Laboratory experiments have shown that the proportional shearing of granular materials along arbitrary strain path directions will lead to stress states that converge asymptotically to proportional stress paths with constant stress ratios. The macro- and microscopic characteristics of this asymptotic behaviour, as well as the existence of asymptotic states exhibiting a constant stress ratio and a steady strain-rate direction, have been studied using the discrete element method (DEM). Proportional shearing along a wide range of strain-rate directions and from various initial stress/density states has been conducted. The simulation results suggest that general contractive asymptotic states (except for isotropic states) do exist but may be practically unattainable. Dilative strain path simulations, on the other hand, result in continuously changing stress ratios until static liquefaction occurs, indicating the absence of dilative asymptotic states. Despite this difference, a unique relationship between the stress increments and the current stress ratio gradually emerges from all strain path simulations, regardless of strain path direction and initial stress/density conditions. At the particle scale, the granular assembly sheared along proportional strain paths exhibits a constant partition ratio between strong and weak contacts. Although general proportional strain paths are associated with changing geometric and mechanical anisotropies, the rates of change in these anisotropies for contractive strain paths are synchronised to maintain a constant ratio of their contributions to the mobilised shear strength of the material, with a higher proportion being contributed by geometric anisotropy for more dilative strain paths.