quantifying and modelling fabric anisotropy of granular...

Yang, Z. X., Li, X. S. & Yang, J. (2008). Geotechnique 58, No. 4, 237–248 [doi: 10.1680/geot.2008.58.4.237]

237

Quantifying and modelling fabric anisotropy of granular soils

Z. X. YANG*, X. S. LI† and J. YANG*

This paper describes an integrated study of the effects offabric anisotropy on granular soil response, in which themicroscopic measurements are properly linked with themacroscopic modelling. Using an image-analysis-basedtechnique and an appropriate mathematical approach,the inherent fabrics of sand specimens prepared in thelaboratory using different sample preparation methodswere measured, quantified and compared at a microscalelevel. It was found that the specimen prepared by the drydeposition method had a more anisotropic microstructurethan the specimen prepared using the moist tampingmethod, which is considered directly associated with theexperimental observation that different sample prepara-tion methods produce samples with distinctive responsesunder otherwise identical conditions. An existing plat-form model was then extended so that the combinedeffects of initial fabric and shear mode dependence wereaccounted for in a simple yet rational manner. To cali-brate and verify the model, a series of laboratory testswas conducted for Toyoura sand under various combina-tions of loading and sample preparation conditions. It isshown that the model is capable of simulating in a unifiedmanner the experimental results reflecting the combinedeffects of sample preparation methods, loading paths, soildensities and confining pressures.

KEYWORDS: anisotropy; constitutive relations; fabric; micro-structure; plasticity; sands

La presente communication decrit une etude integree deseffets anisotropes de la structure sur la reponse du solgranulaire, dans laquelle on met correctement en rapportles mesures microscopiques et la modelisation macro-scopique. En utilisant une technique basee sur l’analysed’images, et une methode mathematique appropriee, on amesure, quantifie et compare, au niveau de la micro-echelle, les structures inherentes de specimens de sableprepares en laboratoire, en appliquant differentes meth-odes de preparation des echantillons. On a releve que lespecimen prepare par retombee seche presentait unestructure plus anisotrope que le specimen prepare avec lamethode de pilonnage humide, et l’on considere que ceciest en rapport direct avec l’observation experimentaled’apres laquelle differentes methodes de preparation desechantillons produisent des echantillons presentant desreponses caracteristiques, les autres conditions etant iden-tiques. On a ensuite developpe un modele a plate-formeexistant, afin de definir, de facon a la fois simple etrationnelle, les effets conjugues de la structure initiale etde la dependance du mode de cisaillement. Pour calibreret verifier le modele, on a effectue une serie d’essais enlaboratoire pour le sable Toyoura, en appliquant differ-entes combinaisons de charges et de preparation desechantillons. On demontre que le modele est en mesurede simuler, de facon harmonisee, les resultats experimen-taux refletant les effets conjugues des methodes de pre-paration des echantillons, des chemins de charge, de ladensite du sol, et des pressions de confinement.

INTRODUCTIONIt has been consistently observed that two specimens of agranular soil, prepared by different methods in the labora-tory, may exhibit quite different responses to applied loadingunder otherwise identical conditions (Miura & Toki, 1982;Tatsuoka et al., 1986; Vaid et al., 1999). This observation ishighlighted in Fig. 1(a), where undrained responses of twoToyoura sand samples, prepared respectively by the drydeposition and moist tamping methods and subjected tomonotonic triaxial compression, are compared in the planeof deviatoric and mean effective stresses. Clearly, the speci-men prepared by the moist tamping method displays astronger resistance to the build-up of pore water pressureand a more dilative response than the specimen prepared bythe dry deposition method. Since both specimens are of thesame density and subjected to the same initial confiningpressure and loading path, the difference observed in Fig.1(a) is considered directly associated with the distinctivefabrics of the two specimens.

On the other hand, a number of experimental investiga-tions have indicated that two specimens of a granular soil,even when prepared using the same method and sheared atthe same initial state (accounting for both density andpressure), may exhibit different responses if the loading pathsapplied to them are different (Vaid & Chern, 1985; Riemer& Seed, 1997; Yoshimine et al., 1998). Shown in Fig. 1(b)are undrained responses of three samples of Toyoura sandprepared using the moist tamping method; one of them issubjected to monotonic triaxial compression, one to triaxialextension, and one to torsional shear. Note that the specimenin extension is much more contractive than that in compres-sion, with the torsional shear in between. This difference isanother indication of anisotropy, which is believed related tothe orientation of the major principal stress direction withreference to the deposition plane of the samples.

