Linear Contact Model Research Articles

Establishing stable time‐steps for DEM simulations of non‐collinear planar collisions with linear contact laws

SummaryThe discrete element method, developed by Cundall and Strack, typically uses some variations of the central difference numerical integration scheme. However, like all explicit schemes, the scheme is only conditionally stable, with the stability determined by the size of the time‐step. The current methods for estimating appropriate discrete element method time‐steps are based on many assumptions; therefore, large factors of safety are usually applied to the time‐step to ensure stability, which substantially increases the computational cost of a simulation. This work introduces a general framework for estimating critical time‐steps for any planar rigid body subject to linear damping and forcing. A numerical investigation of how system damping, coupled with non‐collinear impact, affects the critical time‐step is also presented. It is shown that the critical time‐step is proportional to if a linear contact model is adopted, where m and k represent mass and stiffness, respectively. The term which multiplies this factor is a function of known physical parameters of the system. The stability of a system is independent of the initial conditions. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.

Structural soil crust development from raindrop impacts using two-dimensional discrete element method

The mechanical nature of crust formation as a result of raindrop impacts was simulated within a discrete element modeling environment. Simulations were conducted in two-dimensions (2D) using both linear and non-linear elastic contact models. The 2D approach was found to minimize the computational effort required and maximize the number of particles in the soil profile. For the non-linear model, the effect of the coefficient of restitution (COR) for soil-rain and soil-soil was investigated. Finally, the comparison between the linear and nonlinear elastic contact model was presented. The simulation indicated that the COR for rain-soil had negligible effect on the crust development but the computational time was exponentially increased with increasing coefficient value. In contrast, the COR for soil-soil had a dominant influence on the crust development. To validate the numerical results, a micro computerized tomography (microCT) technique was applied to characterize the changes in pore structure to a USCS SP soil after exposure under a rainfall simulator. Additionally, the effect of cyclic wetting and drying (without rainfall) on the changes in porosity was investigated. The experimental results showed that the rainfall simulator sufficiently densified the soil but the effect of cyclic wetting and drying was negligible. The numerical simulations showed similar changes in porosity along the depth of the soil profile as compared with the experimental results thus validating the DEM technique to simulate crust development.

Rough Surface Contact of Curved Conformal Surfaces: An Application to Rotor–Stator Rub

Rotating machines and associated triboelements are ubiquitous in industrial society, playing a central role in power generation, transportation, and manufacturing. Unfortunately, these systems are susceptible to undesirable contact (i.e., rub) between the rotor and stator, which is both costly and dangerous. These adverse effects can be alleviated by properly applying accurate real-time diagnostics. The first step toward accurate diagnostics is developing rotor–stator rub models which appropriately emulate reality. Previous rotor–stator rub models disavow the contact physics by reducing the problem to a single esoteric linear contact stiffness occurring only at the point of maximum rotor radial deflection. Further, the contact stiffness is typically chosen arbitrarily, and as such provides no additional insight into the contacting surfaces. Here, a novel rotor–stator rub model is developed by treating the strongly conformal curved surfaces according to their actual nature: a collection of stochastically distributed asperities. Such an approach is advantageous in that it relies on real surface measurements to quantify the contact force rather than a heuristic choice of linear contact stiffness. Specifically, the elastoplastic Jackson–Green (JG) rough surface contact model is used to obtain the quasistatic contact force versus rotor radial deflection; differences and similarities in contact force between the linear elastic contact model (LECM) and JG model are discussed. Furthermore, the linear elastic model's point contact assumption is assessed and found to be inaccurate for systems with small clearances. Finally, to aid in computational efficiency in future rotordynamic simulation, a simple exponential curve fit is proposed to approximate the JG force–displacement relationship.

The influence of geometrical and rheological non-linearity on the calculation of rubber friction

We discuss the influence of geometrical and rheological non-linearities on the prediction of rubber friction and true contact area for rough sliding interactions. In particular, we compare the results of a linearly-viscoelastic linear-contact model, formulated in the Fourier space, with those obtained from non-linear finite element calculations. A sinusoidal rigid profile indenting a rubber block is here considered for simplicity, whereas the effects of non-linearity are evaluated by varying the aspect ratio, loading conditions and sliding speed of the contact interface. It is found that accurate friction predictions can be obtained through the linear viscoelastic model, provided that the roughness under investigation features moderate values of root mean square slopes, whereas non-linear finite element computations should be adopted for large root mean square slopes.

