Normal Contact Force Research Articles

Contribution of rolling resistance to the drag coefficient of spheres freely rolling on a rough inclined surface

The drag coefficient (CD) of a sphere freely rolling without slipping on a rough plane is presented in this study. Increasing panel roughness has been found to increase CD, although lubrication theory predicts that the larger gap imposed by the rougher panel should yield a smaller drag. We propose that this increase in drag is due to the effects of rolling resistance, which increases with panel roughness. The total drag on a sphere is decomposed into fluid drag and drag due to rolling resistance, where the fluid drag is predicted using a combined analytical–numerical approach. It is shown that rolling resistance can be modeled as a resistive torque opposing the sphere motion, generated by the offset contact normal force from the sphere center plane. This coefficient of rolling resistance (μr) can be predicted using the root mean square roughness (Rq) of the panel. Additionally, μr is observed to increase with sphere down-slope velocity and an empirical relationship between μr, Rq, and non-dimensional velocity (U∗) is given. A comparison of the drag predicted by the proposed model with measured data indicates good agreement for all the four panels considered. Consistent with previous literature, a non-linear relationship between μr, Rq, and U∗ is proposed. Although increasing panel roughness leads to a smaller fluid drag due to the larger gap imposed by rougher panels, the drag due to rolling resistance increases more rapidly. This leads to an increase in total drag with increase in the panel roughness. Additionally, increasing panel roughness is observed to have a significant effect on the sphere wake, leading to irregular wake shedding and increase in the Strouhal number.

Modeling of the Particle Abrasion Process and a Discrete Element Method Study of Its Shape Effect.

This study introduces a novel method for particle abrasion derived from fundamental natural phenomena and mechanical principles, allowing precise control over the degree of abrasion and more accurately mimicking natural processes. The method's validity is confirmed using a specific shape index. Through conventional triaxial tests, the mechanical behavior of granular aggregates with varying degrees of abrasion was analyzed. The findings indicate that increased particle abrasion leads to a decrease in the average coordination number and sliding amount, while the rotation amount increases. This suggests an inverse relationship between the degree of abrasion and the structural stability and interlocking of the particle aggregate. The fabric anisotropy of the system is mainly attributed to the anisotropy of the contact normal force, which decreases as particle abrasion increases. The partial stress ratio of the particle system is influenced by fabric anisotropy and remains independent of particle shape. Additionally, the internal friction angle may be overestimated in conventional triaxial tests.

Dynamic modeling and analysis of 3-RRCPR parallel pointing mechanism with clearance

In order to study the influence of kinematic pair clearance on the dynamic characteristics and pointing accuracy of 3-RRCPR parallel pointing mechanism, the dynamic model of spatial mechanism with three-dimensional revolute pair clearance is established. Firstly, the revolute pair mode considering three-dimensional clearance is established, and the normal and tangential contact force models between the kinematic pair elements are established based on Flores contact model and modified Coulomb friction model. Secondly, the dynamic model of parallel mechanism with clearance is established according to Lagrange multiplier method. Finally, the effects of different clearance sizes and loads on the dynamic characteristics and pointing accuracy of the mechanism are analyzed by numerical simulation. The results show that the increase of clearance size and load will reduce the pointing accuracy, and the decrease of clearance size and the increase of load will enhance the motion stability of the mechanism. Reasonable clearance and load matching is conducive to improving the accuracy of the mechanism. This study provides a theoretical basis for the study of the nonlinear dynamic characteristics and accuracy of the mechanism considering the clearance.

PC-Kriging-powered parallelizing Bayesian updating for stochastic vehicle-track dynamical system with contact force measurements and Gaussian process discrepancy model

