A toroidal approximation of capillary forces in polydisperse granular assemblies

In this paper we propose a numerical procedure for the prediction of capillary forces in polydisperse granular assemblies at a degree of moisture content that corresponds to the so-called pendular regime. The capillary force model is adopted within the Laplace–Young framework with a toroidal approximation of the liquid bridge geometry. Governing equations are first derived in a form that highlights the role of intrinsic parameters such as inter-particle separation distance, ratio of particle radii and liquid volume. A proper scaling of these equations is adopted so that the solution applies to any particle pair configuration. Numerical integration algorithms are provided in a way that facilitates implementations in macroscopic procedures for computer simulations. A qualitative evaluation is undertaken to highlight model predictions of the effects on capillary force of various intrinsic parameters that characterise the particle pair and liquid bridge. The model is validated against the experimental results provided by Willet et al. (Langmuir 16:9396–9405, 2000) for a wide range of liquid volumes and particle-pair polydispersity.     

Evaluation of energy contributions using inter-particle forces in granular materials under impact loading

A drop-tower based experimental setup was developed for the impact testing of 2D assembly of cylinders with impactor velocity of around 6 m/s. This drop tower setup was used to load 2D granular assemblies of polyurethane and polycarbonate cylinders of 1\(^{\prime \prime }\)–1.25\(^{\prime \prime }\) length with three different diameters of 1/4\(^{\prime \prime }\), 3/8\(^{\prime \prime }\) and 1/2\(^{\prime \prime }\). A high speed camera was used for recording the images at speeds between 10,000 and 55,000 fps to monitor the deformation of the cylinders. Kinematic and strain fields in individual grains during each experiment were measured using digital image correlation. These experimentally measured strain and kinematic fields were used as inputs for the granular element method (GEM) based force inference technique and the inter-particle forces in normal and tangential direction were determined at every contact in each experiment. The inter-particle forces at each contact can facilitate the calculation of frictional work done at each contact. The GEM based inter-particle forces for a simple 2 particle granular assembly were found to be in good agreement with predictions from ABAQUS explicit based FEM simulation. The influence of different model parameters was also characterized such as grain stiffness, frictional co-efficient was investigated qualitatively. The impact response of the various ordered granular assemblies was also investigated using the GEM approach and the effect of local defects such as voids or layering of granular materials on the wave propagation phenomena is also studied. The presence of the point or line defects have significant effects on the wave propagation in the granular assemblies due to wave scattering and attenuation.     

Enhanced continua and discrete lattices for modelling granular assemblies

This article discusses the derivation of continuum models that can be used for modelling the inhomogeneous mechanical behaviour of granular assemblies. These so-called kinematically enhanced models are of the strain-gradient type and of the strain-gradient micro-polar type, and are derived by means of homogenizing the micro-structural interactions between discrete particles. By analysis of the body wave dispersion curves, the enhanced continuum models are compared to corresponding discrete lattice models. Accordingly, it can be examined up to which deformation level the continuum models are able to accurately describe the discrete particle behaviour. Further, the boundary conditions for the enhanced continuum models are formulated, and their stability is considered. It is demonstrated how to use the body wave dispersion relations for the assessment of stability.     

Computer simulated deformation of compact granular assemblies

Summary The paper is concerned with the mechanics of granular material under quasi-equilibrium conditions. The characterisation of the microstructure of granular material is discussed and the relationship between the microstructure and the stress tensor is examined.Results from computer simulated experiments on a large, initially random, dense assembly of different sized discs are used to investigate the evolution of both stress and structure. Two tests are reported: one a constant mean stress test and the other a constant volume test. It is shown that, although the two tests were subjected to different strain histories the degree of induced structural anisotropy evolved in an identical manner.The computer simulated tests show that there also exists a statical anisotropy due to the biased distribution of contact forces within an assembly. This also evolves in the same way for both tests reported. Consequently, the evolution of the angle of internal shearing resistance with strain is found to be unique.With 9 Figures     

Least squares methods for the mechanics of nonhomogeneous granular assemblies

Summary Using least squares methods consistent averages for stress and strain of a nonhomogeneous granular assembly are derived. The expression for the average of the strain increment serves as a constraint in a functional approach to solving the displacements and spins of particles of a granular assembly in a statistical manner. The method shows directly what microscopic angular distributions are needed to describe the internal state of an assembly. They are the distribution of contacts, the distribution of both the averages and the variations in the interactive properties of the grains. The method is then applied to an assembly with normal-interacting circular grains in two dimensions. The incremental stiffness and the average displacement of the contacts as a function of the angle are obtained for the case of non-rotating strains. The results show clearly that for this case the sliding mode of motion is the most relevant one for deviatoric loading.     

