour of granular materi-als. A three-dimensional anisotropic model for granular material is proposed with proper consideration of fabricevolution. An explicit expression for the yield function is proposed in terms of the invariants and joint invariantsof the stress ratio and fabric tensors. The material fabric is assumed to evolve with plastic shear deformationin a manner that its principal axes tend to become co-directional with those of the loading direction and itsmagnitude approaches a critical state value at large deformation. A non-coaxial and associated flow rule in thedeviatoric stress space is employed based on the yield function. The model is capable of characterizing the com-plex anisotropic behaviour of granular materials under monotonic loading with fixed principal stress directionsand meanwhile gives reasonable explanation for the micromechanical mechanism for static liquefaction andnoncoaxiality between the stress and plastic strain increment axes.

1 INTRODUCTION

Transversely anisotropic fabric structure is commonlyobserved in both natural and manmade sand depositsand profoundly influences the mechanical behavi-our of these soils including strength and dilatancy(Yoshimine et al. 1998, Gao et al. 2010). Proper con-sideration of the effect of fabric is important for safedesign and maintenance of relevant key infrastructures(Uthayakumar & Vaid 1998).

There have been many attempts on theoretical char-acterization and modelling of fabric anisotropy insand and its effect on macroscopic sand behaviour.Among many, those models based on the use of rotatedyield and plastic potential surfaces have gained limitedpopularity in the literature (Sekiguchi & Ohta 1977,Pestana & Whittle 1999). However, yield surface rota-tion may not be able to account for the anisotropicnature of sand related to particle orientation, contactnormal and void space distribution properly, as themagnitude and direction of rotation is typically asso-ciated with the initial stress state (Kaliakin 2003).The employment of fabric tensors derived from themicrostructural information of sand has proved to beefficient and physically more realistic in modellingsand behaviour (Oda & Nakayama 1989, Pietruszczak1999, Li & Dafalias 2002, Dafalias et al. 2004). Being

successful to a certain extent, these studies have com-monly ignored the change of the fabric anisotropyduring the deformation of the material, which is atodd with both experimental and numerical observa-tions, as the sand fabric will adjust to sustain theexternal loading in an optimum manner when it isdeformed (Li & Li 2009, Zhao & Guo 2013; Guo &Zhao, 2013).

This paper presents an anisotropic sand modelaccounting for fabric evolution based on the anisotro-pic critical state theory (ACST) by Li & Dafalias(2012). The new model features an explicit yield func-tion expressed in terms of the invariants and jointinvariants of the stress ratio tensor rij and a devi-atoric fabric tensor Fij. Over a typical monotonicloading course, the fabric tensor is assumed to evolvetowards the direction of loading. Based on the pro-posed framework, a non-coaxial flow rule is readilyderived.

2 CONSTITUTIVE MODEL

2.1 Yield function

Based on this micromechanical deformation mecha-nism that the shear resistance of sand is contributed

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by inter-particle friction and fabric anisotropy, wepropose the following yield function,

where R = (3rijrij)0.5 with rij = (σij − pδij)/3 being thestress ratio tensor, in which σij is the stress tensor,p = σij/3 is the mean normal stress, δij is the Kroneckerdelta; H is a hardening parameter whose evolutionlaw depends on the stress as well as internal vari-ables including soil density and fabric; A is a fabricanisotropy variable; kh is a non-negative model con-stant with default value of 0.03; g(θ) is an interpolationfunction based on the Lode angle θ of rij as follows(personal communication, Z.L. Wang 1992)

where c = Me/Mc is the ratio between the critical statestress ratio in triaxial extension Me and that in triaxialcompression Mc.

An important inclusion in the yield function inEquation 1 is a fabric anisotropy variable A that isdefined by the following joint invariant of Fij and nij(Li & Dafalias 2004, Gao et al. 2014)

where Fij is a symmetric traceless tensor whose normF = (FijFij)0.5 is referred to as the degree of fabricanisotropy. For convenience, Fij is normalized suchthat in critical state, F is unity. The deviatoric unitloading direction tensor nij in Equation 3 is defined asfollows (Li & Dafalias 2004)

with

Obviously, nii = 0 and nijnij = 1. Notice that the nij isthe deviatoric unit normal to a yield surface result-ing from Equation 1 with the assumption that A is aconstant (in other words nij is not normal to the yieldsurface of Equation 1).

