A framework for micro–macro transitions in periodic particle aggregates of granular materials Original Research Article

The paper outlines an approach to the modelling of the overall macroscopic response of periodic granular materials based on a numerically evaluated micro-to-macro transition. We consider a homogenized macro-continuum with locally attached microstructure, representing an aggregate of discrete solid granules which possibly come into contact. The micromechanical material response is governed by the interaction of rigid particles based on a Coulomb-type friction law. Central results of the paper are new definitions of homogenized stresses and power-type microheterogeneity conditions for periodic particle aggregates undergoing deformations at finite strains. Here, the critical point is a precise formulation of periodicity conditions for center-displacement fluctuations and rotations of particles associated with a well-defined boundary of the unit cell, inducing the anti-periodicity of fictitious support forces and couples. With these definitions at hand, it is shown that both static and power-type approaches yield the identical symmetric macro-stress tensor. The overall expressions appear in an extremely compact format in terms of a periodicity frame and three support force resultants associated with the three faces of the unit cell. We show that these new representations of overall properties of granular aggregates are identical to those of microstructures for continuous heterogeneous materials, and can formally be obtained in a straightforward manner by a limit of microstresses to microforces on the boundary of the microstructure. On the computational side we outline details of an implicit incremental update algorithm that yields the proposed overall stresses of the particle aggregate in a quasi-static, macro-deformation-driven process. That includes fictitious dynamic relaxation processes based on artificial damping mechanisms in order to overcome the highly complex local instability problems of particle clusters within the quasi-static approach. The performance of the proposed framework is documented by means of a representative set of numerical examples.