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On the stress–force–fabric relationship for granular materials

Li, X.; Yu, H.-S.


X. Li

H.-S. Yu


This paper employed the theory of directional statistics to study the stress state of granular materials
from the particle scale. The work was inspired by the stress–force–fabric relationship proposed by Rothenburg
and Bathurst (1989), which represents a fundamental effort to establish analytical macro–micro
relationship in granular mechanics. The micro-structural expression of the stress tensor rij ¼ 1
f c
j ,
where f c
i is the contact force and vci
is the contact vector, was transformed into directional integration by
grouping the terms with respect to their contact normal directions. The directional statistical theory was
then employed to investigate the statistical features of contact vectors and contact forces. By approximating
the directional distributions of contact normal density, mean contact force and mean contact vector
with polynomial expansions in unit direction vector n, the directional dependences were characterized
by the coefficients of the polynomial functions, i.e., the direction tensors. With such approximations,
the directional integration was achieved by means of tensor multiplication, leading to an explicit expression
of the stress tensor in terms of the direction tensors. Following the terminology used in Rothenburg
and Bathurst (1989), the expression was referred to as the stress–force–fabric (SFF) relationship.
Directional statistical analyses were carried out based on the particle-scale information obtained from
discrete element simulations. The result demonstrated a small but isotropic statistical dependence
between contact forces and contact vectors. It has also been shown that the directional distributions of
contact normal density, mean contact forces and mean contact vectors can be approximated sufficiently
by polynomial expansions in direction n up to 2nd, 3rd and 1st ranks, respectively. By incorporating these
observations and revoking the symmetry of the Cauchy stress tensor, the stress–force–fabric relationship
was further simplified, while its capacity of providing nearly identical predictions of the stresses was
maintained. The derived SFF relationship predicts the complete stress information, including the mean
normal stress, the deviatoric stress ratio as well as the principal stress directions.
The main benefits of deriving the stress–force–fabric relationship based on the directional statistical
theory are: (1) the method does not involve space subdivision and does not require a large number of
directional data; (2) the statistical and directional characteristics of particle-scale directional data can
be systematically investigated; (3) the directional integration can be converted into and achieved by tensor
multiplication, an attractive feature to conduct computer program aided analyses.

Journal Article Type Article
Publication Date May 1, 2013
Journal International Journal of Solids and Structures
Print ISSN 0020-7683
Electronic ISSN 0020-7683
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 50
Issue 9
Institution Citation Li, X., & Yu, H. (2013). On the stress–force–fabric relationship for granular materials. International Journal of Solids and Structures, 50(9), doi:10.1016/j.ijsolstr.2012.12.023
Publisher URL
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf


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