A micromechanics-based methodology for evaluating the fabric ofgranular material

Y.-W. PAN� and J. - J. DONG�

A micromechanics approach can successfullymodel the stress±strain±strength relation of agranular material once the microcharacteristicsof the material have been obtained. However, itis usually dif®cult to determine the fabric ofnatural granular materials. Using a stress-dependent micromechanics model, the elasticproperties can become a function of the geo-metric and kinetic fabric. Also, the wave velocitycan be related to the elastic properties. Conse-quently, this paper proposes a methodology forevaluating the fabric of a granular assemblyfrom a set of measured wave velocities. Themethodology contains three elements: (a) astress-dependent micromechanics elastic model,(b) an anisotropic elastic wave propagation theo-ry and (c) an optimization procedure. It isveri®ed by calibrating available wave velocitydata of a glass ball assembly and washed mortarsand. The methodology is further applied tostudy the microstructural evolution of thewashed mortar sand under biaxial stresses. Twoaspects of fabric change can be observed: (a)concentration of contact normals in the majorprincipal direction and (b) a residual fabricafter subsequent loading/unloading.

KEYWORDS: anisotropy; constitutive relations; fabric/structure of soils; microscopy; sands.

Une approche micromeÂcanique permet de faireune maquette de la relation tension-allongement-reÂsistance d'un mateÂriau granulaire une fois queses micro caracteÂristiques ont eÂte obtenues. Ce-pendant, il est habituellement dif®cile de deÂter-miner la structure des mateÂriaux granulairesnaturels. Si l'on utilise une maquette micromeÂca-nique tributaire de la tension, les proprieÂteÂseÂlastiques peuvent devenir fonction de la struc-ture geÂomeÂtrique et cineÂtique. De plus, la vitessedes ondes peut eÃtre apparenteÂe aux proprieÂteÂseÂlastiques. C'est pourquoi, dans cette eÂtude, nousproposons une meÂthodologie permettant d'eÂva-luer la structure d'un assemblage granulaire aÁpartir d'un lot de vitesses d'ondes mesureÂes. LameÂthodologie contient trois eÂleÂments : (a) unemaquette eÂlastique micromeÂcanique tributaire dela tension, (b) une theÂorie de propagation desondes eÂlastiques et anisotropes et (c) une proceÂ-dure d'optimisation. Nous veÂri®ons en les cali-brant les donneÂes disponibles sur les vitessesd'ondes d'un assemblage de boule de verre etd'un sable de mortier laveÂ. La meÂthodologie sertensuite aÁ eÂtudier l'eÂvolution microstructurale dusable de mortier lave soumis aÁ des tensions bi-axiales. On observe alors deux genres de change-ments de structure : (a) une concentration desperpendiculaires de contact dans la directionmajeure principale et (b) une structure reÂsiduelleapreÁs des charges/deÂcharges conseÂcutives.

INTRODUCTION

On the basis of micromechanics, the microfeaturesof a granular material, including microstructureand contact force distribution, determine the mech-anical behaviour of the material. The developmentof micromechanics for granular materials has beenvery fruitful over the last decade. The successfuldevelopment of the theory offers an attractive ap-proach for understanding the complex behaviour ofgeomaterials from the microscopic point of view.

However, some dif®culties do exist in applying themicromechanics approach for granular materials topractical engineering problems. Among other dif®-culties, the fabric of a granular material and itsevolution are dif®cult to determine. If the fabric ofthe material is unknown, the micromechanics ap-proaches seem unrealistic and useless for practicalapplication.

In this paper, the authors propose a methodologyfor evaluating the fabric of a granular assembly onthe basis of an elastic model and wave velocitymeasurement. The analytical elements involved infabric calibration include (a) a stress-dependentmicromechanical elastic model, (b) a wave propa-gation theory for anisotropic elastic media and(c) a non-linear optimization procedure. The

Pan, Y.-W. & Dong, J.-J. (1999). GeÂotechnique 49, No. 6, 761±775

761

Manuscript received 2 June 1999; revised manuscriptaccepted 24 June 1999.Discussion on this paper closes 30 June 2000; for furtherdetails see p. ii.� National Chiao-Tung University, Hsinchu.

methodology is ®rst veri®ed using test data from aglass ball assembly (Agarwal, 1992). It is thenapplied to evaluate the geometric fabric of washedmortar sand under various stress states in a large-scale triaxial chamber (Lee & Stokoe, 1986). Theevolution of the geometric fabric of the testedspecimen during biaxial loading/unloading is alsodescribed. This study demonstrates that the pro-posed methodology is potentially useful for cali-brating the fabric of a natural granular depositunder speci®ed stress states.

FABRIC CHARACTERIZATION OF NATURAL

GRANULAR MATERIALS

The de®nition of `fabric' for a stressed granularassembly includes the `geometric fabric' and the`kinetic fabric' (Chen et al., 1988). The geometricfabric means the microstructure in a granular as-sembly, while the kinetic fabric indicates the aniso-tropic distribution of interparticle contact forces.The fabrics in a granular assembly govern itsmechanical behaviour. Evolution of the geometricfabric, accompanied by particle sliding, separationand rotation, results in the non-linear behaviour ofa granular material under a large strain (Chang etal., 1992). The induced anisotropy of a granularmaterial related to the fabric evolution has alsobeen veri®ed experimentally and numerically (e.g.Oda et al., 1985; Rothenburg & Bathurst, 1989).In addition, along with the locked-in contactforces, the residual geometric fabric of a granularmaterial after loading and unloading contributes tothe stress path dependence of the material (e.g.Chen & Hung, 1991). The deformation anisotropyof sands under small strains may also be in¯uencedby both the geometric and the kinetic fabric. Con-sequently, the calibration of the initial fabric andfabric evolution provides a possible means to un-derstand the mechanical mechanisms of granularmaterials from the microscopic point of view.