Granular element method (GEM): linking inter-particle forces with macroscopic loading

We present a new method capable of inferring, for the first time, inter-particle contact forces in irregularly-shaped natural granular materials (e.g., sands), using basic Newtonian mechanics and balance of linear momentum at the particle level. The method furnishes a relationship between inter-particle forces and corresponding average particle stresses, which can be inferred, for instance, from measurements of average particle strains emanating from advanced experimental techniques (e.g., 3D X-ray diffraction). Inter-particle forces are the missing link in understanding how forces are transmitted in complex granular structures and the key to developing physics-based constitutive models. We present two numerical examples to verify the method and showcase its promise.     

On the influence of inter-particle friction and dilatancy in granular materials: a numerical analysis

Mechanical behavior of granular soils is a classic research realm but still yet not completely understood as it can be influenced by a large number of factors, including confining pressure, soil density, loading conditions, and anisotropy of soil etc. Traditionally granular materials are macroscopically regarded as continua and their particulate and discrete nature has not been thoroughly considered although many researches indicate the macro mechanical behavior closely depends on the micro-scale characteristics of particles. This paper presents a DEM (discrete element method)-based micromechanical investigation of inter-particle friction effects on the behavior of granular materials. In this study, biaxial DEM simulations are carried out under both ‘drained’ and ‘undrained’ (constant volume) conditions. The numerical experiments employ samples having similar initial isotropic fabric and density, and the same confining pressure, but with different inter-particle friction coefficient. Test results show that the inter-particle friction has a substantial effect on the stress-strain curve, peak strength and dilatancy characteristics of the granular assembly. Clearly, it is noted that apart from the inter-particle friction, the shear resistance is also contributed to the dilation and the particle packing and arrangements. The corresponding microstructure evolutions and variations in contact properties in the particulate level are also elaborated, to interpret the origin of the different macro-scale response due to variations in the inter-particle friction.     

Tolerance analysis and design of nesting forces for exactly constrained mechanical assemblies

In this research we investigate the analysis and design of nesting forces for exactly constrained mechanical assemblies. Exactly constrained assemblies have a number of important advantages over other assemblies including the ability to assemble over a wide range of conditions. Such designs often require nesting forces to keep the design properly seated. To date, little theory has been developed for the analysis and design of nesting forces. We show how the effects of tolerances on nesting forces, a key issue, can be analyzed and then apply the analysis to two example problems. Good agreement is obtained between the method and Monte Carlo simulation.     

Geometrical derivation of frictional forces for granular media under slow shearing

We present an alternative way to determine the frictional forces at the contact between two particles. This alternative approach has its motivation in a detailed analysis of the bounds on the time integration step in the discrete element method for simulating collisions and shearing of granular assemblies. We show that, in standard numerical schemes, the upper limit for the time integration step, usually taken from the average time t _c of one contact, is in fact not sufficiently small to guarantee numerical convergence of the system during relaxation. In particular, we study in detail how the kinetic energy decays during the relaxation stage and compute the correct upper limits for the time integration step, which are significantly smaller than the ones commonly used. In addition, we introduce an alternative approach based on simple relations to compute the frictional forces that converges even for time integration steps above the upper limit.     

