oda fabric

18
INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS. VOI.. 6. 11-94 (19821 A STATISTICAL STUDY OF FABRIC IN A RANDOM ASSEMBLY OF SPHERICAL GRANULES M. ODAt Department of Foundation Engineering, Saitama University, Urawa, Saitama, Japan AND s. NEMAT-NASSERS AND M. M. MEHRABADI~ Department of Civil Engineering, North western University, Evanston, Illinois 60201, U.S.A. SUMMARY It is commonly accepted that the mechanical behaviour of granular masses is strongly affected by their microstructure, namely the relative arrangement of voids and particles, i.e. the granular fabric. Therefore, parameters which characterize the granular fabric are of paramount importance in a fundamental description of the overall macroscopic stresses and deformation measures. In this paper several measures of granular fabric are introduced for a random assembly of spherical granules, using a statistical approach. In particular, a second-order symmetric tensor, ei, emerges from this consideration, which seems to be of fundamental importance for the description of fabric, and which is closely related to the distribution of the contact normals in the assembly. The relation between fabric measures presented here and those discussed by other investigators is also discussed. 1. INTRODUCTION Properties of most materials depend strongly on their microstructure, and granular materials consisting of discrete particles are no exception. The term 'fabric' is used in this paper to identify the microscopic arrangement of the particles and the associated voids. Experimental evidence reported by Oda,1-3 Arthur and Menzies; Arthur and Phillips', Mahmood and Mitchell6 and Mulilis et a[.' suggests that the concept of fabric can be very useful for the understanding of various properties of granular masses, such as dilatancy and anisotropy. On the basis of these experimental results, theories dealing with the corresponding strength and constitutive equations have been developed in recent years, taking into account the fabric (e.g., Horne,"' Mogami," Wilkins," Oda,12 Matsuoka,13 Sadasivan and Raju,14 Davis and Deresiewicz," Murayama,16 Satake17 and Mehrabadi et a1.18). In particular, Oda has examined rather extensively this notion, concluding that the fabric of an assembly of non-spherical particles may be identified by the following two ingredients: (1) the orientation of individual particles, and (2) the mutual relation of the particles. In this paper we shall attempt to define more precisely the concept of fabric for an assembly of spherical particles, using a statistical approach. The result provides further insight into the micromechanics of granular masses, and, therefore, is hoped to serve as a stepping stone for future fundamental developments in this area. f Visiting Scholar at Northwestern University. 1 Professor of Civil Engineering and Applied Mathematics. 5 Post-Doctoral Fellow. 0363-9061/82/010077-18$01.80 0 1982 by John Wiley & Sons, Ltd. Received 25 July 1980 Revised 12 December 1980

Upload: fukufukface

Post on 12-Oct-2014

90 views

Category:

Documents


5 download

TRANSCRIPT

Page 1: Oda FAbric

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS. VOI.. 6 . 11-94 (19821

A STATISTICAL STUDY OF FABRIC IN A RANDOM ASSEMBLY OF SPHERICAL GRANULES

M. ODAt

Department of Foundation Engineering, Saitama University, Urawa, Saitama, Japan

AND

s. NEMAT-NASSERS AND M. M. MEHRABADI~

Department of Civil Engineering, North western University, Evanston, Illinois 60201, U.S.A.

SUMMARY

It is commonly accepted that the mechanical behaviour of granular masses is strongly affected by their microstructure, namely the relative arrangement of voids and particles, i.e. the granular fabric. Therefore, parameters which characterize the granular fabric are of paramount importance in a fundamental description of the overall macroscopic stresses and deformation measures. In this paper several measures of granular fabric are introduced for a random assembly of spherical granules, using a statistical approach. In particular, a second-order symmetric tensor, ei, emerges from this consideration, which seems to be of fundamental importance for the description of fabric, and which is closely related to the distribution of the contact normals in the assembly. The relation between fabric measures presented here and those discussed by other investigators is also discussed.

1. INTRODUCTION

Properties of most materials depend strongly on their microstructure, and granular materials consisting of discrete particles are no exception. The term 'fabric' is used in this paper to identify the microscopic arrangement of the particles and the associated voids. Experimental evidence reported by Oda,1-3 Arthur and Menzies; Arthur and Phillips', Mahmood and Mitchell6 and Mulilis et a[.' suggests that the concept of fabric can be very useful for the understanding of various properties of granular masses, such as dilatancy and anisotropy. On the basis of these experimental results, theories dealing with the corresponding strength and constitutive equations have been developed in recent years, taking into account the fabric (e.g., Horne,"' Mogami," Wilkins," Oda,12 Matsuoka,13 Sadasivan and Raju,14 Davis and Deresiewicz," Murayama,16 Satake17 and Mehrabadi et a1.18). In particular, Oda has examined rather extensively this notion, concluding that the fabric of an assembly of non-spherical particles may be identified by the following two ingredients: (1) the orientation of individual particles, and (2) the mutual relation of the particles.