2.2 Evolution law for H and Fij

Within the hypothesis that sand’s stress-strain responseis incrementally linear, the evolution of the two internalvariables is assumed to be

where ch, n and kf are three model parameters, e isthe current void ratio, ⟨ ⟩ are the Macauley bracketssuch that ⟨L⟩ = L for L > 0 and ⟨L⟩ = 0 for L ≤ 0, ζ isthe dilatancy state parameter defined as below (Li &Dafalias 2012)

where eA is a model parameter, ψ = e − ec is the stateparameter defined by Been & Jefferies (1985) with ecbeing the critical state void ratio corresponding to thecurrent p. In the present work, the critical state line inthe e-p plane is given by (Li & Wang 1998)

where er and λc are two material constants andpa (=101 kPa) is the atmospheric pressure. Theabove evolution law of Fij with plastic deformationexpressed by Equation 7 leads towards coaxiality withthe loading direction nij.

2.3 Dilatancy relation and flow rule

A proper dilatancy relation D defined as below isessential for modelling the sand behaviour

where dεpij is the plastic strain increment and dep

ij isthe plastic shear strain increment. Based on Li &Dafalias (2012), the following dilatancy relation whichaccounts for the effect of density, confining pressureand anisotropy is proposed

where d1 and m are two model constants.By assuming an associated flow rule in the devi-

atoric stress space based on the yield function inEquation 1, the increment of the plastic shear straindep

ij is expressed as

where

Notice that mij is normal to the yield surface expressedby Equation 1. Since ∂f /∂rij consists of two parts withone being coaxial with rij (or equivalently σij itself)and the other involving Fij which is attributed to fabricanisotropy and is in general non-coaxial with rij (Gaoet al. 2014), the flow rule expressed by Equations 12and 13 naturally address the non-coaxiality issue insoil modelling.

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Table 1. Model parameters for Toyoura sand.

Parameter Value

G0 125ν 0.1Mc 1.25c 0.75er 0.934λc 0.02ch 0.90n 3.0d1 0.2m 5.3eA 0.10kf 5.7

2.4 Elastic moduli

As plastic strain dominates sand deformation, theinfluence of elastic anisotropy, if any, is considerednegligible. The following elastic moduli (Richart et al.1970, Li & Dafalias 2012, Gao et al. 2014) areemployed:

where G and K denote the elastic shear and bulk mod-ulus, respectively, G0 is a material constant and ν isthe Poisson’s ratio assumed to be a constant.

3 MODEL SIMULATION FOR ANISOTROPICSAND BEHAVIOUR

3.1 Model parameters

To verify the model capability in simulating theanisotropic sand behaviour, we employ the test data forthe dry-deposited Toyoura sand reported byYoshimineet al. (1998).The model parameters are listed inTable 1and the initial degree of anisotropy F0 is set to be0.45. The procedure for parameter determination isdiscussed in Gao et al. (2014).

3.2 Model simulation

Figure 1 shows the model simulations for theanisotropic sand behaviour under undrained torsionalloading with constant intermediate principal stressvatiable b = 0.25. In this figure, α is the majorprincipal stress direction with respect to the depo-sition direction and Drc is the relative density afterconsolidation. Clearly, the model well captures thetrend that larger value of α generally leads to softer(lower shear stress σ1-σ3 at the same deviatoric strainε1-ε3) and relatively more contractive sand response.

Figure 1. Model simulation for the anisotropic sandbehaviour in undrained torsional shear tests (data fromYoshimine et al. 1998).

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Figure 2. Comparison between the observed and modelpredicted non-coaxial sand behaviour in undrained torsionalshear tests (data from Yoshimine et al. 1998).

Good agreement between the test data and modelsimulations can be observed.

An important feature of the present model is thenon-coaxial flow rule in Equations 12 and 13, result-ing naturally by the introduction of an evolving fabrictensor into the yield function and the associative flowrule assumption in the deviatoric stress space.