High-precision vehicle-track coupled dynamical models play a vital role in assessing the running safety of the vehicle, tracking fatigue behavior, and establishing a digital twin for high-speed railways. This study established a high-fidelity vehicle-track coupled dynamical model as a forward solver, which was then updated in a Bayesian inference framework based on the field tests of rail irregularities and wheel/rail normal contact forces. The prediction errors corresponding to different time steps were represented through a Gaussian process (GP) model characterized by a covariance function with the Gaussian kernel to incorporate the correlation between different time steps. To cope with the considerable expense of the likelihood function calculations with large-scale loop operations and the computational burden due to the large batch of repetitive evaluations of the high-fidelity forward model involved in the stochastic sampling from the posterior probability distribution, a metamodel-powered parallelizing stochastic sampling scheme was adopted. The posterior distribution could be formulated by replacing the expensive explicit model evaluation involved in the likelihood function with a Polynomial-Chaos-Kriging (PC-Kriging) predictor providing a surrogate mapping between the wheel/rail normal contact forces and physical parameters. Finally, a parallelizing computing scheme running across multiple CPU cores on high-performance computers was realized to sample from the vectorized formula of the emulator-powered posterior distribution. Field test data from a high-speed railway in China was adopted to demonstrate that the Bayesian inference scheme can quantify the uncertainties of the core physical parameters of the high-fidelity vehicle-track coupled dynamical system by efficiently trading off accuracy and computational efficiency.

Characterisation of discharge and flow rate predictions for asymmetric wedge-shaped hoppers

It is important to understand the flow behaviour inside wedge-shaped hoppers and accurately predict the discharge rate for the processing and handling of granular materials, regardless of whether the wedge hopper is symmetric or asymmetric. In this paper, the discrete element method (DEM) was used to reveal the flow behaviour of pellet feed in an asymmetric wedge-shaped hopper from the discharge rate, flow pattern, velocity distribution, normal contact force between particles and free-fall arch. The results showed that, with the hopper angle decreasing, the area of the active region of the particles increased, the stagnation zone decreased, and the free-fall arch became unstable. When the unilateral hopper angle was less than 45°, the discharge rate of the wedge-shaped hopper increased, and the average discharge rate reached the maximum value of 1.070 kg s−1 when the left and right hopper angles were equal to 15°. In addition, using the discharging mass proportion coefficient to represent the size of the region in which the unilateral hopper angle affected the flow of particles in the hopper, the hopper angle term in the Brown and Sellers model was corrected. The predictive errors of the corrected discharge rate model were less than 7.6% and 3.3% in the simulated and actual discharging tests respectively, which was better than that of the Brown and Sellers model. The results of the study could provide a theoretical basis for the intelligent upgrading of feed accurate handling equipment, and provide a reference basis for the design of asymmetric wedge-shaped hopper.

Periodic Rolling and Bumping

Abstract We consider a sphere that rolls down an inclined corrugation of identical, frictional circular cylinders in contact. We assume that the motion is a periodic succession of rolling and bumping that results in a velocity that is, on average, steady. We use balances of force and moment and impulse and momentum change to calculate the instantaneous and average velocities, the tangential and normal contact forces, and their ratio in rolling regimes that involve no sliding, some sliding and a transition between these at contact. We compare the predictions for the steady average velocity versus the inclination for different sphere to cylinders radii ratio and the predictions for the angle of inclination versus the radii ratio for stopping, initiation of sliding and loss of contact with the results of numerical simulations.

Force model of ultrasonic empowered minimum quantity lubrication grinding CFRP

As a weight-reducing material in aerospace applications, carbon fiber reinforced polymer (CFRP) requires precision grinding to ensure the accuracy and assembly of mating positioning surfaces. However, achieving low-damage machining of CFRPs remains challenging. Flood cooling reduces the mechanical properties, while dry grinding deteriorates the surface integrity, making minimum quantity lubrication (MQL) a viable alternative. However, nanolubricants struggle to penetrate the grinding area due to gas barrier obstruction. Consequently, a new CFRP grinding approach using multiangle 2D ultrasonic vibration-empowered nanolubricant MQL was proposed. An analytical force model is crucial for optimizing machining strategies and adjusting parameters. The aim is to reveal grinding mechanics and develop a force model under different ultrasonic and lubrication-cooling conditions. First, a uniform random fiber distribution model and a grinding wheel model were established, and the mechanical properties of the interface were predicted. Then, the intermittent grinding behavior and dynamic stiffnesses of the multiangle 2D ultrasonic vibration-assisted grinding (UVAG) system were investigated based on the geometric kinematics of the grains' trajectory. Furthermore, based on the deformation deflection curve, bending fracture force models for 45°, 90°, and 135° fibers were established with different contact and boundary conditions. The material removal mechanisms of shear and tension-compression buckling in 0° CFRP grinding were revealed, and an elliptical contact normal force model was established. The interlayer shear in 45° CFRPs was analyzed. The longitudinal compressive matrix cracking and fiber shear fracture in the 90° CFRPs were elucidated. Fiber microbuckling and fiber bundle removal of 135° CFRPs were investigated. Finally, grinding force models with different fiber orientation angles (FOAs) were interconnected and experimentally assessed. Experimental assessments demonstrated that the grinding force model has acceptable estimation error and effectively captures the mechanics of CFRPs with different FOAs.