In this paper we shall attempt to define more precisely the concept of fabric for an assembly of spherical particles, using a statistical approach. The result provides further insight into the micromechanics of granular masses, and, therefore, is hoped to serve as a stepping stone for future fundamental developments in this area.

f Visiting Scholar at Northwestern University. 1 Professor of Civil Engineering and Applied Mathematics. 5 Post-Doctoral Fellow.

0363-9061/82/010077-18$01.80 0 1982 by John Wiley & Sons, Ltd.

Received 25 July 1980 Revised 12 December 1980

Page 2: Oda FAbric

78 M. O D A . S. NEMAT-NASSER A N D M. M. MEHRABADI

2. STATISTICAL DEFINITION OF GRANULAR FABRIC

2.1. General description

Consider an assembly of spheres with total volume V consisting of the solid volume, V,, and the associated void volume, V,. The-void ratio, e, and the porosity, p (YO), frequently used as index properties, are defined by

e = V,/ V, ( la )

p = lOOV,/V (1b)

Let the distribution of radii, r, in the assembly, be given by a probability density functiont f(rL

where rM and rm are the maximum and the minimum radii of the particles, and f(r) dr is the fraction of the particles with radii ranging from r to r +dr ; Figure 1.

Figure 1 . Probability-density function for the distribution of particle radii

Without loss of generality, the assembly of spheres can be replaced by an assembly of points and lines (SatakeI7); that is: (1) points (. . . , gi, gi, . . .) marking the centres of particles, and (2) lines (. . . , gigi, mk, . . .) connecting the centres of adjacent particles which are in contact at points (. . . , cj-k, . . .); Figure 2. This assumes that the corresponding fabric is represented with sufficient accuracy by the spatial distribution of points, and by the distribution and geometrical arrangement of lines. Satake” calls the line connecting the centres of two contacting particles, ‘branch’. This designation is also used in this paper.

2.2. Spatial distribution of points

particles in the volume V may be estimated by Let S be the density of the points marking the centres of particles. The number, m‘”’, of

t An equation relating f(r) to the probability distribution of the radii of circular sections produced by a cutting plane, is given in Appendix I.

Page 3: Oda FAbric

FABRIC IN RANDOM ASSEMBLY OF SPHERICAL GRANULES 79

j c . . C O S T K T IIIXWEE?: PAKl.ICl.ES gi RSU g 1-j' d . , : YID-POI.:.Il' Oi: BKtLUCH g.g.

1 - 3 1 3

Figure 2. Representation of the granular fabric in a mass of spheres

Since the volume of a sphere of radius r is $v3, and since the number of spheres in V with radii ranging from r to r + d r is m'"'f(r) dr, the total solid volume V, is estimated by

V, = Jrr 377r'm'"'f(r) dr (4)

Furthermore, since the total volume V equals (1 + e ) V,, equation (3) can be rewritten to give the following estimate:t

(5 ) 3 - - 3 ( V )

s=-- - V 4 r ( l + e ) 1;: r 3 f ( r ) dr 4 4 1 + e ) 3

where the following notation is used: 'M

6 = I, @ ( r ) f ( r ) dr (6 )

for the mean of any function @(r) , based on the probability density function f ( r ) . There are a number of published equations which attempt to relate the mean number, ti,

of contacts per particle to the corresponding void ratio e (or porosity p ) ; e.g. Smith et al.," Field," Gray2* and Oda.23 The mean number 5 is called the 'co-ordination number'. Figure 3 shows the relation between 6 and e for random assemblies of spheres; Oda.23 Three assemblies are employed so as to examine the effect of grain size distribution, f ( r ) , on the relation between ii and e. In Figure 3, the homogeneous assembly consists of homogeneous spheres, the two-mixed assembly consists of spheres with two different radii, and the multi-mixed assembly consists of spheres with four different radii. Since all data points fall basically on the same curve, the relation between ti and e seems to be independent of the grain size distribution, f(r); Field" and Oda.23 Hence, it follows that

t An equation relating e and the number of particles which are intersected by a cross-section is given in Appendix 1.

Page 4: Oda FAbric

80 M. ODA, S. NEMAT-NASSER AND M. M. MEHRABADI

Ire 0.5 1 . 0 1.5 0

Figure 3. Experimental relation between the mean co-ordination number, A, and the void ratio, e, (Oda”)

Associated with each contact there are two contact points, one belonging to each contacting , in 1 - (V) particle. Accordingly, the total number, N(v), of contacts (not contact points) is Tnm

which m(v) is given by equation (3). Then, the density of contacts is

Equations (5) and (8) give the average number of particles and contact points in a typical volume. In general, particles and their centres are not distributed homogeneously, so that their density must be defined as a statistical variable with an associated probability density. Equations ( 5 ) and (8) , however, yield first-order information for the description of the granular fabric, as is discussed below.

2.3. Distribution and geometrical arrangement of branches

The mid-point, di-b on a typical branch, a, connecting two neighbouring particles, gi and gj, coincides with the point of their contact, c{-~, if the two particles have equal radii. Moreover, the number N‘v) of contacts in volume V is exactly the same as the number of mid-points, irrespective of particle size. Accordingly, equation (8) not only gives the density N of contacts but also the density of mid-branch points.

