Exact solutions for stress distribution in rhombic disk packings

Theoretical analysis and DEM simulations are combined to study the stress distribution in rhombic lattice packings of disks with friction. In theoretical study, the governing equations for stress transmission in the rhombic lattice packings are derived based on the hypothesis that the friction angles of different contacts have the same value. A method which is called simultaneous diagonalization is proposed to decouple the governing equations and the method of characteristic curves is used to derive the exact solutions. To investigate Janssen effect and the stress response of disk packings subjected to a point load in theory, the exact solutions are used to calculate the stress distribution in rhombic disk packings with lateral and top walls. The theoretical results show that the stress transmission in lattice packings can exhibit features of traveling wave with sources. The square root of Janssen coefficient plays a role of wave speed and the traveling stress wave will be reflected when it meets a wall and will be partially absorbed due to the friction between the wall and grains. DEM simulations show that our theoretical solutions could describe the main features of stress transmission in rhombic disk packings, but there are some differences between the DEM results and the theoretical results. The boundary conditions, the variation of friction angle with spatial position and the slightly distortion of rhombic structure are viewed as the main reasons for the differences.