Applicability of stress–force–fabric relationship for non-proportional loading

Starting from the micro-structural definition of the stress tensor, Rothenburg and Bathurst (1989) [1] derived a stress–force–fabric relationship for granular materials by approximating the directional distributions of the fabric, more specifically, the contact normal density distribution in this paper and the contact forces with Fourier functions and integrating over directions. This paper aims to assess the validity of the two key assumptions made during their derivation using particle-based numerical simulation in the cases of proportional loading and non-proportional loading. These two assumptions are (i) the 2nd-rank Fourier functions adopted are good enough to approximate the directional distributions of contact normal densities and contact forces and (ii) the principal directions of contact forces and contact normal density are coaxial. Numerical simulations have been carried out to conduct virtual experiments on the behaviour of isotropic specimens to monotonic loading, of isotropic specimens to stress rotation, and of anisotropic specimens to monotonic loading. The first one stands for the case of proportional loading while the latter two are non-proportional loading paths involving rotation of the frame of principal stresses and the frame of fabric, respectively. The directional distributions of contact normal density and contact forces are traced during these three typical loading processes. The simulation results indicate that the 2nd-rank Fourier functions give reasonable approximations, while the coaxial assumption is generally not valid in non-proportional loading. In the case that the principal directions of contact normal density and contact force differ, a more general expression of the stress–force–fabric relationship is required. This research can help to improve our understanding of the stress state and hence shear strength of granular materials based on the particle scale investigation.