2.3.1. Angular distribution of branches. For this purpose, it is sufficient to use the probability density function E(n) which has been introduced by Horne’ and Oda’ to describe the angular distribution of contact unit normals, n. At each contact, two unit vectors, n and -n, which

Page 5: Oda FAbric

FABRIC IN RANDOM ASSEMBLY OF SPHERICAL GRANULES 81

for spherical granules are coaxial with the corresponding branch, are considered. These vectors may be identified by angles a and #3 with respect to a fmed rectangular Cartesian coordinate system with coordinate axes xi, i = 1 ,2 ,3 ; Figure 4(a). The function E(n) satisfies

E(n) = E( -n) 2 0 (9b)

where dR is an elementary solid angle equal to sin@ d a d p (Figure 4(b)), and R is the unit sphere defined by Oda <27r and Odp < 7r. In view of the symmetry of E(n) given by (9b) it is convenient to use 2E(n) defined over f11/2, instead of E(n) defined over R, where 0112 is the half-unit-sphere defined by 0 Q < 27r and 0 c /? 7~12. In this manner either n or - n corresponds to one branch.

In the following discussion a fixed rectangular Cartesian coordinate system with coordinate axes xi , i = 1,2 ,3 , corresponding to unit base vectors ei is employed. In addition to this, we

X.

-x 3

i' - dli = sin6 da d6

x 3

( b ) F . I . J l ~ L E ~ 7 ' A K Y SOl.1 I) , \ N l ; L l < dii

Figure 4. Contact unit normal vectors and the elementary solid angle df l

Page 6: Oda FAbric

82 M. O D A , S. NEMAT-NASSER A N D M. M. MEHRABADI

shall have occasion to use another rectangular Cartesian coordinate system, say x I, with unit base vectors el, which is obtained by a rotation of the xi-system about the origin;

x : = ci,xj, i, j = 1, 2, 3 (10)

where repeated indices are summed, and cii = e: . eh the dot denoting inner product. The components of the contact unit normal n, with respect to the primed coordinate system, will be denoted by ni, and the corresponding spherical angles, by a ' and P'. Then defines the half-unit-sphere, 0 s a'< 27r and 0 S @'s 7212.

Finally, note that because of the symmetry condition, equation (9b), n/2 2n [In E(n) dR = lo 1" 2E(a, P ) sin P d a d p = /In,,, 2E(n) d n = 1 (1 1)

If a function P(n) also satisfies the condition P(n) = P( -n), then n/2 277 [I, P(n)E(n) d~ = lo 1" 2P(a, P E ( a , P ) sin P d a dP

where (P(n)) denotes the mean of P(n) with respect to the density distribution function E(n). If on the other hand P(n) = -P( -n) 2 0 over R, then

2P(n)E(n) d R = IP(n)lE(n) dR = (IP(n)l> (13)

2.3.2. Distribution of branch lengths. The branch length 1 may be viewed as a random

2r, s 1 c 2rM (14)

where 2r, and 2rM are branch lengths corresponding to the two smallest (with the radius rm) and the two largest (with the radius rM) particles, respectively. Let g(1) be the probability density function defining the distribution of the branch lengths, I. It is reasonable to expect that g(1) is related to the probability density function f ( r ) which characterizes the statistical distribution of the particle radii.

Consider two groups (A and B) of particles (Figure 5 ) , where the particles in group A have radii ranging from r to r + dr, while the radii of those in group B range from ( I - r ) to (1 - r ) + dr. For the particles in group A, which are in contact with those in group B, the branch lengths fall in the range 1 to 1 + 2 dr. Since each branch is considered to be selected at random, the probability, 2g'AR'(1) dr, of selecting a branch with length in the interval 1 to 1 +2 dr may be assumed proportional to the number of particles belonging to the two groups, obtaining

2g'AB'(1)dr =kf( r )dr f ( l - r )dr (15)

where k is the proportionality coefficient to be determined.t If the radii of group A particles are within the range ( 1 - r M ) to (1-r , ) , then these particles certainly form, as the result of

II, variable with the range,

t Equation (15) assumes that the conditional probability of contact is constant; this, in general, is not the case.

Page 7: Oda FAbric

FABRIC IN RANDOM ASSEMBLY OF SPHERICAL GRANULES 83

f ( r )

g ( 2 )

2dr or , c

r

t t GROUP A GROUP B

Figure 5. Probability density function, g ( l ) , of branch length distribution

their contact with particles in group B, branches with lengths in the range I to I +2 dr. The function g(l) is obtained by adding every combination which yields a branch of length I,

k g(1) = 1 g'AB'(l) = 2 [ f(r)f(l- r) dr

I - rms r s I - r M I - r M

Since g ( l ) is a probability density function, it follows that 2 %

g ( 0 d l = 1

which determines the proportionality coefficient k in equation (16). In this manner it finally follows that

which yields an estimate for the distribution of branch lengths in terms of the particle size distribution.

