Exact solutions for Janssen effect of lattice disk packings

The rhombic lattice packing model proposed here clearly shows the micro causes of the Janssen effect and is solved analytically. Based on the analysis of force equilibrium equations with an exterior product method, a recurrence relation with a force transmission matrix Q is obtained which shows that a contact force will be transmitted and accumulated along the lattice vector direction and then be reflected by lateral walls, and eventually, reach a saturation value if the packing is frictional. A method based on tensor algebra and difference equation is proposed for the diagonalization of Q. Using roundup and remainder functions, we give the exact and approximate expressions which can describe the piecewise smoothness of the pressure function data given by numerical calculations. Numerical results also show the non-uniformity of pressure distribution. These quantitative results have certain deviations with Janssen model except for very small top load or large force reflectivity.

Automated summary

The rhombic lattice packing model proposed here analyzes and solves the micro causes of the Janssen effect. A recurrence relation with a force transmission matrix Q is obtained, showing the transmission and accumulation of contact force along the lattice vector direction. A method for the diagonalization of Q based on tensor algebra and difference equation is proposed. The results reveal deviations from the Janssen model, except for very small top load or large force reflectivity.