Fabric of natural granular materialsGeometric fabric. The geometric fabric of a

granular assembly can be categorized into direc-tion-independent and direction-dependent fabrics.Direction-independent geometric fabrics, such asvoid ratio, coordination number and particle sizeand shape, can be easily described and evaluated.The quantitative description of the direction-depen-dent geometric fabric, resulting from a spatialdistribution of granular particles with randomshape and packing structure, is much more dif®-cult. It is far more challenging to characterize itunder various stress conditions. The important geo-metric fabrics include the distributions of contactnormal, branch vector (vector connecting the cen-troids of two adjacent particles) and particle orien-

tation. Among them, the density function of thecontact normal (packing structure) alone can de-scribe the anisotropic nature of a granular assemblycomposed of equal-sized spherical particles (idea-lized granular assembly). The density function ofthe contact normal E(á, â) can be expressed by aspherical-harmonic expansion (Chang & Misra,1990a). Alternatively, Kanatani (1984) proposed apolynomial expansion in terms of a vector n torepresent the density function of the contact nor-mal E(n). He de®ned three kinds of fabric tensor,namely, the ®rst, second and third kinds of fabrictensor. Fabric tensors of the second rank have beenwidely used to represent the packing structure (e.g.Kanatani, 1984; Oda et al., 1985; Chang & Misra,1990a). The third kind of fabric tensor of thesecond rank, Dij, is adopted in this paper toapproximate the density function of the contactnormal E(n) as follows:

E(n) � (1� Dij ni n j)=4ð (1)

Naturally deposited granular materials are rarelyspherical. Many researchers (Rothenburg & Bath-urst, 1992; Oda et al., 1985) have shown that theshape and orientation of particles signi®cantly in-¯uence the mechanical behaviour of a naturalgranular material. Hence, the geometric fabric as-sociated with particles' shape and preferred orienta-tion should not be neglected in a micromechanicsapproach for modelling a granular material depos-ited naturally. Oda et al. (1985) introduced ananisotropic function l(n) � Sij ni n j to take thesource of anisotropy due to a non-spherical granu-lar assembly into account. Sij, a fabric tensor ofthe second kind, of the second rank, implicitlyre¯ects the combined effects of the shape andpreferred orientation of the particles of a granularassembly composed of elliptical particles. With asimilar idea, this work suggests the followinganisotropic distribution function of averaged branchvector length l(n) to consider the combined effectsof the particles' shape and preferred orientation:

l(n) � le(1� Dsij ni n j) (2)

Dsij, a fabric tensor of the third kind, of the second

rank, approximates the anisotropy of the averagedbranch vector length; le is an equivalent branchvector length (averaging over each branch vectorlength in a speci®c direction). All components ofDs

ij are zero for a granular assembly composed ofeither (a) spherical particles or (b) isotropicallydistributed non-spherical particles with uniformsize and shape. In these cases, the averaged branchvector lengths in various directions are identical.

Kinetic fabric. According to numerical simula-tions and laboratory experiments (Rothenburg &Bathurst, 1989; Konishi, 1978), the distributions of

762 PAN AND DONG

both the averaged normal and the shear contactforce, fn(è) and f r(è) (averaging along a directionè from the horizontal axis), in a stressed granularassembly can be anisotropic in most cases. Thismeans that the kinetic fabric in a stressed assemblyis often anisotropic. A two-dimensionally aniso-tropic kinetic fabric can be represented by a Fourierseries expression of the following form(Rothenburg, 1980):

f cn(è) � f 0

n[1� an cos 2(èÿ èf )] (3)

f cr (è) � f 0

n[aw ÿ at sin 2(èÿ èr)] (4)

in which f 0n is the averaged normal contact force

around all contact points; an, aw, at, èf and èr areconstants. Similarly, a three-dimensionally aniso-tropic distribution of normal contact force fn(n)can be represented in the following form:

fn(n) � f0(1� Dfij ni n j) (5)

in which Dfij, a fabric tensor of the third kind, of

the second rank, represents the direction-dependentfunction of the average normal contact force.

Instead of representing the contact force distri-bution by a fabric tensor, Chang et al. (1995) useda static hypothesis to formulate the averaged con-tact force, in terms of the stress tensor Äó ij andpacking structure of an idealized granular assem-bly, with the following equation:

Ä fc

j � Äó ij Aik nck (6)

in which Ä fc

j � Ä fc

n n j � Ä fc

s s j � Ä fc

t t j is thecontact force at the cth contact plane. The threeterms Ä f

c

n, Ä fc

s and Ä fc

t are the contact forcesalong the directions of n, s and t, respectively,which form the local coordinate system at thecontact point (as shown in Fig. 1). Aik is a tensorrelated to the contact normal distribution of thegranular assembly containing M contacts in arepresentative volume V ; Aik satis®es Aik Fkq �äiq, where Fik � (Aik)ÿ1 � (2rM=V )Nik ; Nik isthe fabric tensor of the ®rst kind (Kanatani, 1984)representing the sample mean of the contact nor-mal distribution, for which Nik � (2=15)Dik �(1=3)äik . As can be observed from equation (6),

the contact force distribution is a function of thestress state and packing structure.

Determination of fabric of a granular materialSeveral laboratory techniques have been devel-

oped for evaluating the geometric and kinetic fab-rics of granular materials. The available techniquesfor fabric characterization can be divided into twocategories based on different concepts: (a) collect-ing and analysing the image data (Oda et al., 1985;Lee & Dass, 1993; Desrues et al., 1996); and (b)measuring some physical quantities and relatingthese to the microfeatures of the material (Arulmoli& Arulanandan, 1994; Santamarina & Cascante,1996). Table 1 lists and appraises the existingtechniques for fabric evaluation.

In situ application of the methods using imageanalysis is hardly possible. Moreover, it is nearlyimpossible to evaluate the kinetic fabric of agranular material and its evolution from an image,unless optically sensitive materials are used. It isreasonable to postulate that some macroscopicphysical quantities, such as electrical resistanceand wave velocity, may re¯ect the microfeaturesof a granular material in a representative volume.Hence, it seems possible to calibrate the geometricor kinetic fabric from measured physical quantitiesif a correct correlation can be established. Unfortu-nately, current data interpretation in these techni-ques almost always relies on empirical models.The fabric calibration, in general, is only qual-itative. Quantitative application of this method toevaluate the fabric of a granular assembly is pre-ferable. Generally speaking, the existing method-ology for fabric characterization is not yetsatisfactory for engineering practice.

Among the methods listed in Table 1, the wavevelocity measurement is commonly used. Theelastic properties of granular materials can beevaluated using the well-developed theory of elas-tic-wave propagation (e.g. Stokoe et al., 1991;Bellotti et al., 1996). Since the initial elastic stiff-ness tensor of a granular material depends on itsfabric, it is therefore logical to calibrate the fabricof a granular material by measuring the velocity ofa wave propagating through the material. Followingthis reasoning, this paper presents a micromecha-nics-based analytical procedure to calibrate thefabric of a granular material from a set of meas-ured wave velocities with a micromechanics-basedelastic model.

