## On the stress–force–fabric relationship for granular materials

Li, X. and Yu, H.-S. (2013) On the stress–force–fabric relationship for granular materials. International Journal of Solids and Structures, 50 (9). pp. 1285-1302. ISSN 0020-7683

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## Abstract

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

V

P

c2Vvci

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.

Item Type: Article https://nottingham-repository.worktribe.com/output/714051 University of Nottingham, UK > Faculty of Engineering > Department of Civil Engineering https://doi.org/10.1016/j.ijsolstr.2012.12.023 de Sousa, Mrs Shona 22 Apr 2014 07:52 04 May 2020 16:36 https://eprints.nottingham.ac.uk/id/eprint/2979

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