In a torsional shear test, the radial stress σr isalways the intermediate principal stress and the radialstrain εr the intermediate principal strain. In this set-ting, it is convenient to use the model to explain thenon-coaxiality in the z-θ plane (Gao et al. 2014).To elaborate on this point and motivated by theapproach in Dafalias et al. (2004), we plot in Figure 2the variation with deviatoric strain of the differenceof the angle α(σ) between the direction of the majorprincipal stress σ1 and the vertical direction, from theangle α(ε) that the major principal strain ε1 forms withthe vertical direction. Such difference is a measureof non-coaxiality. The simulations match the exper-imental observation on non-coaxiality qualitativelywell. When α = 0◦ or 90◦, there is only change ofthe principal values of fabric tensor during the devel-opment of plastic strain, but no fabric rotation is

involved. As such, the two sources of plastic strainincrement due to stress and fabric increments willinfluence its value only, with its direction aligningwith the stress direction during the entire loadingcourse. Thus, the predicted sand response is gener-ally coaxial, which is consistent with the experimentalobservation (Yoshimine et al. 1998). In all the othercases when α is between 0◦ and 90◦, coaxiality isassumed for purely elastic stage (below 0.5% devia-toric strain) due to the employment of isotropic elasticrelation. Beyond this elastic stage to a relative lowstrain level (such as 2%), however, a distinct differencebetween α(ε) and α(σ) of the order of 4 to 5 degrees onthe average is found (Fig. 2), which indicates clearlynon-coaxiality. Upon further loading, the fabric tendsto rotate towards the direction of stress, and the dif-ference between α(ε) and α(σ) predicted by the modeldecreases after the peak, and the non-coaxiality willtotally disappear at large strain levels.

Figure 3 shows the model simulation for the sandfabric evolution in undrained triaxial extension wherestatic liquefaction occurs. As the fabric tensor is ini-tially triaxial-compression like due to the sample’smethod of preparation and coaxial with the loadingdirection, thus, it undergoes only a change of its norm,without any change of its principal directions. In par-ticular the value of its major principal componentdecreases while the value of its minor principal com-ponent increases, which makes the norm F undergoa decrease first until at 7% deviatoric strain. At thispoint, all components of the fabric tensor are 0 so thata transient isotropic state is observed (F = A = 0). Asthe deformation continues, the original minor com-ponent becomes the major one, whilst the originalmajor one turns to be the minor one. The overalldegree of anisotropy Fshows a slight rebound fromzero (Fig. 3c). The anisotropic variable A increasesmonotonically from a negative value through zero toa positive one (Fig. 3c). Nevertheless, both A and Freach a very small positive value at static liquefactionwhere p = 0, which is far smaller than their respec-tive critical state value had liquefaction not occurred,which is also observed in the DEM simulations by Liand Li (2009).

4 CONCLUSIONS

A three-dimensional elasto-plastic constitutive modelhas been proposed to describe the anisotropic behaviourof sand under monotonic loading with fixed princi-pal stress directions. The model constructed withinthe framework of ACST recently presented by Li andDafalias (2012), which emphasizes the role of fab-ric on the characterization of sand response at criticalstate. The model employs a void-based fabric tensorand a physically-based fabric evolution law to accountfor the influence of void size and orientation and theirchange during shear on the sand behaviour includ-ing plastic hardening and dilatancy. At the criticalstate, the fabric tensor has a constant magnitude and

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Figure 3. Model simulation for the sand behaviour inundrained triaxial extension and the fabric evolution (staticliquefaction occurs).

is co-directional with the loading direction. A non-coaxial but associative flow rule in the deviatoric stressspace is used and it can naturally account for the non-coaxial behaviour of initially anisotropic sand samplesunder monotonic loading.

The model has been used to simulate the undra-ined test results for the dry-deposited Toyoura sand

(Yoshimine et al., 1998) under undrained torsionalshear tests with fixed principal stress direction andconstant intermediate principal stress variable. Themodel simulations compare well with the test results.