3. STATISTICAL INDEX MEASURES OF FABRIC

Since granular materials are composed of discrete particles, it is necessary to quantify statisti- cally the distribution of their relevant micromechanical parameters, such as the distribution of the contact forces and that of the sliding velocities at contact points, in order to arrive at a fundamental description of the overall continuum quantities like the stress and the strain rate. When macroscopic constitutive equations for materials of this kind are to be formulated, the microscopic discrete quantities must be averaged. This requires a knowledge of their microscopic distributions, i.e. the fabric.

In this section, some statistical index measures are discussed in detail in an effort to establish an effective measure for the transition from discrete to continuum concepts.

3.1. Statistical index measures on an imagined plane

Consider an imagined plane passing through a granular assembly. Denote by v the unit vector normal to this plane, pointing from the negative (arbitrarily chosen) side towards the positive side of the plane. This plane will be called the v-plane.

Page 8: Oda FAbric

84 M. ODA, S. NEMAT-NASSER AND M. M. MEHRABADI

V

GROUP B P A R T I C L E S t

UNIT AREA OF THE U - P L A N E

b!-- GROUP A P A R T I C L E S

(a) BRANCHES OF T H E \)-PLANE

9

I4ID-BRANCIi P O I N T S

1 v i THE V-PLANE

(b ) ASSEMBLY OF BRANCH VECTORS

Figure 6. Branch vectors, I, which intersect the v-plane

All particles located in the immediate vicinity of the v-plane can be divided into the following two groups (Figure 6(a)): Group A (hatched) are those particles whose centres are located on the negative side of the plane, and Group B (unhatched) are those whose centres are located on the positive side of the plane. Solid lines in Figure 6(a) are branches connecting the centres of the hatched particles with those of the unhatched ones, provided that there is a common contact.

When one attempts to relate the microscopic contact forces (discrete) to the overall macro- scopic tractions transmitted across an interior imagined plane (continuum), then the density, N‘”), of the branches connecting the two groups across this plape, and the angular distribution, E(”)(n), of these branches become of paramount importance. Note that E‘”’(n) is a surface density function? different from E(n) which defines the angular distribution of all branches (or all contact normals) within a given volume.

+ This will be defined more precisely in Subsection 3.1.2.

Page 9: Oda FAbric

FABRIC IN RANDOM ASSEMBLY OF SPHERICAL GRANULES 85

3.1.1. Number of branches on an imagined plane. The branches connecting the two groups A and B intersect with the imagined u-plane, while branches corresponding to each group separately do not. So, the number, N‘”), of branches per unit area of the imagined u-plane equals the density of the branch intersections with this plane.

From all branches associated with the two groups, including those associated with the same group, select those branches whose lengths fall in the range I to I +dl, and whose directions lie within an elementary solid angle dR of the fixed orientation n. The probability of choosing such a branch is

where g ( f ) is estimated by equation (18). Equation (19) is based on the reasonable assumption that the length I and the orientation

n are statistically uncorrelated. Since the density of the mid-branch points is given by N in equation (8),

is the density of the mid-branch points corresponding to (19). Multiplying (20) by the relevant volume (see below), we determine, according to the definition of density, the number of branches with the following three properties: (1) they all fall in the solid angle dR of orientation n; (2 ) their lengths range from I to I+d l ; and (3) their mid-points are contained in the considered volume. If a branch of this kind has its mid-branch point at a distance less than or equal to +In. u from the u-plane, then this branch surely intersects the u-plane (see Figure 7). To find the density of such intersections, multiply (20) by the volume 1 x 1 x In . w, obtaining

%Nfg( l )n . uE(n) dl dR (21)

t

’- 1 - J (a) ( b )

Figure 7 . Number of the branches which intersect the u-plane; note that the vector I intersects the u-plane when its mid-point is inside the region with volume 1 X 1 X I (n . u )

Page 10: Oda FAbric

86 M. ODA. S. NEMAT-NASSER AND M. M. MEHRARADI

Upon integrating over f and R, we arrive at the average number per unit area, N‘”’, of branches which intersect the v-plane,

I g ( l ) df 2 n . vE(n) dfl n1,z

- -Nj2:r f g ( f ) d l j l In. v(E(n) dfl n

where f is the mean branch length defined by

Note that N‘”’, which is a scalar associated with the vector v, is invariant under orthogonal transformation of coordinate systems. Defining the vector

Ni = N f ( ( n i ( ) , i = 1, 2, 3 (24)

it fAlows from (22) that the number of contacts on any plane with the unit normal v is equal to the projection of N on v, i.e.

N‘”’ = N . v (25)

In particular, note that the components N 1 , N2 and N3 of N represent the density of contacts on planes normal to the coordinate axes xl, x2 and x3, respectively.

3.1.2. Angular distribution of branches on an imagined plane. Let dN‘”(n) be the density (per unit area of the v-plane) of branches which intersect the imagined v-plane and lie within dR with orientation n. It is seen that dN‘”’(n) is given by integrating expression (21) with respect to I from 2rM to 2r,,

21M

dN‘”(n) = 2 N n . vE(n) dR l g ( f ) dl I,, = 2Nfn. uE(n) dR (26)

If on the u-plane a probability density function E‘”’(n) is introduced to describe the angular distribution of the branches intersecting this plane,? then dN‘”’(n) is also given by

dN‘”’(n) = N‘”’E‘”’(n) dR (27)

N‘”E‘u’(n) = 2 ~ i n . vE(n)

From equations (26) and (27), it follows that

(28)

Since N‘”’ = NI*(ln . vl), equation (22), equation (28) can be rewritten as

(In . v()E‘”’(n) = 2 n . vE(n) (29)

Since (In. 4) is defined in terms of E(n), we see from (29) that E(“)(n) is completely characterized by the same probability density function. K o n i ~ h i ~ ~ also considered the function E‘”’(n). His result is, however, different from equations (28) and (29).

