Discrete Granular Systems Research Articles

Exact solutions for the vibration of finite granular beam using discrete and gradient elasticity cosserat models

The present study theoretically investigates the free vibration problem of a discrete granular system. This problem can be considered as a simple model to rigorously study the effects of the microstructure on the dynamic behavior of the equivalent continuum structural model. The model consists of uniform grains confined by discrete elastic interactions, to take into account the lateral granular contributions. This repetitive discrete system can be referred to discrete Cosserat chain or a lattice elastic model with shear interaction. First for the simply supported granular beam resting on Winkler foundations, due to the critical frequencies which concern the nature of the dynamic results, the natural frequencies are exactly calculated, starting from the resolution of the linear difference eigenvalue problem. The natural frequencies of such a granular model are analytically calculated for whatever modes. It is shown that the difference equations governed to the discrete system converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation (also classified as a continuous Cosserat beam model) for an infinite number of grains. A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations. This continuous gradient elasticity Cosserat model is obtained from a polynomial or a rational expansion of the pseudo-differential operators, stemming from the continualization process. Scale effects of the granular chain are captured by the continuous gradient elasticity model. The natural frequencies of the continuous gradient Cosserat models are compared with those of the discrete Cosserat model associated with the granular chain. The results clarify the dependency of the beam dynamic responses to the beam length ratio.

Buckling of granular systems with discrete and gradient elasticity Cosserat continua

The stability of a granular column composed of a finite number of grains is investigated through an exact and some approximated continuum models. Shear and rotational interactions are taken into account at the rigid grain interfaces. This system can be considered as a discrete Cosserat chain with two independent degrees-of-freedom, namely the deflection and the rotation of each grain. The buckling of this discrete granular system on elastic foundation with translational and rotational stiffness (to account for some possible transversal grain interactions) is calculated whatever the number of grains. The formulation of the discrete boundary value problem is based on the exact resolution of a fourth-order linear difference equation. This solution is compared to the one of a continuous Cosserat chain asymptotically obtained for an infinite number of grains. In this last case, the asymptotic solution converges towards the one of a Bresse–Timoshenko beam under Winkler–Pasternak foundation. A more refined Cosserat continuum is built by continualization of the difference equations valid for the discrete Cosserat medium. It is shown that this more refine continuous model can be classified as a gradient elasticity Cosserat continuum, which is able to reproduce the scale effects observed for the buckling of the discrete granular system. These scale effects are related to the grain size, as compared to the structural length of the granular system. The key role played by the shear interaction in the instabilities of granular structural system is revealed, especially when the bending interaction can be neglected.

Multiscale modeling of continuous crushing of granular media: the role of grain microstructure

Natural granular materials such as sands often possess complex microstructural features including cleavage and minerals interfaces. Those features bring apparent mechanical anisotropy to particles and are known to have pronounced influence on particle crushing characteristics. This paper presents a multiscale simulation of continuous crushing of granular sand under one-dimensional compression in consideration of particle-scale anisotropy through modeling planes of weakness inside individual particles, with reference to granular materials rich in minerals and containing cleavages. The multiscale modeling is based on a coupled peridynamics and non-smooth contact dynamics method where peridynamics is used to model crushing of individual particles and non-smooth contact dynamics is employed to simulate discrete granular system. Weak microstructural planes are simulated by breaking a fraction of peridynamic bonds as an initial condition. Simulation results show that anisotropic particles containing weak planes result in larger number of fragments and exhibit relatively higher fractal dimension with respect to particle size. Particle shape is found to approach a steady state profile with continuous crushing. Anisotropic particles generally bear smaller sphericity, aspect ratio, elongation and flatness than those isotropic particles. The anisotropy in particles seems to mitigate shape effect on particle strength and crushing energy. Macromechanical yield stress of the sample is related to single particle strength monotonically, but the relationship appears to be nonlinear when different microstructural features are involved.

A Deep Neural Network Model of Particle Thermal Radiation in Packed Bed

Prediction of particle radiative heat transfer flux is an important task in the large discrete granular systems, such as pebble bed in power plants and industrial fluidized beds. For particle motion and packing, discrete element method (DEM) now is widely accepted as the excellent Lagrangian approach. For thermal radiation, traditional methods focus on calculating the obstructed view factor directly by numerical algorithms. The major challenge for the simulation is that the method is proven to be time-consuming and not feasible to be applied in the practical cases. In this work, we propose an analytical model to calculate macroscopic effective conductivity from particle packing structures Then, we develop a deep neural network (DNN) model used as a predictor of the complex view factor function. The DNN model is trained by a large dataset and the computational speed is greatly improved with good accuracy. It is feasible to perform real-time simulation with DNN model for radiative heat transfer in large pebble bed. The trained model also can be coupled with DEM and used to analyze efficiently the directional radiative conductivity, anisotropic factor and wall effect of the particle thermal radiation.