PROPOSED METHODOLOGY FOR EVALUATION OF

FABRIC FROM WAVE VELOCITY

The fabric of a natural granular assembly can beevaluated from the measured wave velocities by ananalytical procedure containing three basic ele-

3

2

l 1c

t

s

cth contact

cth contact

n

Global coordinate system1 Local coordinate system

Fig. 1. The global and local coordinate systems

METHODOLOGY FOR EVALUATING FABRIC OF GRANULAR MATERIAL 763

ments: (a) a stress-dependent micromechanics-based elastic model; (b) a theory of elastic-wavepropagation for an anisotropic material; and (c)an optimization method. Fig. 2 illustrates the con-ceptual procedure. From the wave propagationtheory, the wave velocity of an elastic material iscorrelated with the elastic properties. Using astress-dependent micromechanics model, the elasticproperties can become a function of the geometricand kinetic fabrics. Consequently, the velocity of awave travelling through a granular material canalso be related to its fabric. The three basicelements of the proposed analytical procedure areintroduced in the following context.

Micromechanics-based elastic model of a non-idealized granular assembly

Chang et al. (1995) derived an upper bound anda lower estimate of the elastic constants of anidealized assembly through a static and a kinematic

hypothesis, respectively. The kinematic hypothesiscorresponds to strain localization, and the statichypothesis corresponds to stress localization.Chang & Misra (1990b) showed that the uniform-strain theory is valid for a granular assembly underlow-amplitude cyclic loading. As a consequence,the elastic stiffness of the assembly can be derivedfrom the kinematic hypothesis. The elastic stiffnessCijkl of an idealized granular assembly (Chang etal., 1995; Chang & Misra, 1990a) containing Mcontacts in a representative volume V can bederived as follows:

Cijkl � 1

V

XM

c

lci kc

jl lck �

M

V

�Ù

4rni kcji nk E(n) dÙ

(7)

Here lci � ln l

i � 2rnci is the branch vector connect-

ing the two adjacent particles' centroids at the cthcontact point, where r is the radius of the sphericalparticles; l is the branch vector length; and n l

i isthe unit branch vector, which is identical with thecontact normal nc

i . The term kcjl � kc

n ncj n

cl � kc

s scj s

cl� kc

t lcj l

cl is the local contact stiffness; kc

n, kcs and

kct are the contact stiffnesses along the directions

of n, s and t, respectively. E(n) is the densityfunction of the contact normal in the n direction.The integration

�Ù(:)E(n) dÙ stands for the double

integration� ð

0

� 2ð0

(:)E(á, â) sin â dá dâ.For a non-idealized granular material, the branch

vector length of each contact is no more a constant2r. Inserting the directionally averaged branchvector length l(n) into equation (7) directly, theelastic stiffness becomes a function of E(n) andl(n). The directionally averaged branch vectorlength l(n) accounts for the combined effects ofshape and orientation of a non-spherical-particleassembly. Consequently,

Table 1. Appraisal of available methods for evaluating the fabric of a granular assembly

Data treated Method Main requirement Capability� Limitation{

Solidifying and cutting thesample into thin sections

(1), (3){ (a), (b)

Image for section of X-ray computerized tomography Ef®cient image data- (1), (3) (a), (b), (c)granular assembly processing technique

Photoelasticity method (1), (2),} (3) (a), (b), (c)

Electrical-resistance Proper model for (1) (d), (e)Homogenized method interpreting measured

physical quantities quantities

Wave velocity method (1), (2), (3) (d), (e)

� Capability for fabric evaluation: (1) geometric fabric; (2) kinetic fabric; (3) geometric fabric evolution.{ (a) Huge effort of processing and analysing data for an assembly containing many particles; (b) dif®culty ofreconstructing the 3-D structure of a granular assembly; (c) little potential for applying the method to in situ problems;(d) qualitative evaluation of fabric only; (e) no possibility for evaluating the local heterogeneity of a granular assembly.{ Valid only with destruction of many samples.} Valid only with photoelastic materials.

Optimization

Wave velocity of granular material

Elastic stiffness

Geometric and kinetic fabrics

Anisotropic elastic-wave propagation

Stress-dependent micromechanics, elastic

Fig. 2. Analytical procedure for determining geometricand kinetic fabrics of a granular material frommeasured wave velocity

764 PAN AND DONG

Cijkl � M

V

� � �Ù

li kn

jl l k E(n) dÙ

� �

� M

V

� � �Ù

l(n)l(n)n li k

n

jl(l)n lk E(n) dÙ

� �(8)

in which li � l(n)n li is the branch vector;

kn

ij[l(n)] is the averaged local contact stiffness ofthe grouped contact points in the n direction withan averaged branch vector length equal to l(n).For a non-spherical-particle assembly, the unitbranch vector n l

i is no more identical to thecontact normal ni. A transformation tensor r pqik

can be further introduced into equation (8), withn l

i n lk � r pqik n p nq; r pqik , a fourth-rank tensor,

accounts for the averaged angular deviation of thebranch vector direction and the contact normaldirection in the n direction. As a consequence,the elastic stiffness Cijkl can be expressed by thefollowing equation:

Cijkl �M

V

� � �Ù

[l(n)]2 n p kn

jl[l(n)]nq r pqik E(n) dÙ

� �(9)

In this work, E(n) and l(n) describe thegeometric fabrics of a non-idealized granularmaterial; they are represented by equations (1)and (2), respectively. Hence, the direction-depen-dent geometric fabrics of a non-idealized granularmaterial are described by the three fabric tensorsDij, Ds

ij and r pqik . Dij accounts for the anisotro-pic contact normal distribution; Ds

ij accounts forthe shape and orientation of the particles; andr pqik accounts for the directional deviation of thebranch vector direction and the contact normaldirection. The directional deviation between thecontact normal and the branch vector may resultin a reduction in the global elastic stiffness ofthe granular assembly. It is, in fact, very dif®cultto determine the effect of this deviation rigor-ously without applying some means of micro-scopic observation or discrete-element simulation.Rothenburg & Bathurst (1992) formulated thestress homogenization (averaging over the contactforces) of a biaxially loaded elliptical-particulateassembly. In their formulation, the deviation be-tween the contact normal and the contact vector(the vector connecting the contact point and theparticle centroid) is neglected. They found thatthe discrepancy between the discrete-elementmethod simulated results and the calculated re-sults for the stress was acceptable before failure.The present work does not attempt to explorethis aspect further. Neglecting the directionaldeviation between the contact normal and branchvector, equation (9) reduces to

Cijkl � M

V

� � �Ù

[l(n)]2 n p kn

jl[l(n)]nq E(n) dÙ

� �(10)

For a non-spherical-particulate assembly, equa-tion (10) should be regarded as only an approx-imation to the elastic stiffness since the angulardeviation between the contact normal and thebranch vector is ignored. It should also be notedthat the intermediate scale is not taken into accountin the proposed model; it is assumed that theintermediate scale does not exist and does notdevelop during loading.