Experimental and numerical studies of breakage and fractal characteristics of silica sands in high-pressure triaxial tests

Triaxial compression tests were performed on silica sands with confining pressure ranging from 0.5 to 60 MPa. A multigenerational DEM model was employed parallelly to investigate how micromechanical behaviors evolve with particle breakage. Prediction of the relative breakage considering the influence of axial strain and confining pressure is proposed. A strong correlation between the relative breakage and the input work is proposed regardless of the shear path. A nonlinear relationship is proposed to correlate the relative breakage and the volumetric strain. As the input work goes to infinity, the fractal dimension has an ultimate value approaching 2.66, which is greater than the values from previous theoretical and experimental studies. The mean coordination number in an uncrushable sample is higher than the crushable one during whole shearing. The normal contact force distribution is sensitive to the shear path, and particle breakage makes the contact force more inhomogeneous.

3D DEM investigation of shear behavior and interaction mechanism of woven geotextile-sand interfaces

This paper presents a numerical study on the investigation of microscopic mechanism governing the interaction of woven geotextile and angular sand employing the 3D discrete element method (DEM). The surface texture and tensile properties of the geotextile were simulated using overlapping spherical particles, and the angular sand was simulated using rigid blocks. The DEM models were fully calibrated based on previous experimental data. The shear and dilation zones of sand near the interface were quantitatively determined based on particle displacement gradients. Analysis of contact forces was conducted to explain the microscopic mechanism behind the macroscopic strength evolution. The influence of geotextile surface roughness on the shear strength of the geotextile-sand interface is also discussed. The results show that the failure mode of the woven geotextile-sand interface is a combination of particle sliding failure along the geotextile surface and shear failure of the sand within the shear zone above the interface. There is a rapid redistribution of contact forces prior to reaching peak shear resistance, and the average normal contact force within the shear zone remains relatively constant after the peak shear stress is achieved. A completely developed shear zone stabilizes soil deformation, typically after achieving the peak shear resistance.

DEM analysis of constrained modulus nonlinearity and damping factor of dry granular soils

The seismic site response analysis requires the dynamic soil properties (i.e., the modulus and damping). While it is well understood that the shear modulus and damping ratio are nonlinearly shear-strain dependent, the knowledge on the constrained modulus and damping ratio of compressional waves is still very limited due to lack of laboratory testing equipment. This study aims to simulate cyclic tests of constrained compression for granular specimens by discrete element method (DEM) to understand the dynamic soil properties of compressional waves, i.e., the nonlinearity in constrained modulus and damping ratio with the compression strain. The evolution in microstructure of granular specimens is revealed to provide micromechanical interpretations for the compressional soil nonlinearity. The results show that the dependency of constrained modulus on compression strain is different in compression and extension stages, and the modulus reduction is only observed in the extension path. The damping ratio of samples under cyclic constrained compression is smaller than that of cyclic shear. The nonlinear soil behavior is more obvious at lower confining pressure and for dense sample. The nonlinearity of constrained modulus depends on the coordination number, contact normal force and fabric anisotropy, while the associated damping ratio is only related to fabric anisotropy.

Effect of the Height of a 3D-Printed Model on the Force Transmission and Thickness of Thermoformed Orthodontic Aligners.

