A micro–macro homogenization approach for discrete particle assembly – Cosserat continuum modeling of granular materials

Based on Hill’s lemma for classical Cauchy continuum, a version of Hill’s lemma for micro–macro homogenization modeling of Cosserat continuum is derived. The derived admissible boundary conditions for the modeling are used to direct the proper prescription of boundary conditions on the representative volume element (RVE) in order to ensure the satisfaction of the Hill–Mandel energy condition. A micro–macro computational homogenization procedure for modeling the mechanical response of granular material modeled as heterogeneous Cosserat continuum is presented, taking into account both physical and geometrical non-linear evolution of microstructures of the material. With the link between the discrete particle assembly and its Cosserat continuum equivalent in an individual RVE, the boundary conditions prescribed on the RVE modeled as Cosserat continuum are transformed into those prescribed to the peripheral particles of the RVE modeled as the discrete particle assembly. The average stresses and strains and their rates over the RVE defined for the Cosserat continuum equivalent are then determined by the physical and geometrical quantities of the discrete particle assembly. The consistent macroscopic modulus tensor and the macroscopic constitutive relation defined at the integration point are formulated in terms of the averaged behavior of the associated microstructures. Since the proposed micro–macro computational homogenization procedure is used within the finite element framework, there is no need to specify the macroscopic constitutive relation at the macroscopic integration points.