ACKNOWLEDGEMENT

Y.F. Dafalias would like to acknowledge supportfrom the European Research Council under theEuropean Union’s Seventh Framework Program FP7-ERC-IDEAS Advanced Grant Agreement n◦ 290963(SOMEF), and partial support by NSF project CMMI-1162096.

REFERENCES

Been, K. & Jefferies, M.G. 1985. A state parameter for sands.Géotechnique 35(2): 99–112.

Dafalias, Y.F., Papadimitriou, A.G. & Li, X.S. 2004. Sandplasticity model accounting for inherent fabric anisotropy.Journal of Engineering Mechanics 130(11): 1319–1333.

Gao, Z.W., Zhao, J.D. & Yao, Y.P. 2010. A generalizedanisotropic failure criterion for geomaterials. Interna-tional Journal of Solids and Structures 47(22-23): 3166–3185.

Gao, Z.W., Zhao, J.D., Li, X.S. & Dafalias, Y.F. 2014. Acritical state sand plasticity model accounting for fab-ric evolution. International Journal for Numerical andAnalytical Methods in Geomechanics 38(4): 370–390.

Guo, N. & Zhao, J.D. 2013. The signature of shear-inducedanisotropy in granular media. Computers and Geotechnics47: 1–15.

Kaliakin,V.N. 2003.An assessment of the macroscopic quan-tification of anisotropy in cohesive soils. Prco. 1st Japan-U.S. Workshop on Testing, Modeling, and Simulation,Boston, Massachusetts, USA, 370–393.

Li, X.S. & Dafalias, Y.F. 2002. Constitutive mod-elling of inherently anisotropic sand behaviour. Jour-nal of Geotechnical and Geoenvironmental Engineering128(10): 868–880.

Li, X.S. & Dafalias, Y.F. 2004. A constitutive frameworkfor anisotropic sand including non-proportional loading.Géotechnique 54(1): 41–55.

Li, X.S. & Dafalias, Y.F. 2012. Anisotropic critical state the-ory: the role of fabric. Journal of Engineering Mechanics138(3): 263–275.

Li, X.S. & Li, X. 2009. Micro-Macro quantification ofthe internal structure of granular materials. Journal ofEngineering Mechanics 135(7): 641–656.

Li, X.S. & Wang, Y. 1998. Linear representation of steady-state line for sand. Journal of Geotechnical and Geoenvi-ronmental Engineering 124(12): 1215–1217.

Oda, M. & Nakayama, H. 1989. Yield function for soil withanisotropic fabric. Journal of Engineering Mechanics115(1): 89–104.

Pestana, J.M. & Whittle, A.J. 1999. Formulation of a uni-fied constitutive model for clays and sands. Interna-tional Journal for Numerical and Analytical Methods inGeomechanics 23(12): 1215–1243.

Pietruszczak, S. 1999. On inelastic behaviour of anisotropicfrictional materials. Mechanics of Cohesive-FrictionalMaterials 4(3): 281–293.

Richart, F.E. Jr., Hall, J.R. & Woods, R.D. 1970. Vibrationsof soils and foundations. Englewood Cliffs, NJ: Prentice-Hall.

625

Dow

nloa

ded

by [H

ong

Kon

g U

nive

rsity

of S

cien

ce a

nd T

echn

olog

y] a

t 07:

28 1

2 M

ay 2

015

Sekiguchi, H. & Ohta, K. 1977. Induced anisotropy and timedependency in clays. Constitutive Equations of Soils, Pro-ceedings of the 9th International Conference on Soil Mech.Found. Eng., Special Session 9, Tokyo, 229–238.

Uthayakumar, M. & Vaid, Y.P. 1998. Static liquefaction ofsands under multiaxial loading. Canadian GeotechnicalJournal 35(2): 273–283.

Yoshimine, M., Ishihara, K. & Vargas, W. 1998. Effects ofprincipal stress direction and intermediate principal stresson undrained shear behaviour of sand. Soils Foundations38(3): 179–188.

Zhao, J.D. & Guo, N. 2013. Unique critical state charac-teristics in granular media considering fabric anisotropy.Géotechnique 63(8): 695–704.

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