Local stress-geometry equation of 2D frictionless granular systems

For a static granular system, the constitutive equation of its stress tensor is of great significance for understanding its mechanical behaviors. Under isostatic state, it can have the form of stress-geometry equation. To investigate the force moment tensor and the stress-geometry equation of a two-dimensional (2D) granular system in theory, we propose some algebraic theories such as the decomposition formula of a second-order tensor and the cross-product of two symmetric tensors for the dyadic space $T^2\left(R^2\right)$ . For a 2D frictionless disk packing, the local stress-geometry equation for a disk with three or four contacts is derived based on the definition of force moments tensor and the equilibrium equation of contact forces. The definition of the geometry tensor in the stress-geometry equation shows complex associations between the contact branch vectors of a disk with three or four contacts. For a disk with four contacts, its local Janssen coefficient can be given from the eigenvalues of its geometry tensor. Discrete element method (DEM) simulations for random frictionless disk packings are performed to verify two local stress-geometry equations in this paper, and the numerical results are in good agreement with the theoretical predictions. The local stress-geometry equations are convenient for obtaining some information about the stress tensors according to the contact structures without knowing the details of the deformations and the intergranular interactions.