The total contact number per unit volume(M=V ) in equation (10) can be estimated from theequation suggested by Oda et al. (1982)

M

V� 3N t

(1� e)(8ðr3e)

(11)

in which N t is the average coordination number, eis the void ratio of the assembly and re is anequivalent radius. Originally, Oda et al. (1982)de®ned re as the average radius of all particles inthe graded spherical assembly. In the present work,re is modi®ed into an equivalent radius in order toaccount for arbitrary particle shapes. The equiva-lent radius re satis®es vs � (4=3)ðr3

e Pv, in whichvs is the volume of solid and Pv is the totalparticle number. Oda (1977) found that N t doesnot depend on the grain size distribution. Experi-mental data also show that N t has a good correla-tion with the void ratio. Chang et al. (1989)proposed an empirical equation e � 1:66 ÿ0:125N t correlating the void ratio and the averagecoordination number. This empirical equation isadopted for calculating the averaged coordinationnumber required in equation (11).

It is shown in equation (10) that the elasticstiffness tensor of a non-idealized granular materialderived from the micromechanical elastic model isa function of the geometric fabric. Furthermore, amicromechanical elastic model can also take theeffects of stress level and contact force anisotropy(i.e. the kinetic fabric) into account if a stress-dependent contact stiffness is adopted in equation(10). A stress-dependent contact law producesstress-dependent elastic moduli of granular materi-als. The Hertz±Mindlin contact theory is oftenadopted for correlating the relative contact dis-placement and the contact force (Chang et al.,1989). Although the real contact mechanism isextremely complicated (Mindlin & Deresiewicz,1953; Seridi & Dobry, 1984), a simple but approx-imate local constitutive law seems acceptable inpractice for engineering application. Consequently,the Hertz±Mindlin contact theory is adopted in thiswork. The quantities kn(n) and kt(n), the averagedlocal normal and shear contact stiffness of the

METHODOLOGY FOR EVALUATING FABRIC OF GRANULAR MATERIAL 765

grouped contact points in the n direction, areexpressed as follows (Chang et al., 1989):

kn(n) � C1[ fn(n)]1ÿ2á (12)

kr(n) � C2 1ÿ f r(n)

f (n) tanöì

!â

kn � ëkn(n)

(13)

in which fn(n) and f r(n) are the averaged normaland shear contact forces, respectively; ë is thestiffness ratio; Gs and ís are the shear modulusand Poisson's ratio, respectively, of the solid mak-ing up the particles; and öì is the inter-particlefrictional angle. According to the Hertz±Mindlincontact theory, C1 � (1=2á)(16=9)áRá[Gs=(1 ÿís)]

2á and C2 � 2(1ÿ ís)=(2ÿ ís); á � â � 1=3.The relative curvature of each contact R equalsr=2 for an ideal granular assembly with a particleradius equal to r. For a non-spherical granularassembly, the relative curvature of each contactdepends on the neighbouring particle size, shapeand contact point. In the present work, a direction-dependent equivalent relative curvature R � (1=4)l� (1=4)l(n) is proposed to account for the effectsof the particle shape and preferred orientation ofa non-ideal granular assembly on the contactstiffness.

Both the kinetic (directional contact force dis-tribution) and the geometric fabric can be approxi-mated by fabric tensors. Although it is possible tocalibrate the kinetic and geometric fabrics simulta-neously by the proposed method of calibration,only the parameters related to the geometric fabricare obtained by optimized calibration in the subse-quent development. Therefore, the contact forcedistribution has to be estimated. Except in isotropiccompression, it is dif®cult to derive an analyticalsolution for the contact force distribution in a non-idealized granular assembly. The contact force dis-tribution can be evaluated incrementally using aproper micromechanics model (Chang et al.,1991). However, the integration procedure is rathertime-consuming. The evolution of the packingstructure must also be considered; this results infurther dif®culties. For practical purpose, a simpleapproach is preferable. Neglecting the directionaldeviation between the contact normal and branchvector, the present work approximates the averagedcontact force distribution by the following equa-tion. This equation is modi®ed from equation (6):

f j[l(n)] � 1

(M=V )l(n)ó ij Bik nk (14)

in which f j(l) is the contact force in the n direc-tion with a branch vector length equal to l;f j(l) � fn(l)n j� f s(l)s j � f t(l)t j. The directionalbranch vector length is represented by equation (2)

in order to calculate the averaged normal contactforce in various directions. Bik is a tensor relatingto the contact normal distribution of the granularassembly and satis®es Bik Nkq � äiq, where Nik �(2=15)Dik � (1=3)äik is the fabric tensor of the ®rstkind. Once the stress state and geometric fabric areboth determined, the contact force can be evaluatedfrom equation (14).

It is worth mentioning that the real distributionof contact forces in a stressed granular assembly isstress path dependent (Chen & Hung, 1991) andhighly complicated. The formulation of the presentmicromechanics model is based on the kinematichypothesis, while the estimation of the contactforce distribution is based on a static hypothesis.This aspect may result in a discrepancy betweenthe estimated contact force and the real one. Thecontact force distribution formulated in equation(14) should be regarded as only an estimation ofthe kinetic fabric and used only for evaluating thestress-dependent local stiffness. Besides, equation(10) neglects the antisymmetrical stress and strain.It should be noted that stress and strain symmetry(commonly true in continuum mechanics) is notnecessarily true when particle rotation is taken intoaccount. If the particle rotation is not consistentwith the global rotation ®eld of the granularassembly, an antisymmetric part of the strain ispresent.