+ T h a t is, E‘”(n) di2 is the probability (per unit area) of intersections with the u-plane of branches which are in the elementary solid angle d o of orientation n.

Page 11: Oda FAbric

FABRIC IN RANDOM ASSEMBLY OF SPHERICAL GRANULES 87

3.2. Fabric tensor

To calculate the overall macroscopic tractions transmitted across an imagined plane (Figure 6(a)), it is necessary to add up the contact forces which represent the mutual action of hatched and unhatched particles. This has been discussed in some detail in an accompanying paper." An examination of this kind immediately shows that the vector sum of the branches which intersect a unit area of a plane of arbitrary orientation plays a key role in such calculations. Here, therefore, we consider such a sum, from which emerges in a natural way a second-order tensor which we call the fabric tensor.

Consider the assembly of branches connecting the centres of hatched and unhatched particles, across the ei-planet (Figure 6(a)), and observe that these branches can be replaced by the assembly of unit vectors, l/I as shown in Figure 6(b). The number of such vectors whose directions lie within dR, is

N,E"i' (n) dn (30)

Since 1 = In, it is clear that, in general, E"'(n) =Z?)(I). Now consider the projection, on the xi-axis, of the vector sum of all such I / I = n that intersect

the ei-plane. Since the component of n in the +direction is n . eh this vector sum is given by multiplying (30) by n . ei = ni, i.e.

Nin$ei) (n) dR (31) which stands for a second-order tensor, say, dFi+ The jth component of the total vector sum of all n's, measured per unit area of the plane normal to the ei-direction, is then obtained if we integrate dFii over R112,

Ei = Ni [h,,, n , d e 0 (n) dR

where equation (28) has also been used.

tensor, When new coordinate axes, xl, are introduced, it is clear that Fii transforms as a second-order

F:j = ~ikcjiFk1 (33) This quantity will be called the 'fabric tensor'. It is important to note that, while Fii is defined in terms of the probability density function E(n), its tensorial character is independent of the specific form of this function. As discussed by Oda19 and Oda ef ~ l . , ~ ' experimental results suggest that E(n) may be represented with good accuracy by an ellipsoid. Based on this fact, Oda" has defined the 'fabric ellipsoid'; see Subsection 4.2. It should be noted, however, that since 'fabric' as given by (32) is a second-order symmetric tensor, we may always represent it by an ellipsoid. That fabric may be represented by a second-order tensor, has been previously pointed out by Cowin,26 Jenkins27 and Satake. l7

4. OTHER INDEX MEASURES

The fabric tensor Ei defined by equation (32) appears to represent a fundamental quantity for the characterization of granular material properties. For example, it may be used to define

t That is, the plane normal to the e,-direction.

Page 12: Oda FAbric

88 M. ODA. S. NEMAT-NASSER AND M. M. MEHRABADI

a macroscopic (continuum) stress tensor in terms of (microscopic) contact normals and contact forces, as is discussed in some detail in an accompanying paper” by the present authors. A similar tensor, namely

Jjj = (njnj) (34)

has been introduced by Satake” who also stresses its potential fundamental nature. Gudehus2’ has also introduced a tensor Aij, which he calls ‘Affinitat’, in order to represent macroscopically the microstructure of the soil.

There are, however, other practically useful index measures which have been considered by various researchers. These include the ‘mean projected solid path’ introduced by Horne,’ and the ‘fabric ellipsoid’ proposed by Oda.I9 We shall briefly consider these below.

4.1, Solid path (Horne‘)

‘Diametral plane’ is defined as a plane passing through the centre of a particle. A plane of this kind may be chosen perpendicular to a common reference axis, say XI; Figure 8. Each

DIAMETKAL PLANE

Figure 8. Solid path of Home’

particle is divided by its diametral plane into two portions: the positive portion which is in the positive xl-side, and the negative portion which is on the opposite side of the plane (Figure 8). One particle and one contact on its positive portion are selected at random, say, particle gj and contact c ; - ~ The contact ci-i is on the positive side of particle g , but at the same time, it is on the negative side of particle gi. Again a contact is chosen at random among those on the positive portion of particle gj, and it is connected by a straight line to c ; - ~ This process is continued, connecting consecutively the selected contacts to obtain a zigzag path which is called a ‘solid path’ by Horne.’ If the reference xi-axis is a symmetry axis of E(n), the general trend of the corresponding solid paths would correspond to this direction; otherwise the solid paths would deviate from the xi-direction.

