From Particle to Material Behavior: the Paths Chartered by Jean Biarez - Multiscale Geomechanics: From Soil to Engineering Projects - page 17

2.3. Analysis of geometric anisotropy

Biarez and Wiendieck were the first to use a representation of the anisotropy of
distribution of the orientation of tangent planes between particles in contact
(Figure 2.3). These orientations can be measured in the two-dimensional projection
of a model material composed, for example, of parallel cylinders with circular
sections (Schneebeli rolls) or any other shape (as shown in [BIA 62]). Biarez and
Wiendieck [BIA 63, WIE 64] suggested an approximation of the shape of the
distribution obtained by an ellipse of major axis a and minor axis b. The ratio
A = a − b

a + b gives an overall indication of the value of the geometric anisotropy of the
sample (or texture ), which, combined with the value of the void ratio e , provides a
better description of the material.

Geometric anisotropy

ORIENTATION OF TANGENT PLANES

ANISOTROPY DUE TO
GRAVITY

SEDIMENTATION

Figure 2.3. Definition of geometric anisotropy from the
direction of the tangent planes [BIA 63]

This idea was subsequently taken up by Oda [ODA 72] and adapted to the
distribution of the orientations of normal directions at contact points. This led, in the
1980s, to the definition of several fabric (or texture) tensors, among them those
defined by Satake [SAT 82], Oda, Konishi and Nemat-Nasser [ODA 80], from the
tensorial product of the normal n by itself. The elliptical approximation of the
distribution has also been replaced by an approximation in Fourier series and the
geometric anisotropy can be described by using a second order tensor, for example,
defined by the relation:

H = n ⊗ n C [2.1]

where C is the set of contacts on which the tensorial product is computed.