While there is abundant evidence of the impact of fabricanisotropy on granular soil behaviour as shown above, itremains a challenging task to take the anisotropic effectsinto account in geotechnical analysis and design. The pri-mary difficulties lie in (a) how to effectively quantify thefabric anisotropy and (b) how to rationally introduce theanisotropic effects into the well-established continuum mech-anics framework. To handle the first problem, an appropriateexperimental approach to collecting the information on themicrostructure of a granular soil assembly and a mathema-tical approach to describing the fabric anisotropy objectivelyare required. The second problem requires a proper link tobe established between the microscale measurements and the

Manuscript received 1 August 2006; revised manuscript accepted31 October 2007.Discussion on this paper closes on 1 November 2008, for furtherdetails see p. ii.* Department of Civil Engineering, The University of Hong Kong;currently Department of Civil Engineering, Zhejiang University,China.† Department of Civil Engineering, Hong Kong University ofScience and Technology.

macroscopic stress–strain–strength behaviour and a strategicmodel calibration and verification. For reasons of simplicity,the influence of soil fabric anisotropy has been largelyignored or treated crudely in geotechnical analyses.

A strong motivation for a serious treatment of the fabricanisotropy of granular soils has arisen in recent years fromgreat uncertainties in the analysis of flow liquefaction defor-mation and the determination of the undrained critical stateor steady-state strength (Finn, 2000; Yang, 2002). A back-analysis of the failed Lower San Fernando Dam (Finn, 2000)indicated that the average steady-state strength of sand wasonly 35% of the laboratory triaxial compression value, whichhad already been reduced by a factor of 6.5 to correct fordisturbance (Seed et al., 1989). This observation does not fit

the established theory of critical-state soil mechanics, inwhich the undrained critical-state strength of a soil is afunction only of its void ratio or density. On the contrary, ithas been consistently observed that the undrained responseof a granular soil and its critical-state strength may dependon the stress path and initial fabric as well.

This paper presents an investigation into fabric anisotropyeffects on granular soil behaviour, in which the two primaryproblems mentioned above were addressed in an integratedmanner. First, an image-analysis-based technique combinedwith a mathematical approach was developed to measureand quantify the initial fabric of a granular soil assembly atthe microscale level. The different fabrics formed by twosample preparation methods widely used in the laboratorywere identified and compared. Second, an existing platformmodel (Li & Dafalias, 2002) was extended to account forthe combined effects of inherent fabric and loading directionon granular soil response in a tractable way. Third, a seriesof laboratory tests was conducted on Toyoura sand undervarious loading and sample preparation conditions. Detailedcomparisons of the model simulations and test results aremade to assess the model performance in capturing thefabric anisotropic effects.

MEASUREMENT OF FABRIC ANISOTROPYMeasuring the fabric of a granular soil assembly is of

fundamental importance, yet it is a challenge in the study ofanisotropy effects. Available experimental data on soil fabricare scarce. In recent years image analysis has emerged as apromising technique in geotechnical research, because itallows the soil structure to be characterised at the microscalelevel (Kuo & Frost, 1996; Jang et al., 1999). In doing this,the fabric of a soil specimen should be preserved in amanner with minimum disturbance, and the representativecoupon surfaces are then sectioned for image analysis by ascanning electron microscope (SEM). Generally, there aretwo methods for preservation of the fabric of a granular soilspecimen: one is to impregnate the specimen with resin andcure it, and the other is to saturate the specimen with waterand freeze it. Of these two methods, the former is consid-ered more viable, because a frozen specimen may not bestrong enough to sustain sectioning, grinding and polishingin the process of acquiring high-quality coupon surfaces forimage analysis.

In this study Toyoura sand, a Japanese standard sandconsisting of subrounded to subangular particles, was used.The basic properties of this sand are: mean diameter ¼0.23 mm; uniformity coefficient ¼ 1.32; specific gravity ¼2.65; maximum void ratio ¼ 0.977; and minimum void ratio¼ 0.597. Both the dry deposition and moist tamping methods(Ishihara, 1993) were employed to produce specimens forlaboratory testing. In the dry deposition (DD) method, oven-dried sand is filled into the mould in several layers using afunnel. In each layer the sand is poured by keeping thefunnel’s nozzle slightly above the sand surface so that thesand is deposited in the loosest state. A denser specimen oftarget density can be prepared by tapping the mould using arubber mallet. In the moist tamping (MT) method, moist sand(typically with 5% water content) is placed in five or tenlayers in the mould. In each layer a tamper is used to compactthe sand. The tamping energy applied to the upper layers isgenerally higher than that to the lower layers, so that arelatively uniform density can be achieved. Generally, the drydeposition method is considered suitable for modelling thenatural deposition process, whereas the moist tamping meth-od can model the soil fabric of rolled construction fills better,and has the advantage of preventing segregation of well-graded materials (Kuerbis & Vaid, 1988; Ishihara, 1993).