Integration of geometric variation and part deformation into variation propagation of 3-D assemblies

This paper introduces a novel modelling method for variation propagation calculation of 3-D assemblies taking into account geometric variation and part deformation, which are neglected in most models in tolerance analysis. Initially, numerical studies are carried out in order to illustrate the characteristics of strain distribution in components and contact forces on the mating surfaces of a 3-D assembly. According to these characteristics, a linear equivalent model using springs to represent the elastic mating surfaces with geometric variation was presented. Then, the equilibrium criterions corresponding to actual contact situations and iterative searching algorithm of the equilibrium status of contacting were developed. The proposed modelling and calculation method were finally applied to the assembly of two machined parts, on which finite-element analyses and experimental tests were conducted to validate the effectiveness and accuracy. This linear contact model also shows an important advantage on modelling and calculating efficiency, which enable the practical application to variation propagation calculation in both tolerance design and assembly process.

Dynamic Analysis of the High Speed Train and Slab Track Nonlinear Coupling System with the Cross Iteration Algorithm

A model for dynamic analysis of the vehicle-track nonlinear coupling system is established by the finite element method. The whole system is divided into two subsystems: the vehicle subsystem and the track subsystem. Coupling of the two subsystems is achieved by equilibrium conditions for wheel-to-rail nonlinear contact forces and geometrical compatibility conditions. To solve the nonlinear dynamics equations for the vehicle-track coupling system, a cross iteration algorithm and a relaxation technique are presented. Examples of vibration analysis of the vehicle and slab track coupling system induced by China’s high speed train CRH3 are given. In the computation, the influences of linear and nonlinear wheel-to-rail contact models and different train speeds are considered. It is found that the cross iteration algorithm and the relaxation technique have the following advantages: simple programming; fast convergence; shorter computation time; and greater accuracy. The analyzed dynamic responses for the vehicle and the track with the wheel-to-rail linear contact model are greater than those with the wheel-to-rail nonlinear contact model, where the increasing range of the displacement and the acceleration is about 10%, and the increasing range of the wheel-to-rail contact force is less than 5%.

A New Formula of Impact Stiffness in Linear Viscoelastic Model for Pounding Simulation

The phenomenon of earthquake-induced structural pounding was extensively studied by some researchers using different models for the impact force. The aim of this paper is to provide a new formula of impact stiffness in the linear viscoelastic contact model, based on the assumption that the maximum impact deformation from the distributed mass model should be equal to that from the equivalent lumped mass model. The correctness and accuracy of the proposed formula have been confirmed by comparing the pounding simulation using the present formula of impact stiffness with those using the existing formulae.

An XFEM approach for modelling delamination in composite laminates

A First-order Shear Deformation Theory (FSDT) is chosen to simulate composite laminates in the linear and the geometrically non-linear regimes. The formulation is based on the Equivalent Single Layer (ESL) theory that fails to predict the delamination onset in composite laminates. The lack of resolving three-dimensional states and correct transverse stresses in this model is principally improved using post-processing. In order to precisely compute interlaminar stresses, a non-frictional linear adhesive contact model is applied in the context of the eXtended Finite Element Method (XFEM). The discontinuities are imposed within any arbitrary interface by enriching the displacement field. Thus two sub-domains define the plane of the discontinuity. Thereafter, the aforementioned adhesive contact can be formulated at the discontinued interface. Stress values are retrieved at nodal points using the interface constitutive equation. Consequently, the interface formulations are extended into the softening regime to model the delamination growth as a mixed-mode cohesive effect. The accuracy of the proposed method in predicting the interlaminar stresses and the delamination propagation is demonstrated by comparing the results with the ones available in literature. By combining the lower-order plate theory and the novel XFEM technique, the model is able to accurately calculate the delamination onset and the propagation with less computational effort.

