Rebound behaviour of spheres for plastic impacts

This paper presents a study on the rebound behaviour of spheres impacted normally against a target wall using finite element methods. The emphasis is on the prediction of the coefficient of restitution and the effects of material properties and impact velocities on the rebound behaviour of the sphere. Finite deformation during plastic impact is addressed. The finite element results show that, for impacts of small plastic deformation, the coefficient of restitution is mainly dependent on the ratio of the impact velocity V_i to the yield velocity V_y which is consistent with those predicted by the theory of impact mechanics; while for impacts of finite-plastic-deformation it is also dependent on the ratio of the representative Young's Modulus E^* to the yield stress Y. The FEA results suggest that for impacts of finite-plastic-deformation the coefficient of restitution can be approximated to be proportional to [(V_i/V_y)/(E^*/Y)]^−1/2.     

A coefficient of restitution model for particle–surface collision of particles with a wide range of mechanical characteristics

The coefficient of restitution describes the energy dissipation resulting from particle-particle and particle–surface interactions in solid–fluid flows. The energy loss depends on the mechanical characteristics of the solid phase, therefore, to correctly predict the behavior of these systems it is necessary to use reliable coefficient values based on the properties of the particles. This paper investigated the energy dissipation in particle–surface collisions using 7 types of particles with a wide range of mechanical properties (Young's modulus between 1.38 × 10⁴ and 2.83 × 10⁹ Pa). Three empirical equations have been proposed to calculate the coefficient of restitution based on the impact velocity and the compressional wave velocity. The experimental results presented an inverse relation between the impact velocity and the coefficient of restitution. This effect was more pronounced for less elastic particles. The models presented an accurate fit to the experimental data and statistical analysis showed that the Power model presented the greater capacity to predict the coefficient of restitution from generic data. The experimental results showed the predominant effect of mechanical characteristics on the coefficient of restitution. In addition, the proposed equations are proved to be precise tools for predicting particle coefficients of restitution with a wide range of elasticity modulus at low velocities.     

Measurements of the coefficient of restitution for particle collisions with ductile surfaces in a liquid

This paper presents the approach and rebound of a particle colliding with a deformable surface in a viscous liquid. The complex interaction between the fluid and the solid phases is coupled through the dynamics of the flow as well as the deformation process. For the experiments, steel particles impacted ductile aluminum alloys samples immersed in an aqueous mixture of glycerol and water. The experiments involved particle Stokes number from 5 to 10⁵ and deformation parameters from 10⁻² to 10². From the experiments, the coefficient of restitution is shown to depend on both these parameters. The coefficient of restitution is closest to unity when the deformation parameter is less than one and the Stokes number is greater than 10³. The coefficient of restitution decreases as the deformation parameter increases and when the Stokes number decreases.     

A mixed contact model for an immersed collision between two solid surfaces

Experimental evidence shows that the presence of an ambient liquid can greatly modify the collision process between two solid surfaces. Interactions between the solid surfaces and the surrounding liquid result in energy dissipation at the particle level, which leads to solid-liquid mixture rheology deviating from dry granular flow behaviour. The present work investigates how the surrounding liquid modifies the impact and rebound of solid spheres. Existing collision models use elastohydrodynamic lubrication (EHL) theory to address the surface deformation under the developing lubrication pressure, thereby coupling the motion of the liquid and solid. With EHL theory, idealized smooth particles are made to rebound from a lubrication film. Modified EHL models, however, allow particles to rebound from mutual contacts of surface asperities, assuming negligible liquid effects. In this work, a new contact mechanism, 'mixed contact', is formulated, which considers the interplay between the asperities and the interstitial liquid as part of a hybrid rebound scheme. A recovery factor is further proposed to characterize the additional energy loss due to asperity-liquid interactions. The resulting collision model is evaluated through comparisons with experimental data, exhibiting a better performance than the existing models. In addition to the three non-dimensional numbers that result from the EHL analysis--the wet coefficient of restitution, the particle Stokes number and the elasticity parameter--a fourth parameter is introduced to correlate particle impact momentum to the EHL deformation impulse. This generalized collision model covers a wide range of impact conditions and could be employed in numerical codes to simulate the bulk motion of solid particles with non-negligible liquid effects.     