Wave propagation in an anisotropic elasticmaterial

The Christoffel equation Cijkl nwj nw

l Äk ÿrdv2Äi � 0 correlates the wave velocity v and thestiffness tensor Cijkl of an anisotropic elastic mate-rial. In the Christoffel equation, nw is the directionof the wave velocity, rd is the material density andÄ is the polarization of the wave. By introducingthe Christoffel tensor Ãik � Cijkl n

wj nw

l , the Christof-fel equation can be deduced as follows:

det(Ãik ÿ rdv2äik) � 0 (15)

Since natural granular deposits are often trans-versely isotropic, further elaboration on the relationbetween the elastic stiffness tensor and the wavevelocity of a transversely isotropic elastic materialwill follow (White, 1965). By using the Voigtnotation, the stress increment and strain incrementcan be expressed as Äó m � [Äó11, Äó22, Äó33,Äô12, Äô13, Äô23]T and Äån � [Äå11, Äå22, Äå33,Äã12, Äã13, Äã23]T respectively. Therefore, theglobal constitutive law can be expressed asó m � Emnån, in which m, n are the tensor indices1±6. The indices 4, 5 and 6 denote the planes 12,23 and 13, respectively. Fig. 1 illustrates the co-ordinate system. For a transversely isotropic gran-ular material with axis 3 as the symmetrical axis,the number of independent elastic constants re-

766 PAN AND DONG

duces to ®ve (i.e. E11, E33, E12, E23 and E66). Theother elastic constants are E22 � E11, E13 � E23,E55 � E66, E44 � (1=2)(E11 ÿ E12) and E14 �E24 � E34 � E56 � 0. The elastic wave velocity ofa transversely isotropic material depends on theinclination angle è between the rotation symmetryaxis and the wave propagation direction. Fromequation (15), one primary-wave velocity Vp,è andtwo shear wave velocities Vsh,è, Vsv,è of the elasticwave in the direction inclined at an angle è to therotation symmetry axis can be expressed as fol-lows:

Vsh,è ��������������������������������������������������������(E66 cos2 è� E44 sin2 è)=rd

p(16)

Vp,è ���������������������������������������������(ÿb�

����������������b2 ÿ 4cp

)=2rd

q(17)

Vsv,è ���������������������������������������������(ÿbÿ

����������������b2 ÿ 4cp

)=2rd

q(18)

in which

b � ÿ(E11 sin2 è� E33 cos2 è� E66) (19)

c � (E11 sin2 è� E66 cos2 è)

3 (E66 sin2 è� E33 cos2 è)

ÿ (E12 � E66)2 cos2 è sin2 è (20)

Vsh,è and Vsv,è are de®ned in Fig. 3. In the follow-ing, the elastic wave is assumed to propagate andbe polarized along the principal axes. Sij representsthe shear wave velocity, in which the ®rst index idenotes the principal axis of wave propagation andthe second index j denotes the principal axis of

particle motion. For example, S23 is the shearvelocity of the wave propagating along axis 2 andpolarized along axis 3. Pi represents the primary-wave velocity, in which the index i refers to theaxis of wave propagation. In the remaining section,granular materials are assumed to be transverselyisotropic.

Optimization method for parameter calibrationA non-linear optimization aims to search for the

optimized value of a non-linear object functionÖfxg. Among many different non-linear optimiza-tion methods, the Levenberg±Marquardt methodhas been well established and the required comput-ing routines are readily available (e.g. IMSL). Thismethod is suitable for locating the global optimum(Fletcher, 1987).

The object function Öfxg in this case is de®nedas a non-linear `error square' function, which isthe squared sum of the differences between n(model) calculated data points Ui(fxg) (i � 1, n)and n measured data points of wave velocity Vi

(i � 1, n) as follows:

Öfxg �Xn

i�1

[Ui(fxg)ÿ Vi]2 (21)

in which fxg is the vector containing the undeter-mined parameters. The non-linear optimization,then, aims to search for a set of unknown para-meters fxg corresponding to a series of Ui(fxg)(i � 1, n) such that the object function (i.e. thesquared-error function) is a minimum.

Calibration of the geometric fabric of transverselyisotropic granular materials

The velocity calculated on the basis of theproposed micromechanics-based model depends onseveral parameters including Dij, Ds

ij, e, re, á, ë,Gs and öì. Among them, the fabric tensors Dij

and Dsij are responsible for determining the aniso-

tropy of the wave velocity, while the other onesmainly affect the magnitude of the velocity. Themodel deserves particular elaboration for the deter-mination of Dij and Ds

ij.With the proposed three-element procedure, the

calibration of the packing structure and other mi-crofeatures of natural granular materials from adata set of measured wave velocities becomespossible. This paper focuses on the determinationof the fabric tensors of transversely isotropic gran-ular assemblies. The change in the geometric fabricof a granular assembly due to stresses is alsoexplored. For a transversely isotropic granular ma-terial with axis 3 as the symmetrical axis, thecomponents of the fabric tensor must satisfyDij � 0 and Ds

ij � 0 for i 6� j, and D11 �

1

3

2

Direction ofwave propagation

Directions ofparticle motion

θ

Vsv,θ

Vsh,θ

Vp,θ

Fig. 3. Directions of primary and shear wave propaga-tion and polarization through a transversely isotropicmaterial

METHODOLOGY FOR EVALUATING FABRIC OF GRANULAR MATERIAL 767

D22 � ÿ(1=2)D33 and Ds11 � Ds

22 � ÿ(1=2)Ds33.

Hence, only two parameters, D33 and Ds33, are

necessary for describing a material's fabric. Ahigher D33 means that more contact normals arealong axis 3 than other directions. A negative Ds

33

implies most particles have their longest axesparallel to the 1±2 plane. The absolute value ofDs

33 increases with (a) increasing ¯atness andslenderness of the particles and (b) increasingconcentration of the longest axis on the 1±2 plane.

VERIFICATION OF THE PROPOSED

METHODOLOGY

The experiment results obtained by Agarwal(1992) were adopted to verify the proposedmethodology for calibrating the fabric of a granularassembly. In the laboratory, Agarwal measured theprimary and shear wave velocities propagatingalong different directions of a glass ball assemblycontained in a triaxial cubical box. Since a glassball is spherical, all components of the particle-shape-induced fabric tensor Ds

ij are zero. Theproperties of the glass ball assembly used for theanalysis are as follows: speci®c gravity � 2:472,Poisson's ratio ís � 0:21, equivalent radiusre � 0:118 mm and void ratio e � 0:5744. Theore-tically, the parameters á and â are 1=3, andC2 � 2(1ÿ ís)=(2ÿ ís) is 0´8827 according to theHertz±Mindlin contact theory. As Agarwal did notinclude void ratio updating during a stress change,the void ratio is assumed constant in the calibra-tion. This assumption should not seriously affectthe calibration D33, since the change in void ratiowill not signi®cantly affect the anisotropy of thewave velocity. The void ratio, however, determinesthe contact number per unit volume (M=V ). Itsin¯uence on the stiffness and velocity is mainlyisotropic. This assumption, however, can be waivedif the correct void ratio is determined for eachstress stage.