This research aims to investigate the influence of model height employed in the deep drawing of orthodontic aligner sheets on force transmission and aligner thickness. Forty aligner sheets (Zendura FLX) were thermoformed over four models of varying heights (15, 20, 25, and 30 mm). Normal contact force generated on the facial surface of the upper right central incisor (Tooth 11) was measured using pressure-sensitive films. Aligner thickness around Tooth 11 was measured at five points. A digital caliper and a micro-computed tomography (µ-CT) were employed for thickness measurements. The normal contact force exhibited an uneven distribution across the facial surface of Tooth 11. Model 15 displayed the highest force (88.9 ± 23.2 N), while Model 30 exhibited the lowest (45.7 ± 15.8 N). The force distribution was more favorable for bodily movement with Model 15. Thickness measurements revealed substantial thinning of the aligner after thermoforming. This thinning was most pronounced at the incisal edge (50% of the original thickness) and least at the gingivo-facial part (85%). Additionally, there was a progressive reduction in aligner thickness with increasing model height, which was most significant on the facial tooth surfaces. We conclude that the thermoplastic aligner sheets undergo substantial thinning during the thermoforming process, which becomes more pronounced as the height of the model increases. As a result, there is a decrease in both overall and localized force transmission, which could lead to increased tipping by the aligner and a diminished ability to achieve bodily movement.

Rope–sheave contact transient analysis in hoisting operations with a bristle model and an arbitrary Lagrangian–Eulerian approach

AbstractThis paper describes the development of a computational model for the rope–sheave contact interaction in reeving systems when the ropes are modeled with an arbitrary Lagrangian–Eulerian approach. This discretization approach has been developed in previous publications as a general and systematic method for the modeling and simulation of reeving systems. However, the rope–sheave contact model was avoided assuming the no-slip contact condition. The contact model developed in this paper introduces specialized ALE-ANCF-cubic rope contact elements that are used to discretize the rope segment winded at the sheave. The contact is modeled using a set of virtual discrete bristles attached to material points in the mid-line of the rope in one end and in contact with the sheave in the other end. Therefore, a second Lagrangian mesh, apart of the ALE mesh used to discretize the rope, is used to define the fixed ends of the bristles. The kinematics and dynamics used to calculate the normal and tangential contact forces are described in detail. The contact model is 3D and can be used to analyze the contact with a sheave groove with arbitrary shape. The tangential contact force model can be used to describe stick and slip contact conditions and, to improve the simulation performance of the model, an LuGre regularization tangential contact force model is used. The rope-sheave contact model is used to analyze the behavior of a simple elevator system. The numerical results show that the static rope-sheave contact interaction agrees well with an analytical solution of the problem. Finally, the same elevator system is analyzed dynamically for a cabin ride of 8 meters with a steady velocity of 1 m/s. Results show that the normal and tangential contact forces during the steady velocity period are not so different from the static solution, but very different from the classical Creep Theory and Firbank’s Theory.

Effect of thermomechanical ageing on force transmission by orthodontic aligners made of different thermoformed materials: An experimental study.

Investigating the impact of thermal and mechanical loading on the force generation of orthodontic aligners made from various thermoplastic materials and different compositions. Five distinct materials were utilized including, three multi-layer (Zendura FLX, Zendura VIVA, CA Pro) and two single-layer (Zendura A and Duran). A total of 50 thermoformed aligners (n = 10) underwent a 48-hour ageing protocol, which involved mechanical loading resulting from a 0.2 mm facial malalignment of the upper right central incisor (Tooth 11) and thermal ageing through storage in warm distilled water at 37°C. The force exerted on Tooth 11 of a resin model was measured both before and after ageing using pressure-sensitive films and a biomechanical setup. Before ageing, pressure-sensitive films recorded normal contact forces ranging from 83.1 to 149.7 N, while the biomechanical setup measured resultant forces ranging from 0.1 to 0.5 N, with lingual forces exceeding facial forces. Multi-layer materials exhibited lower force magnitudes compared to single-layer materials. After ageing, a significant reduction in force was observed, with some materials experiencing up to a 50% decrease. Notably, multi-layer materials, especially Zendura VIVA, exhibited lower force decay. The force generated by aligners is influenced by both the aligner material and the direction of movement. Multi-layer materials exhibit superior performance compared to single-layer materials, primarily because of their lower initial force, which enhances patient comfort, and their capability to maintain consistent force application even after undergoing ageing.