0

100

200

300

p: kPa(a)

q: k

Pa

Dry deposition

Moist tamping

0

200

400

600

p: kPa(b)

q: k

Pa Triaxial compression

Triaxialextension

Torsional shear

0 100 200 300

0 200 400 600

Fig. 1. Effects of sample preparation methods and loading pathson undrained response of Toyoura sand: (a) in triaxialcompression (Dr 20%); (b) prepared by moist tamping(Dr 30%)

238 YANG, LI AND YANG

Specimen impregnationA traditional triaxial cell was modified for epoxy impreg-

nation (Fig. 2), which was performed by forcing epoxy resininto the soil specimen under a low differential pressure ofabout 20–30 kPa. The low pressure and slow epoxy flowwere maintained to minimise the disturbance to the soilfabric and to prevent air bubbles from being trapped in thesoil sample. Since the epoxy will not function well in moistconditions, the initial moisture in soil specimens prepared bythe MT method should be completely removed by dryingbefore epoxy impregnation. To achieve this, a system wasdesigned and fabricated to facilitate drying of the specimenswithout causing disturbance (Fig. 3). The drying processused was essentially the same as that used by Jang et al.(1999). Compared with the vacuum pressure of 15 kParecommended by Jang et al. (1999) for the drying process,an even smaller vacuum of about 5 kPa was used to mini-mise potential disturbance.

The desiccant CaSO4 was employed in this study becauseit functions better in a low-moisture environment than thecommonly used silica gel. It turns from blue into pink whenhydration takes place, and can thus serve as a moistureindicator. For the MT specimen the drying process normallyrequired about 24 h to remove all the water in the soil undera vacuum of 5 kPa. Following epoxy impregnation, theepoxy resin needs to be cured until the specimen is hardenedwith sufficient bonding strength.

Coupon surface preparationThe hardened specimen was cut into small patches using

a band saw, and the small patches were then sectioned usinga diamond saw into thin layers with workable dimensions(20 mm 3 20 mm 3 10 mm) for grinding and polishing. Thegrinding and polishing are crucial in the preparation ofcoupon surfaces for soil structure identification, becausequantitative measurement demands a high feature contrastand a high accuracy of the surface images. For statisticalconsiderations, the images captured should be representativeand lie in different locations. In this study, only the horizon-tal and vertical sections were chosen for consideration (Fig.

4) because the samples prepared by the DD and MT meth-ods were essentially transversely isotropic, with the verticalaxis as the axis of symmetry.

Image capture and analysisAfter sectioning, grinding and polishing, the coupon sur-

faces then qualified as SEM specimens. But before imageswere taken, the coupons were placed in a desiccant case forcomplete drying to avoid contamination of the SEM. Theimage capture was ready after a 10 kV voltage was applied.Adjusting the contrast and focusing at large magnification,say 200, the desired image could be obtained. For represen-tative purposes the magnification was fixed at 30, and thecorresponding image had 1024 3 819 pixels.

Figure 5 shows a typical SEM image at a magnification of200. This figure clearly shows the sand particles surroundedby the epoxy matrix. Some air bubbles in dark and circularshapes and lost sand particles due to the damage from couponsurface preparation are observed. Examples of a typical imagecaptured by SEM and black-and-white mask (binary image)used for image analysis are given in Fig. 6. The size ofpatches generally has to be restricted for use in SEM. Toconsider possible local variations in the fabric, a set ofneighbouring images rather than only a single image as shownin Fig. 6 were used in the fabric analysis. The number ofparticles contained by one typical image (Fig. 6) was around300, whereas the number of particles used in the fabricanalysis (i.e. vector magnitude) was up to 1700 (Table 1).