Calibration of packing structureVi denotes the measured sets of primary wave

velocity. Agarwal measures wave velocity in sevencon®ning pressures (ranged from 27´6 to 193´2kPa). In each con®ning pressure level, he measureswave velocities in ®ve different directions. In total,35 wave velocities are measured. The target para-meters fxg include the fabric tensor D33 for thetransversely isotropic material and the shear mod-ulus Gs of the particle.

First, the packing structure corresponding toeach constant con®ning pressure is calibrated. Themeasured data Vi (i � 1, 5) in equation (21) is the®ve wave velocities in different directions for aconstant con®ning pressure. The primary wavevelocity in different directions under a same con-

®ning pressure, Ui(fxg) (i � 1, 5), in equation (21)can be calculated by equation (16)±(20) with thestiffness tensor that depends on the microfeaturesof the particulate assembly. Figure 4 lists thecalibrated packing structure D33 for various con®n-ing pressures.

Next, 35 measured wave velocities, all together,are treated as the measured data Vi (i � 1, 35) inequation (21) for calibration regardless of thedifference in con®ning pressure. Here, the packingstructure of the glass ball assembly remains con-stant under different con®ning pressures. The cali-brated results are D33 � 0:415 and Gs �14:99 GPa. They are obtained using á � â � 1=3.If á and â increase 10%, the calibrated D33 willdecrease less than 2% and vice versa. The D33

from calibration is not largely affected by á and â.To ensure the global optimal, nine pairs of D33 andGs were taken as the initial guesses for searchingthe optimal. The converged answers using differentinitial guesses appear fully consistent. The cali-brated fabric is adopted in the next sub-section tocalculate the primary and shear wave velocities.

Comparison of measured and calculated wavevelocities

Figure 4 shows the comparison between themeasured and calculated results. The solid linesindicate the calculated primary-wave velocities invarious directions. Different symbols indicate themeasured velocities under different con®ning pres-sures. It can be observed that the calculated resultsmatch the experiment results well enough if appro-priate microfeature parameters are used. The pack-ing structure calibrated from all 35 measured wavevelocities was used to demonstrate the velocityanisotropy. Fig. 5 presents the data; the verticalvelocity normalizes the wave velocities along var-ious directions. Both the measured and the cal-culated primary-wave velocities along differentdirections in the sample are shown in Fig. 5 forcomparison.

Figures 6 and 7 present the measured andcalculated shear wave velocities S12, S13 and S32.The calculated results were obtained by using thepacking structure calibrated from the 35 measuredprimary-wave velocities. Since the glass ball as-sembly is assumed to be transversely isotropic, thecalculated shear velocity S13 � S32 is referred toShv. The velocity S12 of shear waves propagatingand polarized on the isotropic plane is referred toShh. The calculated anisotropic ratio Shv=Shh is1´062; it does not change with the con®ning pres-sure, since D33 remains constant in this case(because the void ratio is assumed constant). Themeasured Shv=Shh (ranging from 1´060 to 1´042 forcon®ning pressures of 27´6±193´6 kPa) matchesthe calculated result well. The slightly decreasing

768 PAN AND DONG

Confining pressure 5 165.6 kPa

Confining pressure 5 138.0 kPa

Confining pressure 5 110.4 kPa

Confining pressure 5 82.8 kPa

Confining pressure 5 55.2 kPa

Confining pressure 5 27.6 kPa

Confining pressure 5 193.2 kPa

Measured data (Agarwal, 1992)

7654321

Curves 1 to 7 correspond to confining pressures 27.6 to 193.2 kPa

0.5120.3420.4860.3070.3170.3440.430

D33 5

Calibrated fabric tensor

0 100 200 300 400 500 600 700

Primary-wave velocity along horizontal axis: m/s

0

100

200

300

400

500

600

700

Prim

ary-

wav

e ve

loci

ty a

long

ver

tical

axi

s: m

/s

Fig. 4. Comparison between measured and calculated primary-wavevelocities using the calibrated packing structure

Confining pressure 5 165.6 kPa

Confining pressure 5 138.0 kPa

Confining pressure 5 110.4 kPa

Confining pressure 5 82.8 kPa

Confining pressure 5 55.2 kPa

Confining pressure 5 27.6 kPa

Confining pressure 5 193.2 kPa

Measured data (Agarwal, 1992)

Averaged from measured data

Calculated using calibrated fabric tensor D33 5 0.415Fitted from averaged data

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Normalized primary-wave velocity along horizontal axis

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Nor

mal

ized

prim

ary-

wav

e ve

loci

ty a

long

ver

tical

axi

s

Fig. 5. Comparison between measured and calculated normalizedprimary-wave velocities using the calibrated packing structure

METHODOLOGY FOR EVALUATING FABRIC OF GRANULAR MATERIAL 769

Measured shear wave velocity (Agarwal, 1992)

S32

S13

S12

0 25 50 75 100 125 150 175 200

Isotropic confining pressure: kPa

250

300

350

400

450

She

ar w

ave

velo

city

: m/s

Fig. 6. Measured shear wave velocity

Calculated shear wave velocity

Shv (5 S32 5 S13)

0 25 50 75 100 125 150 175 200

Isotropic confining pressure: kPa

250

300

350

400

450

She

ar w

ave

velo

city

: m/s

Shh (5 S12)

Fig. 7. Calculated shear wave velocity using the calibrated packingstructure

770 PAN AND DONG

tendency of the measured anisotropy for increasingcon®ning pressure may imply that the actual aniso-tropy of the packing structure does slightly changewith the con®ning pressure (this arises from aslight change in void ratio).

