Discrete Particle Assembly Research Articles

Micromechanically informed constitutive model and anisotropic damage characterization of Cosserat continuum for granular materials

The microstructures of granular materials are represented by Voronoi cells generated with a Voronoi tessellation of discrete particle assembly. A Voronoi cell model including not only the reference particle laid inside the Voronoi cell but also its intermediate neighboring particles is presented to formulate micromechanically based macroscopic constitutive relations and constitutive modular tensors of effective Cosserat continuum. The anisotropy of effective Cosserat continuum due to intrinsic characters and deformation-induced evolutions of microstructure of granular materials of the Voronoi cell is quantitatively demonstrated. The derived micromechanically informed macroscopic constitutive relation of effective Cosserat continuum reveals that the Cauchy stresses are not only constitutively related to the strains but also to the curvatures defined in Cosserat continuum, likewise, the couple stresses are not only constitutively related to the curvatures but also to the strains. The derived modular tensors are verified by comparisons of them with those given for classical isotropic Cosserat continuum and are used to identify the elastic constitutive parameters of isotropic Cosserat continuum. The micromechanically informed macroscopic damage factor tensor to characterize anisotropic material damage of effective Cosserat continuum is formulated with no need specifying macroscopic phenomenological damage criterion and damage evolution law. The principal directions and values of the derived damage factor tensor along with numerical results reveal the microscopic mechanisms of macroscopic damage phenomenon, i.e. loss of contacts, re-orientation of contacts of the reference particle with its intermediate neighboring particles and concomitant volumetirc dilatation of the Voronoi cell.

Microscopic modelling of the hydraulic fracturing process

A microscopic perspective is introduced in this study which offers a detailed insight at the inter-particle level to the geo-mechanical responses caused by fluid injection and the resulting pressure build-up. This was achieved by employing the Discrete Element Method (DEM) to model the pressure development and the subsequent fracturing and/or cavity propagation. This technique represents the formation material as an assembly of discrete particles linked to each other through contacts. Numerical experiments were carried out on two sample materials. For the first instance, tests were carried out on a bulk material, representative of a generic intact rock, with the breakage of inter-particle bonds indicating the formation of cracks. The second series of tests was carried out on granular type materials such as sand, where particle separation signified cavity initiation and separation. It was observed from the DEM modelling results that the intact rock material showed a predominance of mode II fracturing at high fluid velocities. However, when the fluid velocity is reduced considerably the fracturing behaviour tended towards more of mode I. Also, records of the pressure development were taken from the numerical results and were used to monitor the fracturing events. The outcome of this study highlights important aspects of the hydraulic fracturing process especially at the particle–particle scale, and thus provides a strong basis for more exhaustive studies involving larger scale reservoir modelling and more complex fracturing scenarios.

A bridging scale method for granular materials with discrete particle assembly – Cosserat continuum modeling

Based on the bridging scale method (BSM) initially presented in nano mechanics [1,2], a new version of the BSM that couples discrete particle assembly modeling using the discrete element method (DEM) and Cosserat continuum modeling using the finite element method (FEM) at both micro–macro scale levels respectively is proposed for multiscale analysis of granular materials. The DEM is only applied to the limited localized regions for accurately simulating discontinuous failure phenomena in microscopic scale, meantime, the FEM that costs much less computational time and storage space covers the whole domain, with permitting to adopt different time step sizes for the time integration schemes in coarse and fine scales respectively. As a consequence, both computational accuracy and efficiency of the proposed BSM are greatly enhanced. With a bridging scale displacement (including translations and rotations) decomposition and based on the virtual work principle applied to the FEM nodes of the Cosserat continuum and the particle centers of the discrete particle assembly respectively, two decoupling sets of equations of motion of the combined coarse–fine scale system are formulated. The interfacial condition between the coarse and fine scale regions in the quasi-static and dynamic loading cases are presented and discussed. The non-reflecting boundary condition and its implementation, which is capable of effectively eliminating spurious reflected waves at the interfaces between the coarse and fine scale regions, are described in detail. The numerical results for 2D example problems demonstrate the applicability and advantages of the present BSM, and the performances of the non-reflecting boundary condition implemented to simulate dynamical responses in geo-structures composed of granular materials.