VECTOR MAGNITUDE AND FABRIC TENSORA random packing of non-spherical particles possesses

statistical characteristics in the spatial arrangement of theparticles and associated voids. Curray (1956) proposed avector magnitude ˜ to characterise the intensity of anisotro-py of the preferred particle orientation. The index ˜ takesthe form

Specimen

Tubeclamp

Low and steadypressure

Pressure chamber

Epoxy resin

Fig. 2. Epoxy impregnation system

Atmosphericair Desiccant

(CaSO )4

Dry air

Low and steadyvacuum

Desiccant(CaSO )4

Fig. 3. Drying system for moist tamped specimen

(a) (b) (c)

H

VI

VII

Fig. 4. Coupon sectioning for image analysis: (a) epoxy-impregnated specimen; (b) sectioning along different locations;(c) enlarged view of a patch used for analysis

Epoxymatrix

Airbubble

Sandparticle

Lostparticle

Fig. 5. A typical SEM image

QUANTIFYING AND MODELLING FABRIC ANISOTROPY OF GRANULAR SOILS 239

˜ ¼ 1

2N

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX2Nk¼1

cos 2�k

!2

þX2Nk¼1

sin 2�k

!2vuut (1)

where �k is the inclination angle of the kth unit vector n,measured with reference to the H1 axis in a representativeV–H1 section (Fig. 7). The value of ˜ depends on theparticle shape and the process of soil deposition. It variesfrom zero, when the material is isotropic, to unity, when the

major axes of all the particles are uniformly distributed inthe horizontal plane H1 –H2.

Through image processing, each particle was numberedand its orientation was identified. By statistical analysis of anumber of images of the vertical sections for both DD andMT coupons, the values of ˜ were computed using equation(1), and the results are summarised in Table 1. It can beseen that in the vertical plane the index ˜ for DD specimensis much greater than zero, indicating that DD specimenspossess obvious inherent anisotropy. By comparison, the MTsamples are slightly anisotropic, since ˜ is much closer tozero. The void ratio of the specimens in Table 1 wasapproximately 0.863, corresponding to the relative density of30% of Toyoura sand. The potential relation between thevector magnitude and the void ratio was not investigated inthis study, nor are there experimental data in the literaturethat allow a definite conclusion. This issue may deserveconsideration in future studies.

The angle � with respect to the horizontal axis in thefrequency histogram and rose diagram representations areshown in Figs 8(a) and 8(b) respectively, for vertical sec-tions. For DD specimens the preferential particle orientationis in the horizontal direction, which was caused mainly bygravitational force during the deposition process in samplepreparation. For the MT specimen, however, the preferredorientation of the particles appears to be more randomlydistributed, owing to the existing initial moisture in the soils.These results suggest that the DD specimens tended to bemore anisotropic than the MT specimens.

Figure 9 shows the preferred orientation of the particlesfor the horizontal sections of both the MT and DD coupons.Compared with Fig. 8, there is no preferential particleorientation in the horizontal plane for either DD or MTspecimens, and the corresponding values of the index ˜ arevery close to zero (Table 1). This observation provides agood justification for the transverse isotropy postulation forboth DD and MT specimens.

(a)

(b)

Fig. 6. Typical SEM image of a thin section: (a) SEMmicrophotograph; (b) B/W mask for image processing

Table 1. Values of vector magnitude for specimens prepared using different methods

Sample ID Plane Number of particles Vector magnitude, ˜ Average value

DD-sample3 Vertical 1754 0.218DD-sample7 Vertical 1015 0.222 0.214DD-sample2 Vertical 601 0.203DD-sample6 Horizontal 1193 0.029 –MT-sample1 Vertical 1636 0.076MT-sample2 Vertical 1438 0.113 0.091MT-sample3 Vertical 1100 0.083MT-sample6 Horizontal 826 0.052 –

Note: The void ratio of the samples was approximately 0.863.

H2

V

H1H1

V

�n

n

φk

kth particle

Fig. 7. Preferred particle orientation in long axis direction


The orientation of a non-spherical particle can be speci-fied by a pair of unit vectors, n and �n, along its major axisof elongation. A fabric tensor of second order (Oda, 1999)can be defined as

Fij ¼1

2N

X2Nk¼1

nki n

kj (2)

where N is the number of particles in a representativevolume, and nk

i and nkj are the components of the kth

vector. The magnitudes of the components in the tensorrepresent the net portion of the particles that are statisticallyorientated towards a particular direction. Note that Fij issymmetric, and therefore it can always be represented bythree principal values, F1, F2 and F3, and three associatedprincipal directions. If the reference frame is chosen to becoincident with the principal directions, the fabric tensor canbe written as

Fij ¼F1 0 0

0 F2 0

0 0 F3

24

35 (3)

In most practical cases soils are transversely isotropic: twoof the principal values, say F2 and F3, are equal to eachother, leaving only two independent principal values, F1 andF3, for the tensor. Furthermore, the tensor Fij possesses aunit trace: that is, F1 ¼ 1 � (F2 þ F3) ¼ 1 � 2F3. Therefore,for a cross-anisotropic soil with a known direction of deposi-tion (usually in the vertical direction), only one scalarquantity is needed to define the fabric tensor. In this case,the fabric tensor can be written as (Oda & Nakayama, 1988)

Fij ¼1

3 þ ˜

1 � ˜ 0 0

0 1 þ ˜ 0

0 0 1 þ ˜

24

35 (4)

where ˜ is the vector magnitude defined earlier. By substi-tuting the ˜ value pertinent to a specific structure (DD orMT specimen) into this expression, the corresponding fabrictensor characterising the material anisotropic property can beobtained.

