Multi-scale Modeling Of Discrete Particle Assembly-Cosserat Continuum Model For Granular Materials Based On The Average-field Theory

Granular materials exist widely in nature and engineering practice, such as rock, soil, food supplies and medicament etc. Generally speaking, granular materials are composed of randomly packed discrete particles partially or fully filled with void fluids and are characterized with highly heterogeneity. Theoretical and numerical studies of mechanical behavior of granular materials are challenging tasks in engineering science and attract the interests of the researchers from a variety of disciplines.Granular materials are modeled in the macroscopic view as continuum, which is discretized by means of the finite element method (FEM) (and/or its alternatives such as the mesh-free method) for the numerical simulation of macro-mechanical behaviors in the material domain. The advantage of the continuum approach lies in that the number of primary unknown variables to be determined is rather limited depending on the FE mesh density so that the approach has been widely and effectively applied to engineering problems. However, the disadvantage of the approach is that phenomenological constitutive relations based on the continuum model include many parameters with no physical meaning and difficult to be identified.On the other hand, granular materials are modeled in the microscopic view as a discrete particle assembly, which is consistent with the physical characteristics of granular materials. The numerical simulation of mechanical behaviors for the particle assembly is performed by using the discrete element method (DEM). The advantages of the approach lie not only in the rationality and simplicity of constitutive relations defined at particle contacts so that the realistic constitutive behaviors of granular materials can be effectively described and predicted, but also in the detection of the mechanisms of different types of material failure processes. However, the motions of a huge number of particles (even billions of particles) need to be quantitatively determined in its applications to engineering problems, which is far beyond of the computer capability even nowadays.Many efforts have been devoted to develop the multi-scale methods for granular materials to fully exert the advantages of both macro and micro approaches and to avoid their respective disadvantages. The present work is focused on the multi-scale modeling of discrete particle assembly Cosserat continuum model for granular materials in the frame of average-field theory, including:(1) Derivation of Hill's lemma of the average-field theory for heterogeneous Cosserat continuum, with which proper boundary conditions for the representative volume element (RVE) are presented; (2) Development of the computational homogenization method for granular materials modeled as discrete particle assembly and Cosserat continuum in micro- and macro- scales respectively; (3) Development of a multi-scale constitutive relation of macro Cosserat continuum description for granular materials in light of the micro-directional model proposed for classical Cauchy continuum in the literature.The presentation of Hill's lemma is first required to properly specify RVE boundary conditions for micro-macro homogenization modeling of heterogeneous materials in the frame of average-field theory. On the basis of Hill's lemma for classical Cauchy continuum, a version of Hill's lemma is systematically derived for micro-macro homogenization modeling of heterogeneous Cosserat continuum. According to the derived Hill's lemma, different types of statically admissible stress boundary conditions and/or kinematically admissible displacement boundary conditions to be imposed on the RVE are extracted and discussed. Then proper RVE boundary conditions are determined to satisfy the Hill-Mandel energy condition and fundamental assumptions of the first-order average-field theory. This work provides the foundation of the following studies of micro-macro homogenization methods for granular materials.In light of the computational homogenization approach and the derived formulae of average-field theory for heterogeneous Cosserat continuum, a micro-macro computational homogenization scheme of discrete particle assembly-Cosserat continuum model for granular materials is developed in the present work.DEM is employed for the microscopic analysis and FEM is adopted for the macroscopic computation. The detailed procedure of the developed computational homogenization scheme is given and discussed. 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 variations over the RVE defined for Cosserat continuum equivalent are then determined by the physical quantities of the discrete particle assembly.. The consistent macroscopic modular tensors and the macroscopic constitutive relations defined at the integration point are formulated in terms of the averaged behavior of associated microstructures. As 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. The results of typical numerical examples for granular materials demonstrate the validity of developed method and its advantages in comparison to existing computational homogenization methods of granular materials in the literature.In accordance with the nature of the granular medium in the sense that each discrete particle at the micro-scale possesses rotational degrees of freedom in addition to translational degrees of freedom in kinematics and is capable of bearing and transmitting couples from one particle to the other in contact in kinetics, a micromechanically based constitutive model is developed for macro Cosserat continuum description of granular materials in light of the micro-directional model proposed to describe the constitutive behavior of classical Cauchy continuum. The effects of micro structures/properties on the anisotropy and heterogeneity of macroscopic constitutive behaviors are embodied with the probability density function of the contact distribution. The micro-macro transitions of kinematical quantities are fulfilled by means of Hill-Mandel condition of the average-field theory for heterogeneous Cosserat continuum. To analyze the asymptotic trend and to validate the proposed model, the micromechanically-based expressions of macroscopic elastic constants for the materials under homogeneous and isotropic assumptions are particularly derived. The validity of the developed model is verified through the comparisons of the theoretical predictions given by the derived formulae and numerical results obtained by using the discrete element method on overall behavior of a regularly packed granular assembly. Moreover, the derived formulae also provide theoretical interpretations to the numerical results obtained by the discrete element method.