Constitutive relations for the mechanical behavior of granular materials based on a four-particle unit cell

Macroscopic constitutive equations are presented for the two-dimensional flow of granular materials based on the mechanics of a microelement consisting of four rigid particles defining an "unit cell". The overall nominal stress and cauchy stress are related to the forces at each of the four contacts in the unit cell as well as to its microstructural geometric parameters such as the contact unit normals. The concept of classes of unit cells is defined based on the direction of the major axis of the microelement with respect to a set of global coordinates. The microstructure, or fabric which continuously evolves in the course of deformation, is represented as the distribution of the unit cells over the granular sample. A probability density function to describe the evolution of fabric is then introduced. Constitutive equations at the microlevel are also developed, relating the rate of change of the contact forces in the microelement to the local deformation rate produced by changes in the microstructure. The local kinematics is modeled by employing the double shearing theory for granular materials and yield is locally described by a Mohr-Coulomb-type condition written in terms of the local normal and tangential forces at each of the contacts in the microelement. A two-step incremental procedure is used to determine the response of a two-dimensional assembly of unit cells. Using a Taylor averaging scheme, macroscopic rate constitutive equations are developed. The stress-volume change-strain behavior, evolution of fabric, yield and distribution of the contact forces are analyzed by means of numerical simulations. A parametric study is performed to determine the influence of the void ratio and material constants on the behavior of the model. The stress-strain-behavior is in qualitative agreement with previous theoretical and experimental data obtained for the behavior of granular media under monotonic and cyclic loads. The volume change-strain behavior for monotonic and cyclic loading also coincide with those obtained from previous experimental and theoretical studies.