(b)

0306090120150

180210240270300330

360

2

4

6

8

10

0

2

4

6

8

10

�150

�120

�90

�60

�30

180

150

120

90

60

0

30

0

2

4

6

8

10

Per

cent

age:

%

Particle preferred orientations(a)

Dry deposited specimenMoist tamped specimen

�90 �60 �30 0 30 60 90

Fig. 8. Characterisation of inherent fabric anisotropy ofToyoura sand with preferred particle orientations for verticalsections: (a) histogram; (b) rose diagram representation

0

2

4

6

8

Per

cent

age:

%

Particle preferred orientations(a)

Dry deposited specimenMoist tamped specimen

0306090120150

180210240270300330

360

2

4

6

8

0

2

4

6

8

�150

�120

�90

�60

�30

180

150

120

90

60

0

30

�90 �60 �30 0 30 60 90

(b)

Fig. 9. Characterisation of inherent fabric anisotropy ofToyoura sand with preferred particle orientations for horizontalsections: (a) histogram; (b) rose diagram representation


INCORPORATION OF ANISOTROPY EFFECTS INMODELLING

The impact of fabric anisotropy on granular soil response,and particularly on the flow liquefaction behaviour, warrantsa serious treatment in geotechnical analysis. Recently, aplasticity platform model has been developed by Li &Dafalias (2002) to tackle this problem. Details of this plat-form model will not be described in this paper. The focushere is, within this constitutive framework, to account forthe combined effects of the initial fabric (associated withdifferent sample preparation methods) and the loading direc-tion (i.e. different shear modes).

Based on the work of Tobita (1988), a stress tensor TTij isintroduced to characterise the anisotropy effects. TTij is givenby

TTij ¼ 16

€� ik F�1kj þ F�1

ik €� kj

� �� pp rrij þ �ij

� �(5)

in which F�1ij is the inverse of the fabric tensor Fij, and €� ij

is a normalised stress given by

€� ij ¼Mc g Łð Þ

Rrij þ �ij (6)

where Mc is a material constant, defined as the critical stressratio under triaxial compression, and g(Ł) is an interpolationfunction that interpolates the stress ratio invariant on thecritical-state failure surface according to the Lode angle Ł(Li & Dafalias, 2002). The tensor rij is defined asrij ¼ sij= p, where sij þ p�ij ¼ � ij is the stress tensor, and Ris the second invariant of rij, defined as R ¼

ffiffiffiffiffiffiffiffi3=2

p ffiffiffiffiffiffiffiffiffirij rij

p.

Note that €� ij represents only the loading direction associatedwith rij, not the loading magnitude.

Being a symmetric second-order tensor, TTij possessesthree independent isotropic invariants, among which the twonon-trivial invariants pertinent to rrij are RR ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 rrij rrij=2

pand

ŁŁ ¼ �[sin�1 (9 rrij rr jk rrki=2RR3)]=3. These two invariants can befurther combined into the following single invariant tocharacterise the anisotropy, called the anisotropic state vari-able,

A ¼ RR

Mc g ŁŁð Þ � 1 (7)

Note that if the material is isotropic, then ˜ ¼ 0 and Fij ¼�ij/3: hence TTij ¼ €� ij and, correspondingly, A ¼ 0. If thematerial is anisotropic, TTij deviates from €� ij, and A can beeither positive or negative depending on the orientation ofthe soil fabric relative to the loading direction and, to alesser degree, on the fabric intensity, as shown in Fig. 10 forToyoura sand samples. In this figure b is the intermediateprincipal stress parameter, defined as b ¼ (�2 � �3)=(�1 � �3), reflecting the shear mode; Æ is the angle betweenthe direction of the major principal stress and the sampleaxis of transverse isotropy; and c is a material constant,defined as the ratio of the critical stress ratio under triaxialextension, Me, to the ratio under triaxial compression, Mc.