The normal and oblique impact of three types of wet granules

Impact tests against a hardened steel plate have been carried out to obtain the coefficient of restitution of three types of spherical granules. The dominant elastic γ-Al₂O₃, the elastic-plastic zeolite 4A and the dominant plastic sodium benzoate have been chosen as granule samples. An electromagnetic canon has been constructed to accelerate the granules and to measure the normal coefficient of restitution. The moisture content of the granules has been varied so that the pore saturation ranges between of S = 0–1. Thereby, the influence of the moisture content on the normal coefficient of restitution could be determined. A free fall apparatus, on which the impact angle is changeable in the range of Θ_A = 0–80°, has been used to investigate the tangential coefficient of restitution. A high speed digital camera has been used to record the events of impact and rebound. The record frequency of the camera has been varied between 4,000 and 8,000 frames per second.     

Experimental study on electrostastic particle deposition from gas–solids two-phase flow based on tribo-electrification

Electrostastic particle deposition on a target embeded in a substrate under a locally applied voltage has been investigated experimentally based on the tribo-electrification of particles. Initially particles deposit mainly on the edge of the target because of the contact potential difference between substrate (poly(methyl methacrylate)) and target (brass). The thickness of particle layer formed by the deposited particles increases with time, but gradually saturates. Since then almost no additional particles deposit on the target. When high voltage is applied to the target, and orange peel phenomenon is observed on the surface of the particle layer. In the central region of the target, the particles under high particles. The larger the Coulombic force parameter K_E, the higher the effective deposition velocity is. The velocity increases dramatically for K_E 6 × 10⁻⁶ and 20 < St < 60. The larger Stokes number makes the coefficient of variation larger for the thickness of the particle layer, i. e. the deposition under higher Stokes number gives a less uniform deposition layer. However, for St < 10, particles deposit almost uniformly. For the deposition under a larger Coulombic force parameter, onset of the rebound and resuspension is suppressed, and the region with a uniform deposition layer is shifted to a higher Stockes number. It is also found that particles can successfully deposit on a non-conductive target of a dielectric substance through setting a grounded guard electrode around the target.     

Effects of specularity and particle-particle restitution coefficients on the recirculation characteristics of dispersed gas-particle flows through a sudden expansion

We numerically investigate the effects of restitution and specularity coefficients on the characteristics of dispersed gas-particle flows through a sudden expansion. The studies are carried out using an indigenous finite volume flow solver in a collocated framework with two-fluid model. Parametric studies are performed to gain insights into the differences in recirculation patterns that arise due to variations in restitution and specularity coefficients. The simulations show that particle-particle interactions, quantified by restitution coefficient (e) have a greater impact on recirculation characteristics than particle-wall interactions, which are quantified by specularity coefficient ($?$). Studies reveal that the recirculation lengths tend to decrease as particle collisions become more elastic (as e tends to unity) while they increase, as the value of $?$ increases. However, the changes in recirculation length are very gradual and less pronounced when only particle-wall interactions are considered as compared to particle-particle interactions. From the range of parametric variations studied in this work, the maximum recirculation length has been found when the value of $?$ is maximum and that of e is minimum.     

