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INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS. VOI.. 6 . 1194 (19821
A STATISTICAL STUDY OF FABRIC IN A RANDOM ASSEMBLY OF SPHERICAL GRANULESM. ODAt
Department of Foundation Engineering, Saitama University, Urawa, Saitama, JapanAND
s. NEMATNASSERS
AND M. M. M E H R A B A D I ~
Department of Civil Engineering, North western University, Evanston, Illinois 60201, U.S.A.
SUMMARYIt 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 secondorder 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. INTRODUCTIONProperties 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,13 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 nonspherical 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 PostDoctoral Fellow.
03639061/82/01007718$01.80 0 1982 by John Wiley & Sons, Ltd.
Received 25 July 1980 Revised 12 December 1980
78
M. O D A . S. NEMATNASSER A N D M. M. MEHRABADI
2. STATISTICAL DEFINITION OF GRANULAR FABRIC 2.1. General descriptionConsider an assembly of spheres with total volume V consisting of the solid volume, V,, frequently and the associated void volume, V,. Thevoid ratio, e, and the porosity, p (YO), used as index properties, are defined bye = V,/ V,p=
(la) (1b)
lOOV,/V
f Let the distribution o 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) d r is the fraction of the particles with radii ranging from r to r + d r ; Figure 1.
Figure 1 . Probabilitydensity 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, m ,. . .) connecting the centres of adjacent particles which are in contact k at points (. . . , cjk, . . .); 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 pointsof Let S be the density of the points marking the centres of particles. The number, m, particles in the volume V may be estimated by
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.
FABRIC IN RANDOM ASSEMBLY OF SPHERICAL GRANULES
79
c. 1j' d . ,: 13
.
C O S T K T IIIXWEE?: PAKl.ICl.ES gi RSU gYIDPOI.:.Il'
Oi: BKtLUCH
gg .. 1 3
j
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 byV, =
Jrr' M
377r'm'"'f(r) d r
(4)
Furthermore, since the total volume V equals (1 + e ) V,, equation (3) can be rewritten to give the following estimate:t
s=
(V)

34 r ( l + e ) 1 r 3 f ( r )d r :;

3
V
4 4 1+ e ) 3
(5 )
where the following notation is used:
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.23The mean number 5 is called the 'coordination number'. Figure 3 shows the relation between 6 and e for random assemblies of spheres; Oda.23Three 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 twomixed assembly consists of spheres with two different radii, and the multimixed 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 crosssection is given in Appendix 1.
80
M. ODA, S NEMATNASSER A N D M. M. MEHRABADI .
0
I r e0.5
1.0
1.5
Figure 3. Experimental relation between the mean coordination number, A, and the void ratio, e, (Oda)
Associated with each contact there are two contact points, one belonging to each contacting 1) particle. Accordingly, the total number, N(v), contacts (not contact points) is Tnm ( V , in of 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 firstorder information for the description of the granular fabric, as is discussed below. 2.3. Distribution and geometrical arrangement of branches The midpoint, dib on a typical branch, 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 midpoints, irrespective of particle size. Accordingly, equation (8) not only gives the density N of contacts but also the density of midbranch points.
a,
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 o contact unit normals, n. At each contact, two unit vectors, n and n, which f
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)
20
(9b)
where d R is an elementary solid angle equal to sin@d a d p (Figure 4(b)), and R is the unit sphere defined by O d a
(13)
2.3.2. Distribution of branch lengths. The branch length 1 may be viewed as a random variable with the range, 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 leng