It follows that the combination of Æ ¼ 0 and b ¼ 0 yieldsa triaxial compression loading condition, whereas Æ ¼ 908and b ¼ 1 give rise to a triaxial extension loading condition.The values of the vector magnitude ˜ have been determined,based on the image analysis, to be 0.091 and 0.214 for theMT and DD specimens respectively. Note that for an MTspecimen the variation of state variable A is less significantthan for a DD specimen: this is because the MT specimen ismore isotropic than the DD specimen.A is an objective measure of the fabric anisotropy effect:

that is, it is independent of the reference frame adopted.Using this objective measure, a varying critical-state line inthe e–p plane can be defined by making it a function of the

parameter A. Considering that the slope of this critical-stateline is influenced mainly by the shape of the grains, whichis an intrinsic property (Poulos et al., 1985), only e, thecritical-state void ratio at intercept p ¼ 0, is made here afunction of A and the intermediate principal stress parameterb as

eˆ ¼ eˆc � kˆ Ac � Að Þ 1 � t � bð Þ (8)

where eˆc (the critical-state void ratio for triaxial compres-sion at intercept p ¼ 0), kˆ and t are three material con-stants, and Ac corresponds to the anisotropic state variable Aat triaxial compression.

Equation (8) creates a sequence of straight critical-statelines running parallel to each other in the e–p plane (Fig.11), which are altered for different loading paths andinherent fabric intensities. Note that the critical-state linehere takes the form ec ¼ eˆ � ºc(p=pa)�, where ºc and �are material constants, and pa is a reference pressure(101 kPa).

Recently, there has been accumulating evidence (e.g.Chapuis & Soulie, 1981; Tobita, 1989; Riemer & Seed,1997; Chen & Chuang, 2001) that the critical-state line inthe e–p plane is dependent on the initial fabric and shearmode. This argument is implicitly reflected by equation (8),because both A and Ac are functions of the anisotropicparameter ˜, and eˆ is also dependent on b. According toequation (8), if b ¼ 0 and A ¼ Ac, then eˆ ¼ eˆc: that is,the critical-state line in triaxial compression is not affectedby the inherent fabric anisotropy. This is in agreement withthe observation of Ishihara (1993). In the case of triaxialextension, because of the dependence of the state parameterA on ˜, eˆ for DD is less than that for MT, with the resultthat the critical-state line for DD specimens is locatedbeneath that for MT (Fig. 11).

An examination of test data in the literature has suggested

�0·2

�0·1

0

0·1

0·2

α: degrees

(a)

α: degrees

(b)

Ani

sotr

opic

sta

te v

aria

ble,

AA

niso

trop

ic s

tate

var

iabl

e,A

∆ 0·091 (MT)�Mc�1·25c � 0·75

Triaxial compression

Triaxial compression

Triaxial extension

Triaxial extension

b � 0b � 0·25

b � 0·50

b � 0·75

b � 1·00

Torsional shear

Torsional shear

�0·5

�0·25

0

0·25

0·5∆ 0·214 (DD)�Mc 1·25�c �0·75

b � 0b � 0·25

b � 0·50

b � 0·75

b � 1·00

0

0

30

30

60

60

90

90

Fig. 10. Anisotropic state variable A for: (a) MT specimen;(b) DD specimen


that the plastic modulus is also appreciably influenced byfabric anisotropy. Thus the plastic modulus is also made afunction of the fabric anisotropy as

Kp ¼ Gh

RMc g Łð Þe�nł � R� �

(9)

where G is the elastic shear modulus, and ł ¼ e� ec is thedifference between the current void ratio e and the critical-state void ratio ec corresponding to the current mean normalstress p. The parameter ł was referred to as the stateparameter by Been & Jefferies (1985), and has been shownto be advantageous in characterising granular soil behaviour(e.g. Wood et al., 1994; Manzari & Dafalias, 1997; Li &Dafalias, 2000; Yang & Li, 2004). The quantity n is a modelconstant, and h is a scaling factor, defined as

h ¼ h1 � h2eð Þ k hAc � Aeð Þ þ 1 � k hð ÞAAc � Ae

(10)

where kh is a new material constant. It can be seen that hvaries linearly with A. When A ¼ Ac (triaxial compressionat Æ ¼ 08), h ¼ h1 � h2e. When A ¼ Ae (triaxial extensionat Æ ¼ 908), h ¼ (h1 � h2e)k h: that is, h is scaled by afactor kh. By including the dependence of the critical-stateline and the plastic modulus on A, this model is capable ofsimulating test results involving various loading directionswith respect to the soil fabric coordinate, as will be shownlater.