In the special case when a symmetry axis, say xl, of E(n) is considered, we can calculate the average number N!”’ of contacts per unit distance measured in the x,-direction, which

Page 13: Oda FAbric

FABRIC IN RANDOM ASSEMBLY OF SPHERICAL GRANULES

fall on the corresponding solid path (Horne,’ Oda”),

NiSp) = 1 2 I;: rf(1) dr I in,,, 2nlE(n) dR

and, in view of equation (12), this can be written as

89

(35)

with similar expressions for NiSp’ and N$sp’.

the granular fabric (e.g., Horne,’ Odai2 and T ~ k u e * ~ ) . The idea of solid path has been frequently used to define strain rate, taking into account

It is of interest to compare equation (24) with equation (36), arriving for example, at

If the assembly consists of uniform spheres, every branch has the length 21. In this special case, f is exactly equal to 21, and equation (37) becomes

1 N 1 = N p i

with similar expressions for NZ and N3.

4.2. Fabric ellipsoid (Oda 29*30)

The probability density function E(n) has been experimentally determined by Oda1-3 and Oda and Konishi3’ for the three-dimensional case, and by K ~ n i s h i , ~ ~ Biarez and W i e n d i e ~ k ~ ~ and W i e n d i e ~ k ~ ~ for the two-dimensional case. By summarizing the experimental results, Oda et al.” conclude that E(n) may be approximated by an ellipsoid. In the following discussion, the orthogonal axes xl, xz and x 3 are selected in such a manner that they coincide with the symmetry axes of E(n). Moreover, the angles LY and p are measured with respect to these symmetry axes.

Based on the assumption that E(n) can be approximated by an ellipsoid, Oda19 has proposed the concept of ‘fabric ellipsoid’ as an index measure for granular materials. A brief description of this is appropriate at this point. Consider an assembly of particles with a large number of contacts. Each particle contacts its neighbour over a small area AS(,). Introduce a closed surface with the following properties (Figure 9): every small element AS;,) on the closed surface corresponds precisely to a contact surface AS(,) in the assembly and, at the same time, n’ and -n‘, unit normals to AS;,,, are, respectively, parallel to the unit normal vectors n and -n at the contact. The closed surface has been called the ‘fabric ellipsoid’, as it approximates an ellipsoid (OdaI9). It is characterized by three measurable quantities SI, SZ and S3, which are the areas of the principal planespf the fabric ellipsoid, respectively. Assuming that each contact has the same contact area, AS, we calculate Si,

si =hS J J 2 n i ~ ( n ) dR n1,2

Page 14: Oda FAbric

90 M. O D A , S. NEMAT-NASSER A N D M. M. MEHRABADI

Figure 9. Fabric ellipsoid of OdaI9

Comparing equation (39) with equations (36) and (24), the following interesting relations are obtained:

with similar expressions for Sz and S3, and -

Accordingly, the two quantities Ni and Si have the same physical meaning. The following points should, however, be noted:

(1) N, is precisely defined in terms of N, f and (Ini\), while the quantitiy AS(,, which does not admit a precise identification, enters the definition of Si. In an assembly of spheres which are made of highly stiff materials like glass, quartz and metal, contact areas are very small, so that AS(,) is not easily identifiable;

(2) Ni is precisely identified only when an assembly is composed of spheres. On the other hand, Si can be defined even for an assembly of non-spherical particles.

5 . DISCUSSION

For a random assembly of spherical granules several measures of the fabric are introduced and their relation with each other and with other index measures previously presented by other investigators is discussed. These measures are the density of contacts, N (see equation (8)), the average branch lengths, f (see equation (23)), the vector N (see equation (24)), whose normal component in a given direction defined by the unit vector, Y, gives the density of the intersection of the contact normals with an imagined plane normal to the v-direction, and, finally, a second-order symmetric tensor, Ei (see equation (32)), which is intimately related to the overall macroscopic stress tensor, as has been discussed in rather great detail in an accompanying paper by the present authors. Indeed, during the flow of a granular medium consisting of spherical particles, the distribution of contact normals changes con- tinuously in response to the change in overall macroscopic stresses. Hence, both the probability density function E(n) which characterizes the distribution of the contact unit normals, as well as the fabric tensor Fij = Nf(nini) which is defined in terms of E(n) and which is a measure of resultant forces transmitted across a unit area of imagined interior planes, change with the

Page 15: Oda FAbric

FABRIC IN RANDOM ASSEMBLY OF SPHERICAL GRANULES 91

overall macroscopic stresses. These observations are supported by experimental results, e.g. Oda e f d2' In fact, as discussed in Subsection 4.2, it has been suggested that the distribution function E(n) has the same symmetry as the stress tensor, and that it approximately resembles an ellipsoid which tends to become coaxial with the stress ellipsoid, as the plastic flow of granular material continues. It is readily seen that if E(n) is indeed an ellipsoid, then the ellipsoid associated with the fabric tensor F.i will be coaxial with it, and hence, one expects that the fabric tensor should tend to become coaxial with the stress tensor.