The measured velocity ratio P3=Shv is 1´64,deduced from the averaged data under differentcon®ning pressures, while the calculated velocityratio P3=Shv is 1´52, by using the theoretical valueof C2 and the parameters calibrated from theprimary-velocity data. The overestimation of theshear wave velocity may originate from the over-estimation of the shear stiffness from C2 � 0:8827,deduced from the Hertz±Mindlin contact theory.An overestimated C2 implies an overestimated con-tact stiffnes ratio ë. In the following, VP and VS

denote the primary and shear wave velocities,respectively. On the basis of a micromechanicsmodel (Chang et al., 1995) derived from a kine-matic hypothesis and elastic wave theory, VP=VS

(� P3=Shv) is equal top

2 for ë � 1:0 andp

3 forë � 0, for a granular assembly with an isotropicpacking structure. Obviously, the ratio between thevelocities of primary waves and shear waves de-pends on the selected value of ë. The lower thevalue of ë chosen, the higher the value of VP=VS

obtained will be.Although ë is a predominant factor in¯uencing

VP=VS, it is by no means a sensitive factor forcalibrating the packing structure of a granularassembly. The evolution of contact shear forces cancontribute to the non-linearity and irrecoverablestrains of a granular material. The stiffness ratio ëmay account for this locked-in stress effect. How-ever, the sensitivity of ë deserves special concernfor fabric calibration. The sensitivity of ë to theanisotropy of the primary-wave velocity is furtherexamined in the following. Here, the normal con-tact stiffness is assumed constant and stress-independent. For D33 � 0:44, the velocity ratioVP,90=VP,0 changes by only ÿ2:5% when ë dropsfrom 0´8827 to 0´0. For ë � 0:8827, VP,90=VP,0

changes by �13:8% when D33 rises from 0´0 to0´44. This reveals that ë is an insensitive parameterfor determining the velocity anisotropy of primarywaves and for calibrating the fabric of granularmaterials from the measured anisotropy of theprimary-wave velocity.

The discrepancy between the measured and cal-culated shear wave velocities shown in Figs 6 and7 can be reduced by choosing a lower C2 (than thetheoretical value 0´8827). As a reference for cali-bration, a 10% reduction in C2 results in a 1´7%decrease in D33 and a 14´7% decrease in Gs (forthe case illustrated). It appears that the change inC2 does not alter the calibrated magnitude of D33

signi®cantly, nor does C2 affect the anisotropy ofthe stiffness and wave velocity substantially. Actu-ally, C2 can also be a calibrated parameter if both

the primary and the shear wave velocities are avail-able.

To explore the convergence of the calibratedparameters, the changes in the object functionÖfxg due to individual perturbations of D33 andGs were evaluated. The calibrated result (i.e.D33 � 0:415 and Gs � 14:99 GPa) was taken asthe centre of the perturbation. It was noted thatÖfxg increases by 4%, for a 10% increase in D33,while Öfxg increases by 74%, for a 10% increasein Gs. The parameter Gs appears relatively sensi-tive for Öfxg.

EVALUATION OF FABRIC EVOLUTION

The role of fabric evolution in the mechanicalbehaviour of granular materials has been discussedin the foregoing. It is possible to use the proposedmethodology to study the fabric evolution of agranular material. For purpose of demonstration,the geometric fabrics corresponding to variousstress states were back-calculated from the meas-ured shear wave velocity (Lee & Stokoe, 1986). Intotal, 16 stress stages in the experimental pro-gramme of Lee & Stokoe (1986) were selected forthe fabric calibration. The stress stages numberedfrom 1 to 9 were a series of isotropic loadings andunloadings. The stress stages from 13 to 19 were aseries of biaxial loadings and unloadings. Theevolution of the calculated microstructure was ana-lysed. The contact stiffness ratio ë was assumed tobe zero since ë is insensitive to the fabric calibra-tion. The properties of the washed mortar sand inthe large-scale triaxial chamber (Lee & Stokoe,1986) were as follows: speci®c gravity � 2:67;Poisson's ratio ís � 0:25; equivalent radius re �0:23 mm; void ratio e � 0:64. Since the horizontalplane (the 1±2 plane) is assumed to be the sym-metric plane for a transversely isotropic material,the measured velocities were averaged as Shh �(1=2)(S12 � S21) and Shv(1=4)(S13 � S23 � S31 �S32).

Calibration of the microfeatures of an isotropicallycompressed specimen

The measured velocity of shear waves propagat-ing through an isotropically compressed specimenwas ®rst used to calibrate the microfeatures of thewashed mortar sand. For the stress stages 1±5, theapplied isotropic stress started from 69´0 kPa andwas raised to 103´4, 137´9, 206´9 and 275´8 kPasubsequently. The stress stages 6±9 were an un-loading series corresponding to the isotropic stres-ses 206´9, 137´9, 103´4 and 69´0 kPa in sequence.In the calibration, the measured velocity for load-ing and unloading under the same stress state wasaveraged. In total, ten measured data points wereanalysed. The measured data Vi (i � 1, 10) (in

METHODOLOGY FOR EVALUATING FABRIC OF GRANULAR MATERIAL 771

equation (21)) contain ®ve values of Shh and ®veof Shv. Three parameters fxg (in equation (21))were calibrated using the proposed procedure. Theywere (a) the fabric tensor D33 representing thecontact normal distribution, (b) the shear modulusGs of the sand particles and (c) the coef®cient á.The particle-shape-induced fabric tensor Ds

ij de-scribes the particle shape and the distribution ofpreferred particle orientations. Since the material isassumed to be transversely isotropic, a single Ds

33

was taken as the independent parameter represent-ing the distribution. A negative value of Ds

33

implies that most particles have their longer axis inthe horizontal direction.

Table 2 lists the calibrated results using themeasured shear wave velocities (under con®ningpressures of 69±275´8 kPa) corresponding to var-ious assumed values of Ds

33. The calibrated resultfor á equals 0´301, somewhat smaller than thevalue derived from the Hertz theory, i.e. 0´333.The parameter á controls the stress dependence ofthe shear wave velocity according to VS �C2ó

(1ÿ2á)=20 for isotropic granular materials under

an isotropic stress ó0. From the linear regressionof the experimental results, á is about 0´31 forstress stages 1±9, fairly close to the calibratedvalue 0´301 (Lee & Stokoe, 1986). Fig. 8 showsthe calculated and measured shear wave velocitiesunder various isotropic stress conditions.Jamiolkowski et al. (1995) collected data for wavevelocity (Lee & Stokoe, 1986; Stokoe et al., 1991;

Bellotti et al., 1996); they found that, for isotropi-cally stressed sand in the laboratory, the velocity ofshear waves polarized in the horizontal plane islarger than for waves polarized vertically. It is seenfrom Fig. 8 that the proposed micromechanicsmodel correctly describes the above-mentionedphenomenon. The parameters á and Gs calibratedfrom the data of stress stages 1±9 were treated asconstants in the subsequent study for evaluatingthe contact normal evolution of the sand particlesduring loading and unloading.