Effect of interfacial contact forces in single layer cable assemblies

The mechanical behaviour of cables and ropes used in engineering applications, though studied for the past three decades, varies widely depending on the numerical models adopted. Though these models predict the global response reasonably well, they differ widely in modelling the local contact conditions, the frictional effects at the interfaces and predicting the loss of stiffness of the cable assemblies. Three modes of contact can exist among the wires in a stranded cable assembly, i.e. the contact among the wires in the same layer (known as hoop or lateral contact), the contact among the wires in adjoining layers (radial contact) and the combined contact of all the wires (combined lateral and radial contact). The cables are hitherto modelled on the assumption of the presence of one of the contact modes only, though in reality the contact of modes change from one to the other, depending on the loading and the nature of contraction of the wires. The behaviour of the cable can be well understood if the appropriate mode of contact prevalent at every stage of loading is adopted in the model. This paper analyses the contact modes present in a single layer cable assembly and considers its response under an axial tensile load and an axial twisting moment. Based on the initial geometry, the contact mode is determined and depending on successive loading, the contraction of all the wires in the radial and lateral directions are ascertained and the threshold limits at which the contact modes change from one to the other are established. The overall response of the cables under the cascading effects of the presence of different contact modes, is compared with the works of the other authors who have adopted one type of contact mode only during their study. This has resulted in an overall reduction in the stiffness of the cable assembly, compared to the existing models. The force and moments in the individual wires are studied and the contact forces and the resulting contact stresses are established as a function of applied loads. The effect of the friction and the associated slip of the wires have been included. Apart from consideration of the radial contraction of the wires due to the Poisson effect, as accounted by few authors, this paper considers the radial deformation due to contact forces, as a special feature. This has resulted in refined expressions for the curvatures and twist of the wire and the associated forces in the normal and binormal directions. The predictions with these inclusions are compared with the existing works and the importance of the refinements to the cable designers are highlighted.     

An algorithm to generate random dense arrangements for discrete element simulations of granular assemblies

A discrete element simulation of a mechanical problem involving granular materials begins with the definition of the geometry of the sample to be analyzed. Since the dynamic sample preparation methods typically used in the practice are very time-consuming, constructive algorithms are becoming increasingly popular. This paper introduces a novel constructive method for the preparation of random, isotropic assemblies of contacting circular discs with a user-defined grain size distribution. The proposed approach is compared with other currently applied sample preparation methods.     

Computer simulation of evolving capillary bridges in granular media

A numerical method for the simulation of spatially evolving liquid–vapour interfaces in arbitrary two dimensional granular media is presented. Solid- and liquid-phase objects are described by polynomials whose edges evolve according to surface tension forces until a prescribed equilibrium contact angle at three-phase contact points and a constant mean curvature on two-phase contact lines is achieved. The main advantage of the method is the possibility to account for topological transitions (interface coalescence or rupture) and direct calculation of the force acting on solid interfaces due to liquid bridges. The method has been validated by comparing numerical and analytical results for a single pendular liquid bridge and then demonstrated on the simulation of transition from the pendular to funicular and capillary state in a wet particle assembly.     

Quantitative evaluation of the effect of irregularly shaped particles in sheared granular assemblies

Assemblies of irregularly shaped particles exhibit higher shear strengths than assemblies of circular particles. We performed a series of 2D discrete element simulations to demonstrate that this particle shape effect is related to the induced moment and the additional dilation at the contacts between particles. We proposed a mechanically based particle shape index that is closely related to such contact behavior. A simple structural model is also investigated to clarify the micromechanical role of the particle shape.     

Homogenization modeling of capillary forces in selective laser sintering

We study the initial stage of liquid phase sintering when the particles rearrangement due to densification forces is over. The heterogeneous medium is a mixture of powders, connected by liquid necks. It gives a fluid-structure problem built on a periodic domain, with three phases: an isotropic elastic solid, a newtonian, incompressible, weakly viscous liquid and a barotropic viscous gas. We study this equilibrium under the action of capillary forces on solid-liquid boundary. We penalize the divergence free condition in the liquid and by the energy method, we obtain the homogenized problem satisfied by the macroscopic displacement field in the homogenized medium. The result gets close to a Stokes problem where the capillary forces act as a density of body forces and the viscosity effects vanish.     

Exponential distribution of force chain lengths: a useful statistic that characterizes granular assemblies

The distribution of the lengths of force chains in 2D granular assemblies of photoelastic disks was found to decay exponentially, with the decay length a quantitative measure of the way force is applied to the system. A plausibility argument is provided for why this statistic displays an exponential decay.