Investigation of the influence of impact velocity and liquid bridge volume on the maximum liquid bridge length

Several models exist to predict the capillary forces due to liquid bridges during static particle contacts. However, for dynamic impacts, there is still a lack of knowledge to accurately describe the geometry and rupture of liquid bridges. This is essential in order to calculate correctly the energy dissipation in numerical models. Therefore, the liquid bridge volume during the rebound phase of collision, the restitution coefficient, the rupture time and the maximum liquid bridge length were analyzed for a wide range of contact velocities from 0.0001 to 4.0 m?s^?1. To perform experiments at different velocities three experimental setups were developed. First, free-fall experiments with spherical particles colliding on a glass pane covered with a water layer were performed at different impact velocities from 0.3 to 1.5 m?s^?1 and water layer thicknesses. Secondly, a pneumatic setup was developed for investigating the maximum liquid bridge length at constant liquid bridge volume and contact velocities from 0.3 to 4.0 m?s^?1. A third setup was used to investigate the liquid bridge behavior during low velocities from 0.0001 to 0.04 m?s^?1. Based on the experimental results, a new model is presented to account for the significant influence of impact velocity on the maximum liquid bridge length.     

The oblique impact of a hard ball against ductile,semi-infinite target materials—experiment and analysis

Six ductile target materials were impacted with hardened steel balls over a range of impact velocities (50–200 m s⁻¹) and impact angles (15–90°). The dimensions of the resulting impact craters and also the rebound velocity and angle were experimentally measured. These experimental values were compared with the predictions of a simplified model which assumes a constant dynamic yield pressure and friction coefficient. The model correctly predicts the magnitude of the crater dimensions and volume and also their variation with impact angle and velocity. In contrast, the rebound angle and the energy absorbed per impact are not predicted correctly by the simple model.     

Energy absorption during compression and impact of dry elastic-plastic spherical granules

The discrete modelling and understanding of the particle dynamics in fluidized bed apparatuses, mixers, mills and others are based on the knowledge about the physical properties of particles and their mechanical behaviour during slow, fast and repeated stressing. In this paper model parameters (modulus of elasticity, stiffness, yield pressure, restitution coefficient and strength) of spherical granules (γ-Al₂O₃, zeolites 4A and 13X, sodium benzoate) with different mechanical behaviour have been measured by single particle compression and impact tests. Starting with the elastic compression behaviour of granules as described by Hertz theory, a new contact model was developed to describe the force-displacement behaviour of elastic-plastic granules. The aim of this work is to understand the energy absorption during compression (slow stressing velocity of 0.02 mm/s) and impact (the impact velocity of 0.5–4.5 m/s) of granules. For all examined granules the estimated energy absorption during the impact is found to be far lower than that during compression. Moreover, the measured restitution coefficient is independent of the impact velocity in the examined range and independent of the load intensity by compression (i.e. maximum compressive load). In the case of repeated loading with a constant load amplitude, the granules show cyclic hardening with increasing restitution coefficient up to a certain saturation in the plastic deformation. A model was proposed to describe the increase of the contact stiffness with the number of cycles. When the load amplitude is subsequently increased, further plastic deformation takes place and the restitution coefficient strongly decreases.     

弹性圆环冲击弹性柱的碰撞回弹行为研究

目的研究弹性薄壁圆环低速冲击理想弹性柱的碰撞回弹行为。方法实验方面,使用高速数码显微系统记录铜环冲击3种不同橡胶柱的碰撞相互作用的全过程。理论方面,应用模态叠加法和拉格朗日方程法导出圆环约束运动阶段和自由运动阶段的运动控制方程,并建立2组方程间的迭代关系,据此描述碰撞相互作用的全过程。结果碰撞次数随着刚度比的增大而增加,无量纲回弹时间和恢复系数随着刚度比的增大而减小。多次碰撞的临界刚度比随着圆环厚径比的增大而增大。结论理论方法成功预测了弹性薄壁圆环低速冲击理想弹性柱的碰撞回弹行为,可为防撞结构的几何与材料参数选择,以及实验模态分析中的锤头选择提供理论指导。     