Implementation and Verification of Linear Cohesive Viscoelastic Contact Model for Discrete Element Method

PURPOSES: Implementation and verification of the simple linear cohesive viscoelastic contact model that can be used to simulate dynamic behavior of sticky aggregates. METHODS: The differential equations were derived and the initial conditions were determined to simulate a free falling ball with a sticky surface from a ground. To describe this behavior, a combination of linear contact model and a cohesive contact model was used. The general solution for the differential equation was used to verify the implemented linear cohesive viscoelastic API model in the DEM. Sensitivity analysis was also performed using the derived analytical solutions for several combinations of damping coefficients and cohesive coefficients. RESULTS : The numerical solution obtained using the DEM showed good agreement with the analytical solution for two extreme conditions. It was observed that the linear cohesive model can be successfully implemented with a linear spring in the DEM API for dynamic analysis of the aggregates. CONCLUSIONS: It can be concluded that the derived closed form solutions are applicable for the analysis of the rebounding behavior of sticky particles, and for verification of the implemented API model in the DEM. The assumption of underdamped condition for the viscous behavior of the particles seems to be reasonable. Several factors have to be additionally identified in order to develop an enhanced contact model for an asphalt mixture.

Railway ground vibrations induced by wheel and rail singular defects

Railway local irregularities are a growing source of ground-borne vibration and can cause negative environmental impacts, particularly in urban areas. Therefore, this paper analyses the effect of railway track singular defects (discontinuities) on ground vibration generation and propagation. A vehicle/track/soil numerical railway model is presented, capable of accurately predicting vibration levels. The prediction model is composed of a multibody vehicle model, a flexible track model and a finite/infinite element soil model. Firstly, analysis is undertaken to assess the ability of wheel/rail contact models to accurately simulate the force generation at the wheel/rail contact, in the presence of a singular defect. It is found that, although linear contact models are sufficient for modelling ground vibration on smooth tracks, when singular defects are present higher accuracy wheel/rail models are required. Furthermore, it is found that the variation in wheel/rail force during the singular defect contact depends on the track flexibility, and thus requires a fully coupled vehicle/track/foundation model. Next, a parametric study of ground vibrations generated by singular rail and wheel defects is undertaken. Six shapes of discontinuity are modelled, representing various defect types such as transition zones, switches, crossings, rail joints and wheel flats. The vehicle is modelled as an AM96 train set and it is found that ground vibration levels are highly sensitive to defect height, length and shape.

Computation of minimum contact forces of multifingered robot hand with soft fingertips

In the human hand, the softness of a fingertip plays a significant role on stable grasping and dexterous manipulation. This paper presents a method of computing minimum grasp forces of a multi-fingered hand with soft fingertips. In the first, we built up a simple linear contact model of a soft fingertip. Then, based on Pontryagin's principle, the problem of minimizing contact forces for realizing the stable grasping was formulated and the forces were computed. Finally, the experiments were carried out using a multifingered robot hand, called "Allegro Hand" with hard fingertips and soft ones, and the effectiveness of the proposed method was validated.

Simulation of Collapse of Granular Columns Using the Discrete Element Method

AbstractIn this study, a three-dimensional (3D) numerical investigation of axisymmetric collapse of granular columns has been conducted using the discrete element method (DEM). The simulated granular columns have a constant initial radius of 5.68 mm and three aspect ratios: 0.55, 1.0, and 2.0. The columns consist of uniform spherical quartz particles with a diameter of 0.32 mm. In the DEM model, rotational velocities of particles are reduced by a factor at every time step to partially account for the additional rolling resistance due to the effect of particle shape and hysteretic contact behavior. The simple linear contact model is used; however, its performance is improved by using different stiffness values calculated by nonlinear Hertz–Mindlin contact model for each aspect ratio. The simulated final deposit heights, runout distances, and energy dissipation values are in good agreement with experimental observations reported in the literature. The effects of initial porosity and rotational resistance on...

Discrete element simulation of impact disaggregation for wet granule agglomerate

Based on the combination of linear contact model, Coulomb slip contact model and parallel bond contact model, a discret element model (DEM) of wet granule agglomerates with coating structure is constructed. Disaggregation processes of wet agglomerates in impacting to a horizontal plate are performed by applying particle flow code (PFC). Three failure patterns are obtained corresponding to those in experiment. The variation of velocities and rupture characteristics of liquid bridge in disaggregation process are investigated. Effects of impact velocity, gravity of adhered granules, and rotation of core granule are analyzed. DEM simulations show that there are three disaggregation patterns in the coating structure of agglomerates: impact disaggregation, gravity-impact disaggregation and shear-impact disaggregation, depending on the size of primary particles and the rotation of the core granules. With the enlargement of size, gravity plays an increasingly important role and the impact disaggregation pattern shifts to gravity-impact disaggregation. The rotation of core can generate a shear force to separate the fine and disaggregation pattern to turn to shear-impact disaggregation. Impacting results in a heterogeneous distribution of granule velocities and a tendency of relative movement in agglomerates. Relative movement will bring about the stretch of liquid bridge between granules. If the maximum separation distance of wet granules exceeds the rupture distance of liquid bridge, disaggregation happens. The ruptures of liquid bridge start from impact point and expand to outward, from bottom to up, from inside to outside in coating agglomeration. It is found that the rupture of liquid bridge needs time for accumulation and goes through three stages termed as slow rupture stage, quick rupture stage and entire rupture stage. With the increase of impact velocity, particle gravity, and rotating speed of core granules, disaggregation processes of wet granule agglomerates become fast and thorough. Impact velocity plays a primary role in disaggregation. DEM simulations are consistent with the experimental results.

