Interparticle forces and displacements in granular materials

In a micromechanics framework, the main issue is the relationship between the microscale variables and the macroscale variables. These variables are used to describe either the statics or kinematics of the system. The relationships can be classified in two ways, namely, the “averaging” relationships and the “tracking” relationships. The averaging relationships express the macroscale variable as an averaging of the microscale variables; for example, the stress as a function of contact forces. The “tracking” relationships express the microscale variable as a function of the macroscale variables; for example, the contact force at a given orientation as a function of the stress. Based on fundamental premises, a unique averaging relationship exists for either the statics or the kinematics case. However, it is generally impossible to have a unique expression of the “tracking” relationship because they are generally derived with certain assumptions. In this paper, we will present expressions of the “tracking” based on three different approaches, namely, (1) energy conservation principle, (2) representation theory, and (3) indirect scheme. The assumptions used in each approach are discussed. The results are compared among the three approaches as well as that obtained from the Discrete Element Method (DEM).