Collision and rebound of ping pong balls on a rigid target

The collision and rebound behavior of ping pong balls impinging onto rigid target are studied. Three dimensionless dominant parameters are identified: (1) the ratio of the wall-thickness to the average radius of the ball; (2) the dimensionless initial velocity; and (3) the yield strain of the material.Depending on the dimensionless initial velocity, various collision and rebound behaviors of the ball are revealed: (1) When the initial velocity is low, the deformation of the ball remains purely elastic, for which the characteristic duration is theoretically obtained; (2) With the increase of initial velocity, the ball's cap begins to buckle and multiple impacts occur, leading to the increase of the restitution duration and the reduction of the coefficient of restitution (COR); (3) With the higher initial velocity, the ball's cap buckles permanently, leading to the disappearance of multiple impacts and a sudden drop of the restitution duration; consequently the COR decreases from 0.5 to 0.3; and (4) When the initial velocity is close to the material's yield velocity, the ball buckles into a non-axisymmetric mode.The simulation results are also compared with experimental ones. Furthermore, the effects of thickness-to-radius ratio, yield strain and coefficient of friction are also discussed.     

Experimental investigation of dry,wet and cryogenic boring of AA 7075 alloy

In this research work, an attempt has been carried out to examine (investigate) and study the dry, wet and cryogenic boring of AA 7075 alloy, which is predominantly used in transport applications in defense (aeronautical parts), oceanic and automaker industries. To ensure direct supply of the coolant, and real-time measurement of cutting temperature a modified boring bar is used (modification is carried out using EDM to accommodate placement of a thermocouple to obtain real-time measurement of temperature readings during the boring cycle). It is observed that during cryogenic boring of AA 7075 alloy there is a considerable reduction in the cutting force (F_c), cutting temperature (T_c) and surface roughness (R_a) by 56.16%, 84.70%, 58.98% compared to dry boring and 48.43%, 80.70%, 34.70% compared to wet boring, respectively. Decrease in F_c and T_c leads to a reduction in high stresses at localized points during machining and in turn curtail wear in workpiece and tool. Lubrication provided by cryogenic fluids also plays a sizable role in reduction of F_c and T_c. Reduction in lower F_c and T_c has a glaring effect on the surface characteristics of the hole produced during the boring process. Tool wear is reduced in cryogenic boring by 36.96% and 17.57% compared to dry and wet boring, respectively. Taguchi and ANOVA was carried out which helped in determining feed as an important parameter with respect to F_c and R_a during boring of AA 7075 under dry, wet and cryogenic conditions whereas speed as an important parameter in determining T_c in dry and wet conditions and feed for T_c in cryogenic boring condition. TOPSIS analysis highlighted speed of 770 rpm and feed of 0.055 mm/min as the most closest to ideal solution for all three different cutting conditions. Surface morphology study after boring of AA 7075 highlighted better surface characteristics in cryogenic bored surface compared to dry and wet boring. Roughness measured in AFM for tool used in boring highlighted a decrease in 86.79% and 66.01% in cryogenic boring in juxtaposition with dry and wet boring, respectively. A surge in compressive residual stress is observed in cryogenic bored surface by 10.41% and 3.5% in juxtaposition with dry and wet boring, respectively, highlighting an abatement in tensile residual stress and better workpiece integrity as compared to dry and wet boring conditions.     

Aging in velocity autocorrelations in granular gas of viscoelastic particles in two dimensions

We use large-scale molecular dynamics simulations to study aging of the velocity autocorrelation function of a force-free granular gas consisting of viscoelastic particles. We study the velocity autocorrelation function for a simplified model where the coefficient of restitution is constant for all collisions, but depends on current temperature of granular gas, it is called quasi-constant coefficient of restitution ε_eff. From our simulation results, it is observed that A(τ_w, τ) depends independently on both τ and τ_w. Initially, A(τ_w, τ) decays exponentially but later as τ_w increases, A(τ_w, τ) decays slowly due to emergence of correlations in velocity field. The explicit dependence of A(τ_w, τ) on τ_w implies that the system exhibits aging property.     

