Bridging Scale Method Coupling Multiscale DEM-FEM Analysis For Granular Materials

Granular material is composed of a large number of solid particles and void spaces surrounded by the particles and fully filled with fluid phases, such as air, water etc. Granular materials, such as soil, rocks, sand, pharmaceuticals and industrial powders etc., widely exist in the nature and in a variety of engineering practices. Granular material is characterized with highly heterogeneity and complicated mechanical behavior. In the recent years, theoretical studies and numerical simulations for mechanical behavior of granular material have received a great deal of attentions from many international and domestic researchers.Investigating the mechanical behaviors of granular material from the microscale and macroscale views respectively brings on two distinct modeling approaches. In the microscopic scale, a discrete particle assembly model with finite size that characterizes the microstructure of granular material is adopted to the detection of its micro-mechanically-based constitutive behavior. While in the macroscale level, granular material is modeled as an equivalent porous continuum in which the constitutive model is generally constructed in light of the macroscopic phenomenological theory.The microscopic discrete particle assembly model not only effectively describes its microstructure in geometry and material properties, but also provides the realistic kinematic and kinetic analysis for each pair of moving particles in contact. The discrete element method (DEM) based on discrete particle assembly modeling able to efficiently simulate microscopic uncontinuous failure responses of granular material, has been universally acknowledged as an effective way to simulate mechanical responses of granular materials in micro-scale level. However, a huge number of particles exist in the DEM calculation for most practical engineering problem, thus computational time and storage space required by the DEM are always unaffordable.The FEM calculation based on the macroscopic continuum model of granular material has unparalleled advantages in computational efficiency as compared to the DEM. At first, the number of governing equations to be solved in the FEM is much smaller than that required in DEM. Secondly, a time step size used in the FEM calculation can be much larger than the critical time step size required by the DEM calculation. However, the disadvantages of the macroscopic continuum model lie in a number of parameters with no physical meaning are involved and hardly properly determined in the constitutive description of granular materials. Especially, the FEM based on the macroscopic continuum model is unable to simulate microscopic uncontinuous failure phenomena and capture the micromechanical mechanism of granular materials.To combine the advantages of the micro and macro scale models, many efforts have been devoted to develop a multiscale numerical method for analysis of mechanical behavior of granular materials, with highly efficiency and highly accuracy, coupling the two scale models of granular material. In discrete particle assembly modeling rotations of each individual particle are defined as independent variables. It is in accordance with the Cosserat continuum modeling, in which the rotations, in addition to translational displacements, as independent field variables are defined at each mathematical point in the continuum. Hence, it is reasonable and advisable to combine the Cosserat continuum and discrete particle assembly models in macro-micro scale analysis for granular materials. The research contents of the present bridging scale method (BSM) for multiscale analysis of granular materials include:(1) Based on the BSM coupling the Cauchy continuum and molecular dynamics models at both micro-macro scale for nano-materials, a new version of the BSM that couples discrete particle assembly modeling and Cosserat continuum modeling at both micro-macro scale levels respectively for granular materials is developed.(2) The coulping numerical methods that implement the multiscale quasi-static interficial condition and the dynamic non-reflecting interfacial condition between the fine scale region of discrete particle assembly and the coarse scale region of Cosserat continuum are proposed in the present BSM.(3) The BSM for multiscale hydro-mechanical analysis of three phase wetting granular material, coupling the discrete particle assembly model and the Biot-Cosserat continuum model at both micro-macro scale levels is developed.The present BSM applies the DEM only to limited local regions of the whole computational domain for the purpose of accurate simulation of material failure with discontinuous deformation characteristics in microscopic scale, and meantime applies the FEM that costs much less both computational time and storage space to the whole domain. With the coarse and fine scale decomposition of translational and rotational displacements, in light of the principle of virtual work applied to the FEM nodes of Cosserat continuum and the particle centers of discrete particle assembly respectively, two decoupling sets of equations of motion of the combined coarse-fine scale system are derived. According to the decoupling of the fine and coarse scale calculations, it is permitted to use different sizes of time step to the coarse and fine scale time integration schemes respectively. As a consequence, The present BSM not only greatly enhances the computational efficiency of numerical analysis for granular materials, but also is capable of simulating microscopic uncontinuous failure phenomena occurring in a local region of granular material. The numerical results for example problems illustrate the performances and advantages of the present BSM.The simplified and efficient interfacial condition between the coarse and fine regions in the case of quasi-static loading, and the non-reflecting interfacial condition, which is capable of effectively eliminating spurious reflective waves at the interface between coarse and fine regions under dynamic loads are presented and discussed. As dynamic response problems are concerned, according to the distinct difference between both spatial scales attributed respectively to the FE mesh in the MS region and the mesh generated by connecting the particle centers in the DEM region, and the distinct difference between both time step sizes (associated with temporal scales) used for numerical integrations in the temporal domain for the fine and coarse scales, the waves with high frequencies originated from the DEM region, which cannot pass through the interface and cannot travel into the region outside the DEM region in the numerical simulation, return into the DEM region and become spurious reflected waves. Indeed the present nonreflecting interfacial condition for the discrete particle assembly Cosserat continuum modeling is able to eliminate the spurious reflected waves with high frequencies and spurious numerical oscillations efficiently. The multiscale interfacial condition in the present work takes into account the effect of the eliminated (virtual) fine scale degrees of freedom collocated in the coarse scale region near the interface on the absorption of the wave energy. The convolution integral expression for the interfacial impedance forces applied to the DEM peripheral particles with respect to their fine scale displacements is derived and used to compute the interfacial impedance forces at the current time step with the help of recorded fine scale displacement history of the peripheral particles.Based on mass, momentum and moment of momentum conservation laws for the three phases in unsaturated soils, i.e. the solid skeleton, the pore water and the pore air, the governing equations for unsaturated porous Biot-Cosserat continuum model are derived. In light of the passive air phase assumption, along with the reduction of semi-discretized mass conservation equation for pore water, the pore water pressure is further eliminated as the primary unknown from the formulation for numerical solution procedure of unsaturated soils. Instead, the pore water pressure is only taken as the internal state variable defined and evaluated at the quadrature points of the FE mesh. Consequently, a reduced finite element model for unsaturated porous Cosserat continuum, in which the translational and rotational displacements of the solid phase of porous Cosserat continuum are only taken as primary unknows is proposed. According to the decoupling of the fine and coarse scale calculations in the numerical procedure of the bridging scale method, the present reduced finite element model for unsaturated porous continua and the discrete element model for granular media are combined. The BSM that couples the Biot-Cosserat continuum modeling and the discrete particle assembly modeling in both coarse and fine scales respectively is proposed to study the hydro-mechanical coupling problem for three phase wetting granular materials.