Contact normal evolution during loading/unloadingCalibration of the microfeatures from the wave

velocity measured during biaxial loading/unloadingtests on the same specimen was then carried out.Stress stages 13±16 were lateral loading (denotedby LL), while stress stages 16±19 were lateralunloading (denoted by LU). The axial stress (inthe direction of axis 3) remained constant(� 275:8 kPa) during both LL and LU. For LL, thelateral stress (in the direction of axis 1 and axis 2)increased from 103´4 to 278´5 kPa; for LU, thelateral stress decreased from 278´5 to 103´4 kPa.

Both the experimental results of Oda et al.(1985) and the numerical simulation of Rothenburg& Bathurst (1992) show that the preferred particleorientation of a non-spherical assembly does notchange signi®cantly during loading/unloading un-less the stress state is close to failure. Hence, the

Table 2. Calibrated microfeatures of washed mortar sand under isotropic compres-sion (ó0� 69±275:8 kPa)

Hypothetical shape-induced fabric

Ds33 0 ÿ0:1 ÿ0:2 ÿ0:3

Gs: GPa 83´3 82´9 82´8 83´0Calibrated parameters á 0´301 0´301 0´301 0´301

D33 ÿ0.646 ÿ0.419 ÿ0.188 ÿ0.040

Measured Shv

Measured Shh

Calculated shear wave velocity

50 60 80 100 200 500400300

Isotropic stress: kPa

220

240

260

280

300

320

340

360

380

She

ar w

ave

velo

city

: m/s

Fig. 8. Measured and calculated velocities of shear waves propagatingthrough the washed mortar sand specimen under isotropic compression

772 PAN AND DONG

contact normal distribution of the specimen forstress stages 13±19 were calibrated assuming var-ious constant values of Ds

33, unchanged duringloading and unloading. Fig. 9 presents the cali-brated results. The various symbols denote differentassumed values of Ds

33. Fig. 10 presents the meas-ured and calculated shear wave velocities. Thecalculated shear wave velocities are shown witherror bars for various values of Ds

33.Figure 9 reveals two interesting phenomena.

First, it is generally known that the contact normalsconcentrate progressively in the major principaldirection of a stressed granular assembly. Fig. 9clearly displays this trend for both LL and LUconditions. Next, the contact normal distribution ofstressed sand is dependent on the stress history. A`residual fabric' can be observed in Fig. 9 for asand specimen subjected to subsequent loading/unloading, no matter what Ds

33 is assumed. Thisobservation agrees with the conclusion drawn byChen & Hung (1991) from numerical tests.

Figure 9 also reveals another interesting aspect.If the sandy material is modelled as an idealizedgranular assembly (i.e. Ds

33 � 0), a denser distribu-tion of contact normals in the horizontal directionthan in the vertical direction (Ds

33 , 0) is requiredfor reproducing the measured results. However, thisdoes not agree with Feda's (1982) ®nding. Feda(1982) reported that most of the contact planesorientate horizontally for deposited granular materi-als. The condition Ds

33 ,ÿ0:1 seems more reason-able since the calibrated packing structure showsmore contact normals in the vertical direction forstress ratios less than 0´75, which is a likely rangeof in situ K0 conditions.

The introduction of an anisotropic branch vectorlength distribution is not just a mathematical trick.Arthur & Menzies (1972) summarized studies onthe inherent anisotropy of sands and concluded thatthe particles of a granular sediment tend to havetheir shortest axes in accordance with the gravity

direction. Their conclusion supports the proposedrepresentation of the fabric.

SUMMARY AND CONCLUSIONS

Evaluating the fabric of a granular material isessential for studying its mechanical behaviourfrom a microscopic viewpoint. A micromechanics-based methodology has been developed for charac-terizing the microstructure of a granular material.This methodology contains three basic elements:(a) a micromechanics-based elastic model, (b) atheory of elastic-wave propagation for an anisotro-pic material and (c) an optimization method. Themicromechanical elastic model has two features.First, the adoption of a force-dependent contactstiffness accounts for the stress dependence of theelastic properties. The contact force distribution isapproximated, using a localization process, fromthe global stress. Next, the shape and preferredorientation of non-spherical assemblies is describedby the distribution of branch vector length. Conse-quently, the micromechanics model presented herecan simulate the velocity anisotropy of a granularmaterial by considering the anisotropic distributionof microstructure and contact force.

For veri®cation, the proposed methodology wasused to determine the contact normal distributionof a glass ball assembly. The fabric of washedmortar sand was also evaluated. This demonstratesthat the fabric of a natural granular material can becalibrated from the measured wave velocities. Thefabric evolution of the granular material was alsoevaluated. Two interesting aspects of fabric changecan be observed: (a) a concentration of contactnormals in the major principal direction, and (b) aresidual fabric after subsequent loading/unloadingsequences. The enhancement of the character-ization of the fabric of natural granular materialsmakes the microstructural continuum approachmore attractive and applicable.

LL

LU

20.1

20.2

20.3

0.0

Shape-inducedfabric tensor D33

s

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Stress ratio (minor principal stress/major principal stress)

20.6

20.4

20.2

0.0

0.2

0.4

0.6

Cal

ibra

ted

fabr

ic te

nsor

D33

Fig. 9. Contact normal evolution of washed mortar sand specimenunder lateral loading and unloading

METHODOLOGY FOR EVALUATING FABRIC OF GRANULAR MATERIAL 773

ACKNOWLEDGEMENTS

The National Science Council of the Republicof China ®nancially supported this work underContract NSC86-2621-E009-12. This support isgratefully acknowledged.

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Fig. 10. Measured and calculated velocities of shear waves propagatingthrough the washed mortar sand under (a) lateral loading; (b) lateralunloading

50 60 80 100 200 500400300

Minor principal stress parallel to axes 1 and 2: kPa

220

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She

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ave

velo

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: m/s

Measured Shv

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(a)

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Minor principal stress parallel to axes 1 and 2: kPa

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She

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Measured Shv

Measured Shh

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METHODOLOGY FOR EVALUATING FABRIC OF GRANULAR MATERIAL 775