Stiffness pathologies in discrete granular systems: Bifurcation, neutral equilibrium, and instability in the presence of kinematic constraints

SummaryThe paper develops the stiffness relationship between the movements and forces among a system of discrete interacting grains. The approach is similar to that used in structural analysis, but the stiffness matrix of granular material is inherently nonsymmetric because of the geometrics of particle interactions and of the frictional behavior of the contacts. Internal geometric constraints are imposed by the particles' shapes, in particular, by the surface curvatures of the particles at their points of contact. Moreover, the stiffness relationship is incrementally nonlinear, and even small assemblies require the analysis of multiple stiffness branches, with each branch region being a pointed convex cone in displacement space. These aspects of the particle‐level stiffness relationship give rise to three types of microscale failure: neutral equilibrium, bifurcation and path instability, and instability of equilibrium. These three pathologies are defined in the context of four types of displacement constraints, which can be readily analyzed with certain generalized inverses. That is, instability and nonuniqueness are investigated in the presence of kinematic constraints. Bifurcation paths can be either stable or unstable, as determined with the Hill–Bažant–Petryk criterion. Examples of simple granular systems of three, 16, and 64 disks are analyzed. With each system, multiple contacts were assumed to be at the friction limit. Even with these small systems, microscale failure is expressed in many different forms, with some systems having hundreds of microscale failure modes. The examples suggest that microscale failure is pervasive within granular materials, with particle arrangements being in a nearly continual state of instability.

Granular jet impact: probing the ideal fluid description

AbstractWe investigate the impact of a granular jet on a finite target by means of particle simulations. The resulting hydrodynamic fields are compared with theoretical predictions for the corresponding flow of an incompressible and rotation-free fluid. The degree of coincidence between the field obtained from the discrete granular system and the idealized continuous fluid flow depends on the characteristics of the granular system, such as granularity, packing fraction, inelasticity of collisions, friction and target size. In certain limits we observe a granular–continuum transition under which the geometric and dynamic properties of the particle jet and the fluid jet become almost identical.

On buckling of granular columns with shear interaction: Discrete versus nonlocal approaches

This paper investigates the macroscopic behaviour of an axially loaded discrete granular system from a stability perspective. The granular system comprises uniform grains that are elastically connected with some bending and shear interactions and confined by some elastic supports. This structural system can then be classified as a discrete repetitive system, a lattice elastic model or a Cosserat chain model. It is shown that this Cosserat chain model is exactly tantamount to the finite difference formulation of a shear-deformable Timoshenko column in interaction with a Winkler foundation. The buckling of the discrete column with pinned ends is first analytically investigated through the resolution of a finite difference equation. The solution is compared to a nonlocal approach derived by continualizing the discrete problem. The approximated Timoshenko nonlocal approach appears to be efficient with respect to the reference lattice problem and highlights some specific scale effects. This scale effect is related to the grain size with respect to the total length of the Cosserat chain. Finally, the paper shows the key role played by the shear interaction in the instabilities of granular structural system, especially when the bending interaction can be neglected.

Exner equation: A continuum approximation of a discrete granular system

In contrast to using a standard control volume approach, general statements of sediment mass balance are derived herein from spatial averaging of the sub‐particle‐scale differential equation of solid mass conservation. The general form of the Exner equation for sediment continuity that is obtained enables analyses in terms of size fractions and also in terms of individual or successive layers, where layer interfaces (e.g., for the bed surface and for bed and suspended loads) are defined on the basis of isosurfaces of sediment concentration (volume fraction) or other sediment properties (e.g., densities or transport rates) within regions of constant concentration. The presented expressions highlight the averaged nature of variables and also the effects of the scales of consideration on definition and interpretation of the macroscopic (mixture‐scale) sediment and layer properties (e.g., averaged densities, volume concentrations or fractions, velocities, transport modes and rates, interfaces, and bed layers). For appropriate simplifications, the general form of the Exner equation is shown to reduce to give more specific conventional expressions, revealing the assumptions implicit in these equations.