Elastic Behavior of 2D Grain Packing Modeled as Micromorphic Media Based on Granular Micromechanics

Abstract

Discrete simulation and continuum modeling are two competing methods for analyzing the behavior of complex material systems and granular assemblies. In this paper, a two-dimensional (2D) granular micromechanics model for micromorphic media is presented. The granular micromechanics method is enhanced by incorporating nonclassical terms in the kinematics. These include, in addition to the usual macroscale displacement gradient, the fluctuations in displacement gradient and also the second gradient of displacement. Microscopic intergranular force vectors and macroscopic stress tensors conjugate to the kinematic measures are defined, and the macroscopic strain energy density function is set equal to the volume average of grain-pair energies. As a result, continuum stiffness tensors are formulated on the basis of intergranular stiffness coefficients and fabric parameters defining the geometry of grains and their contacts. To demonstrate the applicability of the continuum method, a random granular assembly is analyzed with both the developed model and discrete modeling method. Results of the discrete analysis then are used to identify the macroscale (or continuum) and the microscale (or grain-pair) stiffness coefficients for a randomly generated assembly of disks. Effects of the three kinematic fields, macrodisplacement gradient, fluctuations in displacement gradient, and its second gradient, are compared. It is found that the derived grain-scale stiffness coefficients are not equal to the ones used in the discrete simulation. This implies that the intergranular stiffness coefficients used for continuum modeling do not represent stiffness of an isolated grain-pair, rather they represent a collective behavior from the grain-pair and its surroundings. An advantage of the granular micromechanics method is that the grains locations and their contacts are not needed. Conversely, for discrete modeling, one needs exact data about grains’ positions and contacts in addition to their contact properties, which can be very complicated, perhaps intractable, for real materials and complex grain assemblies.