To develop a complete set of constitutive relations for granular materials, taking into account fabric and its evolution, one requires two essential considerations: (1) one must relate at each instant (at each loading state) the overall macroscopic stress to the fabric measures, and in particular, to the fabric tensor; and (2) one must relate the overall macroscopic deformation rate tensor and the stress rate tensor to the rate of change of fabric, and hence, describe the stress rate in terms of the strain rate. In an accompanying paper, item (1) has been discussed in considerable detail by the present authors. The examination of item (2), on the other hand, demands an understanding of the manner by which the distribution of contact normals changes as new contacts are continuously developed while some existing ones are lost. Hence, the next major outstanding problem is to develop an evolutionary equation for the fabric tensor, which is being pursued by the present authors.

ACKNOWLEDGEMENT

This work has been supported by U.S. Air Force Office of Scientific Research, Grant No. AFOSR-80-0017 to Northwestern University.

APPENDIX I

If an assembly of spheres is cut along a plane by a saw, an assembly of circles would be observed in the resulting cross-section; Figure 10. We wish to establish some statistical measures relating to the arrangement of these circular sections.

l / CUTTlNG PLANE \ I I

Figure 10. Circles on a cutting plane

Number of particles intersected by a cross-section

Let us consider a cross-section of unit area. Since the spheres are small, a large number of them are cut by this section. Moreover, since f ( r ) dr is the number of particles whose radii range from r to r+dr, and since 6 is the density of all particles which are distributed homogeneously in space, we observe that 6 f ( r ) dr gives the density of the particles whose radii

Page 16: Oda FAbric

92 M. ODA. S. NEMAT-NASSER AND M. M. MEHRABADI

range from r to r +dr. If the centres of particles with radii of r to r +dr are located within the distance r from the cross-section, then these particles can be intersected. Therefore, the particles contained in the volume 1 x 1 X 2r will be sectioned; Figure 10. Since the number of particles with radii of r to r +dr in this volume is 26rf(r) dr, the total number mccs) of circular sections on the cross-section is

‘M

m‘cs’ = 26rf(r) dr I, = 2si (Al)

It is worth noting that mCCs’ is independent of the orientation of the cross-section, while the number N‘” in equation (22) depends linearly on the orientation of the considered imaginary plane. Hence equation (Al) provides no information regarding the anisotropic arrangement of the particles.

Since the number m(cs) can easily be obtained experimentally, it can be used, instead of (3, to define the density S,

(CS)

s=- 2F

From equations (5) and (A2), the void ratio e can be estimated in terms of the averages 7 and 7, and m‘CS’,

Distribution of radii of circular sections on a cross-section

Since radius R of a circular section observed on a cross-section has a random variation, a probability density function h (R) is required for its specification.

First, consider an assembly composed of uniform spheres of common radius ro. Let x be the distance between the considered cross-section and the centre of a typical particle. If x s ro, the particle is intersected, where

r i = x z + R 2 (A4)

dR x J(rE-R2) dx R R

- -

We see that the probability of an intersection which forms a circular section of radius, R, is equal to the probability that the particle’s centre be located within the distance x to x +dx from either side of the cross-section. Hence,

(A6)

where the minus sign shows that R decreases with increasing x. From equations (A4), (AS) and (A2), h(R) is determined,

m‘CS’h(R) dR = -26 dx

Since R can range from 0 to ro,

lor” h(R) dR = 1

Page 17: Oda FAbric

FABRIC IN RANDOM ASSEMBLY OF SPHERICAL GRANULES 93

In the assembly of spheres whose radii are distributed according to f ( r ) , the function h ( R ) is given by (see Kendall and Moran3’)

1.

2.

3.

4. 5.

6.

7.

8.

9.

10.

11. 12. 13.

14.

REFERENCES

M. Oda, ‘Initial fabrics and their relations to mechanical properties of granular material’, Soils and Foundations,

M. Oda. ‘The mechanism of fabric changes during compressional deformation of sand’, Soils and Foundations,

M. Oda, ‘Deformation mechanism of sand in triaxial compression tests’, Soils and Foundations, 12, No. 4, 45-63 (1972). J. R. F. Arthur and B. K. Menzies, ‘Inherent anisotropy in a sand‘, Gebtechnique, 22, 115-128 (1972). J. R. F. Arthur and A. B. Phillips, ‘Homogeneous and layered sand in triaxial compression’, Giotechnique, 25,

A. Mahmood and J. K. Mitchell, ‘Fabric-property relationships in fine granular materials’, Clays and Clay Minerals, 22, 397408 (1974). J. P. Mulilis, C. K. Chan and H. B. Seed, ‘The effects of method of sample preparation on the cyclic stressstrain behavior of sand’, Report No. HERC 75-18, Earthquake Engineering Research Center, University of California, Berkeley (1975). M. R. Horne, ‘The behavior of an assembly of rotund, rigid, cohesionless particles (I and II)’, Proc. Roy. SOC.