DEM simulation of effects of particle size distribution on meso-mechanical properties of slip zone soil

Particle Size Distribution (PSD) exerts a substantial influence on the mechanical properties of geological materials such as rocks and soils, which can be viewed at a microscale as an assembly of discrete particles. An exploration into the effects of particle gradation on the properties of these materials provides valuable insights into their nature. In the study, the Discrete Element Method (DEM) was used to conduct numerical shear tests on eight distinct groups of slip zone soil, each characterized by a different particle gradation. The aim was to examine the meso-mechanical properties and shear evolution laws of slip zone soil numerical samples with both optimal and sub-optimal PSDs. Findings underscore the pivotal role that PSD plays in various aspects, including dilatancy, the evolution of the displacement field, the network of contact force chains, the principal stress, and the distribution of normal and tangential contact forces within the slip zone soil. It was observed that the network of contact force chains in the numerical samples with an optimal PSD was more complex than in those samples with a sub-optimal PSD. Additionally, the distribution of principal stresses before and after shear was more uniformly balanced. This particle size-based study offers significant reference value for future investigations into the impact of PSD on the macroscopic and meso-mechanical properties of slip zone soil. By augmenting this knowledge, a more comprehensive understanding of the fundamental behavior of these materials can be attained, leading to improved prediction and management of geological risks.

Modeling Density Waves and Circulations in Vertical Cross-Section in Adhesive Contacts

This work continues the study of the process of friction between a steel spherical indenter and a soft elastic elastomer previously published in our paper. It is done in the context of our previous experimental results obtained on systems with strongly pronounced adhesive interaction between the surfaces of contacting bodies during the process of friction between a steel spherical indenter and a soft elastic elastomer. In the present paper, we concentrate on the theoretical study of the processes developing in a vertical cross-section of the system. For continuity, here the case of indenter motion at a low speed at different indentation depths is considered as before. The analysis of the evolution of normal and tangential contact forces, mean normal pressure, tangential stresses, as well as the size of the contact area is performed. Despite its relative simplicity, a numerical two-dimensional (2D = 1 + 1) model, which is used here, satisfactorily reproduces experimentally observed effects. Furthermore, it allows direct visualization of the motion in the vertical cross-section of the system, which is currently invisible experimentally. Partially, it recalls two-dimensional (2D = 1 + 1) models recently proposed to describe the “turbulent” shear flow of solids under torsion and in cellular materials. The observations extracted from the model help us to understand better the adhesive processes that underlie the experimental results.

Stiffness Anisotropy and Micro-Mechanism of Calcareous Sand with Different Particle Breakage Ratios Subjected to Shearing Based on DEM Simulations

Stress-induced anisotropy in calcareous sand can cause an uneven displacement in island reef engineering. In this study, stiffness, as a quantitative indicator, is explored to reveal the stress-induced anisotropy in calcareous sand. Based on the discrete element method, the stiffness anisotropic characteristics of calcareous sand during shearing, as well as the impact of particle breakage, are investigated by numerical simulations. Both the macro and micro responses, i.e., the maximum shear modulus, contact normal, strong and weak contact normal force, and the direction of particle breakage, are explored for calcareous sand with different particle breakage ratios. The results show that calcareous sand exhibits notable anisotropy during shearing, with the maximum shear modulus in the vertical direction (deviatoric stress direction) being significantly greater than that in the horizontal direction. Moreover, the higher the particle breakage rate, the lower the stiffness and its anisotropy. The micro-mechanism results indicate that the primary particle breakage during the shearing process occurs in the vertical direction. That is, the particle breakage weakens the strong contact force in the vertical direction, leading to a redistribution of the strong contact forces from the vertical direction to other directions. This redistribution mainly manifests in a decrease in the anisotropy of contact normal and contact vector within the sample, as well as a decrease in the proportion of strong contact forces in the overall contacts. This, in turn, reduces the shear strength and stiffness of calcareous sand, particularly in the vertical direction, and results in a decrease in the maximum shear modulus and its anisotropy. The maximum reduction can be up to 50% of the original value. These insights can provide a certain theoretical support for the uneven displacement and long-term stability of calcareous sand for islands and reefs.