Effects of specularity and particle-particle restitution coefficients on the hydrodynamic behavior of dispersed gas-particle flows through horizontal channels

Specularity coefficient ($?$) and particle–particle restitution coefficient (e) are two important parameters governing the flow physics of dispersed gas-particle flows. In this work, a detailed numerical analysis is carried out to get an insight into the effects of these two parameters in the flow hydrodynamics of dispersed gas-particle flows through horizontal channels. Investigations have also been carried out to find the $?$-e pair for which the phase velocities become an extremum. It has been found that at a particular value of e, both gas and particle velocities at the centerline of the channel increase with increase in the value of $?$, whereas near the wall, they tend to decrease. At a fixed non-zero value of $?$, both gas and particle velocities tend to increase with increase in the value of e. For $?$ equal to zero, which corresponds to free-slip boundary condition for particle velocity, there is no significant variations in gas and particle velocities with changes in e. Out of all combinations of values of $?$ and e investigated herein, it is found that both gas and particle velocities attain a maximum value when both the values of $?$ and e are maximum.     

Stokes shape factor for lactose crystals

The alpha-lactose crystal has a tomahawk shape which needs to be accounted for when designing settling equipment. A shape factor can be used to achieve this. A variety of shape factors have been used for lactose crystals in the literature. This paper sets out to experimentally determine a shape factor for lactose. Large undamaged tomahawk shaped lactose crystals were grown in a lactose agar gel and then recovered for use in settling experiments. Typical industry produced crystals were also tested for comparison. Settling experiments enabled the calculation of a Stokes settling diameter, the diameter of a sphere with the same density and settling velocity as the tomahawk shaped lactose crystal. Using crystal mass to calculate equivalent particle volume and Stokes diameter, the Stokes shape factor for lactose gel-grown crystals was calculated to be 0.99. A Stokes height factor (B_St) was formulated which, when multiplied by the height of the lactose crystal, gives the Stokes settling diameter. The lactose B_St value was determined to be 0.595 ± 0.007 and 0.643 ± 0.008 for gel-grown and plant-grown lactose crystals, respectively.     

Immersed-boundary/soft-sphere method for particle–particle-fluid interaction in a viscous flow: An OpenFOAM solver

We developed a stable OpenFOAM solver for Immersed Boundary Method based on direct forcing and regularized delta function. The soft-sphere model and a lubrication model were implemented to consider particle–particle collision in a viscous flow. We proposed a fluid–structure interaction (FSI) coupling method to accurately calculate the fluid forcing term and particle velocity. Our solver was validated for fixed and moving bodies, including rotation. The accuracy of various FSI schemes was evaluated in predicting the solid and fluid flow behavior in a viscous flow. It was demonstrated that neglecting or simplifying the fluid momentum change affects the accuracy of the solid velocity and fluid flow dynamic; for higher solid-to-fluid density ratios, a larger deviation was predicted. Furthermore, the FSI schemes highly influenced the behavior of the formed vortices.The solver was validated to predict the effective restitution coefficient of particles in a viscous flow as a function of the Stokes number. We also thoroughly analyzed the dynamic flow behavior of colliding particles through the pressure and velocity field and fluid force. This analysis helped us accurately determine the rebound velocity of particles in case of high Stokes numbers when the effect of viscous force is significant.     

A Random Pairing Collision Model (RPCM) for Improving the DEM Simulation of Particle-Bed Collisions in Aeolian Sand Transport

The impact of sand particles with bed is the key process in aeolian sand transport. In this article, the discrete element method (DEM) is embedded to the large-eddy simulation (LES) method for simulating the wind-blown sand two-phase flow. A numerical model is developed to stochastically describe collisions between sand particles and bed within the DEM approach. Statistical features of major simulated motion parameters are undertaken and compared with measured data to verify the model. The probability distribution of the kinetic energy restitution coefficient is normal, and the probability density functions of the impact and lift-off angles are consistent with reported results. The probability density functions of three-dimensional impact and lift-off velocities are addressed in further detail.