The loading path dependence of sand constitutive behaviour and its relationship with pressure dip in sand piles

The subject of granular materials attracts substantial interests from both the academic and engineering communities. Their ubiquitous existences in nature and industries certify their engineering importance. Although relatively straightforward at the particle scale, the collective response of granular materials could be very complicated. The pressure dip under a sand pile is one among many intriguing phenomena that granular materials exhibit. It refers to a small dip in pressure appearing underneath the sand pile with the peak pressure observed away from the centre of the sand pile. Since first reported by Hummel and Finnan (1920), this counter-intuitive phenomenon has attracted extensive research endeavours in the past few decades. It is now generally believed that the construction history is crucial to the pressure dip formation, but a quantitative explanation is not yet available. An in-depth explanation in pressure dip formation would range three different scales: the particle-scale, the continuum-scale when the material could be treated as a representative volume element (RVE) and the system scale, in this case, the scale of the sand pile. Discrete element method (DEM) is a great tool in the study of granular materials by modelling it directly from the particle scale. This research sets out to investigate the pressure dip formation of granular materials using the discrete element method. It is however noted that the dimension of the system that DEM can practically simulate is limited. Therefore, the research project is achieved by conducting DEM simulation of sand pile and DEM simulation of elementary behaviour in parallel. Numerical experiments have been carried out using discrete element method to simulate the formation of sand pile in the three-dimension (particle diameter of 1.2 ± 0.4 mm) with the base plane diameter of 260mm , following three different procedures: a) boundary removal method; b) rain deposition and c) point deposition. Pressure dip phenomenon is observed following the point deposition but not the boundary removal method and rain deposition method. The stress distribution of invariants p,q,b are studied within the sand pile at the final stage. The boundary removal method and rain deposition method result in a larger confining pressure and deviatoric stress in the centre than the point deposition method, with b value approximately 0.2 for the three methods. This differences in pressure profiles suggest that the history dependence and the loading path dependence of granular materials play a central role in explaining the pressure dip formation. Discrete element simulations have also been conducted to investigate the RVE stress-strain responses of the granular materials. The numerical algorithm previously proposed by Li et al. (2013; 2016) has been implemented in LIGGGHTS. Numerical experiments have been conducted to study the effect of void ratio and the intermediate principal stress ratio b on the material response to proportional loading, as well as to non-proportional loading using the radius expansion method. It shows the peak and critical stress ratio are both decreased with the increase of b value. With regards to the non-proportional loading path, significant plastic deformation is observed especially with a larger stress ratio. The packing following all the three construction methods are in a loose state. And their initial structures are different from that following the radius expansion method. A sample is extracted from the deposited specimen under the boundary removal method to study the impact of the initial state. A smaller shear stress with a larger void ratio is observed for the deposited sample than the radius expansion prepared sample for the triaxial shear. Furthermore, under the rotation shear, the size of strain trajectory is increased larger for the deposited sample when increases the stress ratio. The “stress-force-fabric” (SFF) relationship is employed to interpret the effect of internal structure on material constitutive behaviour. Due to the higher value of average particle size and larger probability of strong contact force for the deposited sample, it produces a larger average coordination number and directional averaged contact force, accompanied with a smaller degree of fabric anisotropy and contact force anisotropy. This may be the reason of the lower shear resistance during shearing. Post-processing of the particle-level stress data from the sand pile simulations suggests that granular materials form different internal structure and experience different stress history following different construction methods. For the point deposition, a larger magnitude of σrz and a higher degree of fabric anisotropy are observed in the central region at the progressive stages. It implies that the constitutive phenomenon has a relationship with the path of the construction history. In addition, the point deposition test also generates a larger rotation of the major principal stress direction than the boundary removal and rain deposition tests. According to RVE simulations, it suggests the granular materials are under a triaxial compression loading path and a rotational loading path underneath the apex for the rain deposition and point deposition respectively. This increases the probability of forming arches and granular materials may experience a higher plastic deformation underneath the central apex during the point deposition. The particle-scale mechanism has been carried out to study the material behaviour with the loading path dependence of sand pile. Different stress field is observed with various construction histories. It demonstrates the pressure dip phenomenon may relate to the memory of the fabric anisotropy and the potential effect of plastic deformation. Moreover, a better understanding in this problem is believed to improve the problems of silos, retaining wall and embankments. It may be useful for the development of constitutive modelling of the stress field