Following the work of Li & Dafalias (2000), the state-dependent dilatancy is given as

D ¼ d1

Mc g Łð Þ 1 þ R

Mc g Łð Þ

� Mc g Łð Þemł � R� �

(11)

where d1 and m are two material constants. Note that avarying critical-state line in the e–p plane will make thestate parameter ł a function of both the major principalstress direction and the soil fabric, which in turn makes thesoil dilatancy both fabric and stress path dependent.

LABORATORY TESTS AND MODEL SIMULATIONSA series of undrained tests, including triaxial compression

(b ¼ 0), triaxial extension (b ¼ 1.0) and torsional shear (b

¼ 0.5) tests, was performed on Toyoura sand. Both the DDand MT methods were used to prepare soil specimens. Twodifferent void ratios and three initial effective confiningpressures (100, 200 and 400 kPa) were considered. A combi-nation of different conditions gave rise to a total of 30 tests,as listed in Table 2.

All test results are shown in Figs 12–16 with the labelExperiment. The data are presented in the q–p and q–�qplanes, where �q is deviatoric strain. It can be seen that,compared with triaxial compression, the sand behaviourunder the triaxial extension condition is much more contrac-tive and softer, with the torsional shear in between. Further,the sand responses in triaxial extension and torsional shearmodes were found to be more significantly affected by thespecimen preparation method than the response in triaxialcompression. The contractive response and softening behav-iour are observed in all the tests performed on the DDspecimens under either triaxial extension or torsional shearconditions, whereas for the MT specimens dilative behaviourappears to be dominant when the shear strain exceeds acertain level. The underlying reasoning for the above obser-vations is that the critical-state line determined by thetriaxial extension tests lies to the left of that determined bythe compression tests in the e–ln p plane (Fig. 11), and thedifferences between compression and extension lines for theDD samples are much larger than for the MT samples.

All the test results have been simulated by the anisotropicmodel described in the preceding section, with a unified setof model parameters (Table 3). These parameters can bedivided into four groups according to their functions: elasticparameters; critical-state parameters; anisotropic parameters;and parameters associated with dilatancy. Most of the cali-brated model constants are consistent with those used in Li& Dafalias (2002), except for those that are related to thesample preparation methods and the shear-mode-dependentcritical-state line. The model predictions are presented to-gether with the experimental data in Figs 12–16.

It is seen that the model can capture the general trend ofthe various sand behaviours associated with the anisotropiceffects and the density and pressure dependence. The ob-served discrepancies are considered to be related mainly to

(a) the uncertainty in experimental determination of thecritical-state line in the e–p plane

0·70

0·75

0·80

0·85

0·90

0·95

0 2 4 6 8

Voi

d ra

tio, e

Final point (Triaxial compression, DD)

Final point ( , MT)Triaxial compression

Final point (Triaxial extension, DD)

Final point ( , MT)Triaxial extension

CSL ( of DD and MT)Triaxial compression

CSL ( of DD)Triaxial extension

CSL ( of MT)Triaxial extension

( / )p pa�

Fig. 11. Final points in experimental tests and critical-state lines (CSL) used inmodelling


Table 2. Summary of laboratory tests conducted

Sample preparationmethod

Relative density:%

Confining pressure:kPa

Loadingpath

Series I-a DD 30 100 TCDD 30 200 TCDD 30 400 TCDD 41 100 TCDD 41 200 TCDD 41 400 TC

Series I-b MT 30 100 TCMT 30 200 TCMT 30 400 TCMT 41 100 TCMT 41 200 TCMT 41 400 TC

Series II-a DD 30 100 TEDD 30 200 TEDD 30 400 TEDD 41 100 TEDD 41 200 TEDD 41 400 TE

Series II-b MT 30 100 TEMT 30 200 TEMT 30 400 TEMT 41 100 TEMT 41 200 TEMT 41 400 TE

Series III-a DD 30 100 TSDD 30 200 TSDD 30 400 TS

Series III-b MT 30 100 TSMT 30 200 TSMT 30 400 TS

Notes: DD ¼ dry deposition; MT ¼ moist tamping; TC ¼ triaxial compression; TE ¼triaxial extension; TS ¼ torsional shear.

q: k

Pa

q: k

Pa

εq: %

εq: %

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(b)

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Pa

Experiment 100 kPap �Simulation 100 kPap �Experiment 200 kPap �Simulation 200 kPap �Experiment 400 kPap �Simulation 400 kPap �

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Fig. 12. Comparison between undrained triaxial compression test results and model responses for (a) MT and (b) DD specimens withDr 30%


εq: %

εq: %

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(b)

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p: kPa

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0

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400Experiment 100 kPap �Simulation 100 kPap �Experiment 200 kPap �Simulation 200 kPap �Experiment 400 kPap �Simulation 400 kPap �

