Governing equations for stress distribution in rhombic disk packings

The governing equations of stress tensor fields are significant to understand the mechanical behaviors of static granular matter. A multiscale method based on the algebra of translation operators is proposed to derive the governing equations of stress transmission in rhombic lattice packings of disks with friction. Except for the two well-known equations, i.e. the symmetry of stress tensor and the stress equilibrium equation, we also obtain a stress-structure equation as the constitutive relation of the lattice packings. A key variable called friction-geometry tensor is introduced for the expression of the constitutive equation and the Janssen coefficient is given from the eigenvalues of this tensor. DEM (Discrete Element Method) simulations are performed to calculate the numerical values of Janssen coefficient and the results are in good agreement well with theoretical predictions.