A generalized Hill’s lemma and micromechanically based macroscopic constitutive model for heterogeneous granular materials

Based on the Hill’s lemma for classical Cauchy continuum, a generalized Hill’s lemma for micro–macro homogenization modeling of gradient-enhanced Cosserat continuum is presented in the frame of the average-field theory. In this context not only the strain and stress tensors defined in classical Cosserat continuum but also their gradients are attributed to assigned micro-structural representative volume element (RVE), that leads to a higher-order macroscopic Cosserat continuum modeling and enables to incorporate the micro-structural size effects. The enhanced Hill–Mandel condition for gradient-enhanced Cosserat continuum is extracted as a corollary of the presented generalized Hill’s lemma. The derived admissible boundary conditions for the modeling are deduced to direct the proper presentation of boundary conditions to be prescribed on the RVE in order to ensure the satisfaction of the Hill–Mandel energy condition.With the link between the discrete particle assembly and its effective Cosserat continuum 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 micromechanically based macroscopic constitutive model and corresponding rate forms of the macroscopic stress–strain relations taking into account the local microstructure and its evolution are formulated with neither need of specifying the macroscopic constitutive relation nor need of providing macroscopic material parameters.

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.

Mean steady granular force on a wall overflowed by free-surface gravity-driven dense flows

We studied free-surface gravity-driven recirculating flows of cohesionless granular materials down a rough inclined plane and overflowing a wall normal to the incoming flow and to the bottom. We performed two-dimensional spherical particle discrete element simulations using a linear damped spring law between particles with a Coulomb failure criterion. High-frequency force fluctuations were observed. This paper focuses on the mean steady force exerted by the flow on the obstacle versus the macroscopic inertial number of the incoming flow, where the inertial number measures the ratio between a macroscopic deformation time scale and an inertial time scale. A triangular stagnant zone is formed upstream of the obstacle and sharply increases the mean force at low incoming inertial numbers. A simple hydrodynamic model based on depth-averaged momentum conservation is proposed. This analytical model predicts the numerical data fairly well and allows us to quantify the different contributions to the mean force on the wall. Beyond this model, our study provides an example of the ability of simple hydrodynamic approaches to describe the macroscopic behavior of an assembly of discrete particles not only in terms of kinematics but also in terms of forces.

Simulation Study on Effects of Hematocrit on Blood Flow Properties Using Particle Method

The effects of hematocrit (Hct) on blood flow in microcirculation were investigated by computer simulation using a particle method. Deformable red blood cells (RBCs) and blood plasma were modeled by assembly of discrete particles. It was assumed that an RBC consisted of an elastic membrane and inner viscous fluid, and that plasma was viscous fluid. The particles for the RBC membrane were connected with their neighboring membrane particles by stretch/compression and bending springs. The motion of all the particles that was subjected to incompressible viscous flow was solved by the moving particle semi-implicit (MPS) method based on Navier-Stokes (NS) equations. The forces induced by the springs to express the elastic RBC membrane were substituted into the NS equations as the external force, which enabled coupled analysis of elastic RBC motion and plasma fluid flow. Two-dimensional simulations of blood flow between parallel plates were carried out for various Hct values. As a result, it was shown that at higher Hct, RBCs were less deformed into a parachute shape during their downstream motion, indicating that mechanical interaction between RBCs restricted the RBC deformation. Mechanical interaction between RBCs had a significant influence on RBC deformation and the velocity profile of plasma flow when the Hct value was more than 0.20∼0.30. Apparent blood flow resistance increased with Hct, corresponding to previously reported in vitro experimental results compiled to an empirical formula.

On the application of a discrete model to the fracture process of cohesive granular materials

We present a two-dimensional model of heterogeneous cohesive frictional solids where the material structure is idealized by a discrete granular particle assembly. Our discrete model is composed of convex polygons which are linked together by simple beams accounting for cohesive effects. Varying its parameters the model naturally interpolates between a continuous solid state and a discontinuous dry granular state of the material. We performed simulations of the quasi-static uniaxial loading with two different boundary conditions, and shearing of a solid. To provide a geometrical description of the damage state, and of the microscopic fracture events, different characteristic quantities will be introduced and their relevance will be examined. The results obtained from simulations are in satisfactory agreement with experimental observations.

APPLICATION OF THE EXTENDED DISTINCT ELEMENT METHOD FOR COLLAPSE SIMULATION OF A DOUBLEDECK BRIDGE

The Distinct Element Method (DEM), in which soil is represented as an assembly of numerous discrete particles, does not account for the continuity of the medium. The extended method has another physical structure which represents the effect of the internal material between the particles as mortar of concrete. We simulated the two-dimensional dynamic fracture process of the Cypress Double-Deck Bridge on Interstate 880-Line suffered by the 1989 Loma Prieta earthquake using this method. Results agree well with the real damage qualitatively and confirmed that this method can simulate the collapse of structures involving discontinuous behavior.