Micromechanics of granular media Part I: Generation of overall constitutive equation for assemblies of circular disks

Many problems of special interest to engineers and earth scientists involve characterization of the overall engineering behavior of granular media. Granular materials respond quite differently from polycrystalline materials such as metals due to their particulate nature and random microstructure. Since the particle-to-particle interaction is frictional in nature, contacts in granular media can easily form or break as the particles move towards their new equilibrium positions. From a modeling standpoint, the prediction of the new microstructure of a granular material is a major source of numerical difficulty. Consequently, the effects of microstructure on the overall response of granular media remain hardly understood. This work attempts to capture the effects of microstructural changes on the overall response of granular media. A mathematical model is presented in which the response of a macroscopic point is derived from the overall response of the particle assembly. The conventional hypothesis in computational plasticity is employed, i.e. the overall stress response of the assembly is determined from the micromechanical responses of the constituent granules to a given overall displacement gradient. An important feature of the present formulation is the quasi-static nature in which the micromechanical responses are computed. A quasi-static approach is more appropriate for a wide class of engineering applications where inertia effects are negligible. To assess the performance of the model, numerical results obtained from two-dimensional plane-strain simulations with regular and random initial packings of circular disks are presented.