M. R. Horne, ‘The behavior of an assembly of rotund, rigid, cohesionless particles (111)’. Proc. Roy. SOC. A , 310,

T. Mogami, ‘A statistical approach to the mechanics of granular materials’, Soils and Foundations, 5, No. 2,

J. K. Wilkins. ‘A theory for the shear strength of rockfill’, Rock Mech., 2, 205-222 (1970). M. Oda, ‘A mechanical and statistical model of granular material’, SoilsandFoundations, 14, No. 1.13-27 (1974). H. Matsuoka, ‘A microscopic study on shear mechanism of granular materials’, Soils and Foundations, 14, No. 1 , 2 9 4 3 (1974). S. K. Sadasivan and J. S. Raju, ‘Theory for shear strength of granular materials’, J. Geotech. Div., Proc. Am. SOC. Civ. Enn.. 103.837-861 (1977).

12, NO. 1, 17-36 (1972).

12, NO. 2, 1-18 (1972).

799-815 (1975).

A, 286,62-97 (1965).

21-34 (1969).

26-36 (1965).

15. R. A. Davis‘and H: Deresiewicz, ‘A discrete probabilistic model for mechanical response of granular medium’, Acta Mech., 27.69-89 (1977).

16. S. Murayama, ‘Constitutive equation of particulate material in the plastic state’, 9th Inr. Conf. Soil Mech. Found. Engrg, Specialty Session 9, 175-182 (1977).

17. M. Satake, ‘Constitution of mechanics of granular materials through graph representation’, Theorefical and Applied Mechanics, 26, University Tokyo Press,,257-266 (1978).

18. M. M. Mehrabadi, S. Nemat-Nasser and M. Oda, ‘On statistical description of stress and fabric in granular materials’, Int. J. Numer. Anal. Methods Geomech., 6 , 95-108 (1982).

19. M. Oda, ‘Significance of fabric in granular mechanics’, Proc. US.-Japan Seminar on Continuum-Mechanical and Sfatistical Approaches in the Mechanics of Granular Materials, Gakujutsu Bunken Fukyukai, Tokyo, Japan,

20. W. 0. Smith, P. D. Foote and P. F. Busang, ‘Packing of homogeneous spheres’, Phys. Rev., 34.1271-1274 (1929). 21. W. G. Field, ‘Towards the statistical definition of granular mass’, Proc. 4th A . and N.Z. Conf. on Soil Mech.,

22. W. A. Gray, The Packing of Solid Particles, Chapman and Hall, 1968. 23. M. Oda, ‘Co-ordination number and its relation to shear strength of granular material’, Soils and Foundations,

17, No. 2, 2 9 4 2 (1977). 24. J. Konishi, ‘A microscopic consideration on deformation and strength behavior of granular material like sand’,

Doctorate thesis at Kyoto University (1978). 25. M. Oda, J. Konishi and S. Nemat-Nasser, ‘Index measure of granular materials’, Earthquake Research and

Engineering Laboratory, Technical Report No. 80-3-26, Dept. of Civil Engineering, Northwestern University. Evanston, Ill. (1980); ‘Some experimentally based fundamental results on the mechanical behavior of granular materials’, Giotechnique, 30, 479-495 (1980).

26. S. C. Cowin, ‘Microstructural continuum modes for granular materials’, Proc. US.-Japan Seminar on Continuum- Mechanical and Sfatistical Approaches in the Mechanics of Granular Materials, Gakujutsu Bunken Fukyukai, Tokyo, Japan, 162-170 (1978).

7-26 (1978).

143-148 (1963).

Page 18: Oda FAbric

94 M. ODA, S. NEMAT-NASSER AND M. M. MEHRABADI

27. J. T. Jenkins, ‘Gravitational equilibrium of a model of granular media’, Proc. US.-Japan Seminar on Continuum- Mechanical and Statistical Approaches in the Mechanics of Granular Materials, Gakujutsu Bunken Fukyukai, Tokyo, Japan, 181-188 (1978).

28. G. Gudehus, ’Gedanken zur Statistischen Boden Mechanik’, Baningenieur, 43, 320-326 (1968). 29. T. Tokue, ‘A stress-dilatancy model of granular material under general stress conditions‘, Soils and Foundations,

30. M. Oda, ‘Fabrics and their effects on the deformation behavior of sand’, Report of Dept. of Foundation Eng., Saitama University, Japan, Special Issue (1976).

31. M. Oda and J. Konishi, ‘Microscopic deformation mechanism of granular material in simple shear‘, Soils and Foundations, 14, No. 4, 15-29 (1974).

32. J. Konishi, ‘Microscopic model studies on the mechanical behavior of granular materials’, Proc. US.-Japan Seminar on Continuum-Mechanical and Statistical Approaches in the Mechanics of Granular Materials, Gakujutsu Bunken Fukyukai, Tokyo, Japan, 27-45 (1978).

33. J. Biarez and K. Wiendieck, ‘La comparison qualitative entre I’anisotropie mecanique et I’anisotropie d e milieux pulvCrulents’, C.R. Acad. Sci., 256, 1217-1220 (1963).

34. K. Wiendieck, ‘Zur Struktur Korniger Medien’, Bautechnik, 6, 196-199 (1967). 35. M. G. Kendall and P. A. P. Moran, Geometrical Probability, Griffin’s Statistical Monographs and Courses, Ed.

19, NO. 1 , 6 3 4 0 (1979).

M. G. Kendall, 86-89, 1963.