Investigation of the dense granular soil behaviour inside and outside the shear band: a three-dimensional DEM-FDM study

The present paper aims to compare the behaviour of dense granular soils inside and outside the Shear Band (SB) during the Plane Strain Compression (PSC) test using the Discrete Element Method (DEM). The flexible membrane in the PSC test was modelled by adopting a discrete element–finite difference (DEM-FDM) hybrid approach. In contrast with previous 2D studies, the present study employed 3D simulations to consider the effects of out-of-plane characteristics such as the intermediate principal stress. According to the findings, the minimum and intermediate principal stresses outside the SB were twice as high as those inside at the residual state. It was also found that mechanical coordination numbers and redundancy indexes both inside and outside the SB decreased up to the onset of the residual phase, with a higher reduction rate inside the SB. As far as fabric anisotropy is concerned, the degree of anisotropy increased both in and out the SB until the peak state, with a somewhat higher rate inside the SB. A further finding was that the distribution of normal contact forces outside the SB remained symmetrical around the vertical axis after shear banding, however, it became asymmetrical inside the SB.

Micromechanical modeling of hollow cylinder torsional shear test on sand using discrete element method

Previous studies on the hollow cylinder torsional shear test (HCTST) have mainly focused on the macroscopic behavior, while the micromechanical responses in soil specimens with shaped particles have rarely been investigated. This paper develops a numerical model of the HCTST using the discrete element method (DEM). The method of bonded spheres in a hexagonal arrangement is proposed to generate flexible boundaries that can achieve real-time adjustment of the internal and external cell pressures and capture the inhomogeneous deformation in the radial direction during shearing. Representative angular particles are selected from Toyoura sand and reproduced in this model to approximate real sand particles. The model is then validated by comparing numerical and experimental results of HCTSTs on Toyoura sand with different major principal stress directions. Next, a series of HCTSTs with different combinations of major principal stress direction (α) and intermediate principal stress ratio (b) is simulated to quantitatively characterize the sand behavior under different shear conditions. The results show that the shaped particles are horizontally distributed before shearing, and the initial anisotropic packing structure further results in different stress–strain curves in cases with different α and b values. The distribution of force chains is affected by both α and b during the shear process, together with the formation of the shear bands in different patterns. The contact normal anisotropy and contact force anisotropy show different evolution patterns when either α or b varies, resulting in the differences in the non-coaxiality and other macroscopic responses. This study improves the understanding of the macroscopic response of sand from a microscopic perspective and provides valuable insights for the constitutive modeling of sand.

Investigation on the rheological properties of dense granular inertial flow based on the contact features of four dual-partitioned subnetworks

The rheological properties of shear granular system under inertial flow regimes are investigated by fully considering both strong/weak and sliding/non-sliding contact features of four dual-partitioned subnetworks. Discrete element simulation results reveal that strong non-sliding subnetwork plays a dominant role in bearing external load. For strong sliding and weak non-sliding subnetworks, their anisotropy principal directions of normal contact forces undergo dynamic deflections during evolution, resulting in the lateral support of microstructures being primarily borne by unstable sliding contacts. Although the subnetworks display substantial variations in stability, their strong spatial correlation makes these variations complement each other, forming an intrinsic coupling relationship.

Numerical study on the shear strength of granular materials under the low confining pressure

The relationship between stress and strength in granular soil differs under low confining pressure as compared to high confining pressure, which is of great importance for the geotechnical engineering. Numerical simulations using discrete element method (DEM) are conducted to model consolidated-drained triaxial compression tests, with the confining pressure σc ranging from 1 to1000 kPa. From the analysis of the shear strength of granular soils under various confining pressures and void ratios, it is found that the peak stress ratio ηp under a low σc is significantly lower than that under the high σc, particularly for samples with the same initial void ratio. As the confining pressure increases, ηp for samples with the same initial void ratio increases up to 29 % under the low σc while stabilizing under the high σc. Conversely, ηp for samples with the same void ratio after the consolidation can increase up to 13.5 % under the low σc while declining under the high σc. In addition, the critical confining pressure distinguishing the low and high confining pressures is about 200 kPa for dense samples with initial void ratio of 0.605 used in the study, and it increases as the sample void ratio increases. From the microscopic analysis, the variation of ηp under the low σc can be attributed to the enhanced anisotropy of normal contact force and contact normal under the low confining pressure. Finally, a modified Mohr-Coulomb failure criterion is proposed, which can determine the strength of granular soils under the low confining pressure with different void ratios.