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0

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Fig. 13. Comparison between undrained triaxial extension test results and model responses for (a) MT and (b) DD specimens withDr 30%

εq: %

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Fig. 14. Comparison between undrained torsional shear test results and model responses for (a) MT and (b) DD specimens withDr 30%


(a)

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Fig. 15. Comparison between undrained triaxial compression test results and model responses for (a) MT and (b) DD specimens withDr 41%

(a)

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εq: %

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Fig. 16. Comparison between undrained triaxial extension test results and model responses for (a) MT and (b) DD specimens withDr 41%


(b) the simplified assumption that the critical-state line intriaxial extension is parallel to that in triaxial com-pression

(c) the influence of non-uniformity of soil samples at largestrains on test data

(d ) the evolution of the fabric tensor and the influence ofthe fabric anisotropy on plastic yielding.

These discrepancies suggest the need for more experimentaldata at both the microscopic and macroscopic levels, andfor further improvements of the constitutive model in thefuture.

CONCLUSIONSAccumulating evidence has shown that the undrained

response of a granular soil depends very much on thesample preparation method as well as on the shear mode,calling for a serious treatment of the impact of fabricanisotropy in geotechnical analysis. This paper describes anintegrated study of the problem in which a microscopicquantification of the inherent fabric of granular soil speci-mens is properly linked with a macroscopic modelling ofvarious anisotropic responses.

Using an image-analysis-based technique, the distinctlydifferent fabrics of Toyoura sand specimens prepared in thelaboratory using the dry deposition (DD) and moist tamping(MT) methods were measured and quantified at the micro-scale level. It is found that the sand samples prepared by theDD and MT methods can be reasonably assumed to betransversely isotropic, with the vertical direction as the axisof symmetry. In the vertical plane, the DD specimenspossess obviously inherent anisotropy, whereas the MT sam-ples tend to be more isotropic. The vector magnitude andthe fabric tensor are shown to be useful indices for char-acterising the soil fabric.

An existing platform model has been extended so that theinherent fabric anisotropy and shear mode dependence areaccounted for in a rational yet simple manner. This isachieved by rendering the location of the critical-state linein the e–p plane and the plastic modulus functions of theanisotropic state variable, the major principal stress directionand the intermediate principal stress parameter.

A structured testing programme has been carried out forToyoura sand to investigate various combined effects ofprincipal stress directions, intermediate principal stress va-lues, sample preparation methods, soil densities and confin-ing pressures. A detailed comparison of the model responsesand experimental results shows the capability of the consti-tutive model in capturing the complicated anisotropic effectsof granular soils.

ACKNOWLEDGEMENTSThe financial support provided by the Research Grants

Council of Hong Kong (600202) is gratefully acknowledged.

NOTATIONA, Ac, Ae anisotropic parameters

b intermediate principal stress parameterc intrinsic material constant

CSL critical-state lineD dilatancyDr relative density

e, ec void ratio, critical-state void ratioFij, F�1

ij fabric tensor, inverse of fabric tensorG elastic shear modulus

g(Ł) interpolation functionKp plastic modulus

Mc, Me critical stress ratios at triaxial compression andextension respectively

p mean effective stresspa atmospheric pressure ¼ 101 kPapp hydrostatic component of TTij

RR invariant of rrijrij stress ratio tensorrrij deviatoric stress ratio tensor of TTij

sij deviatoric stress tensorTTij modified stress tensorÆ angle between principal stress and fabric˜ vector magnitude�ij Kronecker delta

ŁŁ Lode angle of rrij�1, �2, �3 principal stress components

�ij stress tensor€� ij normalised stress tensor� angle of long axis of particle to horizontal directionł state parameter

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Table 3. Model constants for Toyoura sand

Elasticparameters

Critical-stateparameters

Anisotropicparameter

Parameters associated withdilatancy

G0 ¼ 125 Mc ¼ 1.25 ˜DD ¼ 0.214 hDD1 ¼ 3.45

� ¼ 0.25 c ¼ 0.75 ˜MT ¼ 0.091 hMT1 ¼ 3.5 3 hDD

1

eˆc ¼ 0.933 kh ¼ 0.2 hDD2 ¼ 3.34

kˆ ¼ 0.22 n ¼ 1.1 hMT2 ¼ 3.5 3 hDD

2

t ¼ 0.45 d1 ¼ 0.38ºc ¼ 0.02 m ¼ 1.0� ¼ 0.7

Note: The superscripts MT and DD denote constants for MT and DD specimensrespectively.


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