Dynamic responses of a saturated poroelastic half-space generated by a moving truck on the uneven pavement

A truck–pavement–ground coupling model was established to study the dynamic responses of a saturated poroelastic half-space generated by a moving heavy truck on the uneven pavement. The ground was simulated as a fully saturated poroelastic half-space governed by Biot’s theory. The overlying pavement was simplified as a Kirchhoff thin plate. With the assumption of a sinusoidal pavement surface, the dynamic wheel–pavement force was obtained through a linear Hertizian contact model. The numerical results showed that this dynamic load could make considerable contributions to the stress and excess pore water pressure responses in the ground. Furthermore, the effective stress path of the soil unit beneath the pavement caused by the moving truck was firstly calculated and presented. It was found that the differences between the total stress path and the effective stress path became significant as the truck speed increased, thus the effective stress path was more suitable than total stress path to reflect the stress history of soil elements in the saturated ground during the passage of high-speed traffics.

Frequency filtering in disordered granular chains

The study of disorder-induced frequency filtering is presented for one-dimensional systems composed of random, pre-stressed masses interacting through both linear and nonlinear (Hertzian) repulsive forces. An ensemble of such systems is driven at a specified frequency, and the spectral content of the propagated disturbance is examined as a function of distance from the source. It is shown that the transmitted signal contains only low-frequency components, and the attenuation is dependent on the magnitude of disorder, the input frequency, and the contact model. It is found that increased disorder leads to a narrower bandwidth of transmitted frequencies at a given distance from the source and that lower input frequencies exhibit less sensitivity to the arrangement of the masses. Comparison of the nonlinear and linear contact models reveals qualitatively similar filtering behavior; however, it is observed that the nonlinear chain produces transmission spectrums with a greater density at the lowest frequencies. In addition, it is shown that random masses sampled from normal, uniform, and binary distributions produce quantitatively indistinguishable filtering behavior, suggesting that knowledge of only the distribution’s first two moments is sufficient to characterize the bulk signal transmission behavior. Finally, we examine the wave number evolution of random chains constrained to move between fixed end-particles and present a transfer matrix theory in wave number space, and an argument for the observed filtering based on the spatial localization of the higher-frequency normal modes.

Three-dimensional discrete element modelling of tillage: Determination of a suitable contact model and parameters for a cohesionless soil

Accurate modelling of soil–implement interaction is vital to optimise the design of tillage implements. Discrete element modelling (DEM) is a numerical method used for modelling the mechanical behaviour of granular materials. When DEM is used for tillage the soil is assumed to consist of distinct particles with the interactions between the particles controlled by contact models that are governed by physical laws. In previous three-dimensional (3D) DEM tillage studies the soil particles were assumed as having elastic contacts and the plastic deformation behaviour of the soil particles was not taken into account. Using EDEM software, a non-linear elastic contact model was compared with a linear hysteretic spring contact model that considers the plasticity of the soil. The DEM parameters required to run the simulations were determined from angle of repose and penetration tests, matching the simulation results to test results using a modelled spherical particle with radii of 10 mm. The simulation results of both the non-linear elastic and linear hysteretic spring (plastic) contact models were compared with experimental results from a sweep tillage tool. An improved correlation was obtained between the measured draft and vertical force results using the linear hysteretic spring (plastic) contact model. Further simulations using the linear hysteretic spring (plastic) model showed there was a quadratically increasing trend with particle size from 4 to 10 mm for the parameters of coefficient of restitution of soil–soil, friction of soil–soil and integration time step.

