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 View the MathML sourceσij=1V∑c∈Vvicfjc, where View the MathML sourcefic is the contact force and View the MathML sourcevic 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 nn, 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 nn 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.

Keywords: Stress–force–fabric (SFF) relationship, Directional statistics, Anisotropy, Multi-scale investigations, Discrete element method (DEM)

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