Multi-Visco-Elastic Contact Model in Discrete Element Method - Numerical Simulations of Margarine’s Dynamics during Manufacturing Process -

The multi-visco-elastic contact model embedded in the Discrete Element Method (DEM), which was developed by authors previous study (Shimizu et al., 2012)1) is applied to simulations of margarine’s dynamics in a pin machine, in order to improve margarine’s uniformity during manufacturing process. The results are compared with those using the linear contact model. Also, parametric studies are conducted by changing the operational conditions. As a result, it was found that the new developed multi-visco-elastic contact model represents margarine’s dynamics well during manufacturing process. The following results are considered from the numerical simulations, that is, (1) the migration process is slower than that of granular material, which is often modeled by using the linear contact model in the DEM, also the loaded torque on the machine is around 30 % from the case of using the linear contact model, (2) the migration proceeds mainly in the circumferential direction rather than in the radial direction, as movement of the bulk in the circumferential direction is much larger than that in the radial one, (3) performance of migration in the circumferential direction improves by increasing rotational speed, which is not expected in the radial direction though. Further studies to examine load characteristics of the machine are needed to use the model as a design tool for improvement of the machine’s performance and development of new machines.

Discrete element simulation of mechanical properties of wet granular pile

Discrete element method (DEM) simulations for pile-up processes of different particle systems were performed based on linear cohesion contact model. Effects of particle shape and liquid bridge force between wet particles on the piling form were analyzed. The significant central dip profiles of normal force acting on the base surface, normal force and tangential force between particles were predicted. Effects of particle shape and cohesion energy density on the forces on the base surface and inter-particles were described. The results show that particle shape and the liquid bridge force have significant impacts on the piling form. With the increase of the cohesion energy density the angle of repose for each granular pile increases. But the angle of repose of cubical particles is bigger than that of spherical particles under the same condition. Particle shape and the liquid bridge force also significantly affect the change and the maximum amplitude of the forces acting on the base surface and the forces between the particles. The maximum amplitude of the forces increases with the increase of the cohesion energy density, and the value of the maximum force on cubical particles is bigger than that on spherical particles. When the value of cohesion energy density is very large, the mechanical properties of granular piles become more complicated, so that the liquid bridge force has a larger impact on the packing characteristic of particles than the impact on particle shape.

Dry granular avalanche down a flume: Choice of discrete element simulation parameters

This paper presents a method to assign soft-sphere contact model parameters in a discrete-element simulation with which we can reproduce the experimentally measured avalanche dynamics of finite dry granular mass down a flume. We adopt the simplest linear model in which interaction force is decomposed along or tangent to the contact normal. The model parameters are chosen uniquely to satisfy theoretical models or to meet experimental evidences at either the particle or the bulk size level. The normal mode parameters are chosen specifically to ensure Hertzian contact time (but not its force-displacement history) and the resulting loss of particle kinetic energy, characterized by a measured coefficient of restitution, for each pair of colliding surfaces. We follow the literature to assign the tangential spring constant according to an elasticity model but propose a method to assign the friction coefficient using a measured bulk property that characterizes the bulk discharge volume flow rate. The linear contact model with the assigned parameters are evaluated by comparing the simulated bulk avalanche dynamics down three slopes to the experimental data, including instantaneous particle trajectories and bulk unsteady velocity profile. Satisfying quantitative agreement can be obtained except at the free surface and the early-time front propagation velocity.

The effect of base roughness on the development of a dense granular flow down an inclined plane

The development of the flow of a granular material down an inclined plane starting from rest is studied as a function of the base roughness. In the simulations, the particles are rough frictional spheres interacting via the Hertz contact law. The rough base is made of a random configuration of fixed spheres with diameter different from the flowing particles, and the base roughness is decreased by decreasing the diameter of the base particles. The transition from an ordered to a disordered flowing state at a critical value of the base particle diameter, first reported by Kumaran and Maheshwari [Phys. Fluids 24, 053302 (2012)]10.1063/1.4710543 for particles with the linear contact model, is observed for the Hertzian contact model as well. The flow development for the ordered and disordered flows is very different. During the development of the disordered flow for the rougher base, there is shearing throughout the height. During the development of the ordered flow for the smoother base, there is a shear layer at the bottom and a plug region with no internal shearing above. In the shear layer, the particles are layered and hexagonally ordered in the plane parallel to the base, and the velocity profile is well approximated by Bagnold law. The flow develops in two phases. In the first phase, the thickness of the shear layer and the maximum velocity increase linearly in time till the shear front reaches the top. In the second phase, after the shear layer encompasses the entire flow, there is a much slower increase in the maximum velocity until the steady state is reached.