Hertzian Contact Mechanics Research Articles

Compressive elastic moduli and polishing performance of non-rigid core/shell structured PS/SiO2 composite abrasives evaluated by AFM

The core/shell structured polystyrene (PS)/SiO2 composite microspheres with different silica shell morphology were synthesized by a modified Stöber method. As confirmed by transmission electron microscopy (TEM), the rough discontinuous shell consisted of separate SiO2 nanoparticles for composite-A, while the smooth continuous one was composed of amorphous silica network for composite-B. Atomic force microscopy (AFM) was employed to probe the compressive Young's moduli (E) and chemical mechanical polishing (CMP) performances of the as-prepared PS/SiO2 composite microspheres. On the basis of the Hertzian contact mechanics, the calculated E values of the PS microspheres, composite-A and composite-B were 2.9±0.4, 5.1±1.2 and 6.0±1.2GPa, respectively. Compared to traditional abrasives, thermally grown silicon oxide wafers after polished by the core/shell PS/SiO2 composite abrasives obtained a lower root mean square roughness and a higher material removal rate value. In addition, there is an obvious effect of shell morphology of the composites on oxide CMP performance and structural stability during polishing process. This approach would provide a basis for understanding the actual role of organic/inorganic core/shell composite abrasives in the material removal process of CMP.

Mechanical characterization of living and dead undifferentiated human adipose-derived stem cells by using atomic force microscopy

In this article, to map the mechanical properties of undifferentiated human adipose-derived stem cells, local mechanical characterization is carried out on the adipose-derived stem cells. In addition, to distinguish the living and dead human adipose-derived stem cells, mechanical characterization is also implemented on both living and dead adipose-derived stem cells. In this study, Young's modulus of the cell membrane is used for representing the mechanical properties of cells. To obtain Young's modulus of cell membrane, the force-spectroscopy mode of atomic force microscopy is employed to measure the atomic force microscopy tip indentation depth and force on the cells. Then, Young's modulus is obtained through fitting these experimental data to the Hertzian contact mechanics model. The global Young's moduli of living and dead undifferentiated adipose-derived stem cells are about 1.27 and 18.61 kPa, respectively. This displays obvious gap of Young's modulus between the living and dead undifferentiated adipose-derived stem cells. Finally, comparison of the local Young's modulus shows deviation of the local Young's modulus for either living or dead undifferentiated adipose-derived stem cells, and the root-mean-square errors of the global Young's modulus of living and dead undifferentiated adipose-derived stem cells are about 0.48 and 5.05 kPa, respectively.

Relationship between single and bulk mechanical properties for zeolite ZSM5 spray-dried particles

In this work typical mechanical properties for a catalyst support material, ZSM5 (a spray-dried granular zeolite), have been measured in order to relate the bulk behaviour of the powder material to the single particle mechanical properties. Particle shape and size distribution of the powders, determined by laser diffraction and scanning electron microscopy (SEM), confirmed the spherical shape of the spray-dried particles. The excellent flowability of the material was assessed by typical methods such as the Hausner ratio and the Carr index. This was confirmed by bulk measurements of the particle–particle internal friction parameter and flow function using a Schulze shear cell, which also illustrated the low compressibility of the material. Single particle compression was used to characterize single particle mechanical properties such as reduced elastic modulus and strength from Hertz contact mechanics theory. Comparison with surface properties obtained from nanoindentation suggests heterogeneity, the surface being harder than the core. In order to evaluate the relationship between single particle mechanical properties and bulk compression behaviour, uniaxial confined compression was carried out. It was determined that the Adams model was suitable for describing the bulk compression and furthermore that the Adams model parameter, apparent strength of single particles, was in good agreement with the single particle strength determined from single particle compression test.

Model for the resistance force acting on circular bodies in the imminence of rolling

The laws of friction are reasonably well understood for the case of blocks in contact with rough plane surfaces. However, as far as bodies with circular sections are concerned, the physics of friction becomes more involving and it is not possible to adopt a simple conceptual framework to explain all phenomena. In particular, there is no approach so far to the problem of the resistance force that opposes to circular bodies that remain at rest while acted upon by small forces. Here we fill this gap by introducing a mechanical model based on both the elasticity theory and Hertz contact mechanics. Our approach furnishes a quantitative expression for the critical force beyond which rest can no longer be maintained. Besides confirming the expected proportionality of the resistance force with the load, our result contains no free parameters and is expressed solely in terms of physical properties of the problem, such as the pressure of the body per unit of superficial area, a relation between the Young modulus of the surface and its Poisson ratio, and the symmetry of the contact.

The Impact of Fibre Surface Morphology on the Effective Thermal Conductivity of a PEM Fuel Cell Gas Diffusion Layer

In this work, the preliminary findings for determining the effect of fibre surface morphology on the effective thermal conductivity of the gas diffusion layer of a polymer electrolyte membrane fuel cell is presented. Atomic force microscopy was used to measure the fibre surface roughness and asperity height distribution for various fibres and locations along each fibre for Toray carbon paper TGP-H-120. The thermal contact resistance was measured using preliminary methods, which will be further advanced in the near future. Hertzian contact mechanics was used to validate the results obtained, and to compare with smooth fibre thermal contact resistance for rough and smooth fibres. The thermal contact resistance was determined for various angles of fibre orientation and contact forces; it was determined that values for rough fibre contact are on average twice that of the contact of smooth fibres.

Material removal during abrasive impregnated brush deburring of micromilled grooves in NiTi foils

While abrasive impregnated brushes have been used extensively for deburring of machined parts, understanding of the associated mechanics of material removal is lacking. This paper investigates the dependence of brush deburring rate, surface roughness and change in foil thickness on spindle speed and brush nominal penetration during brush deburring of 25μm thick Nitinol (NiTi) foils. Analysis of the surface after brushing shows that both ploughing and chip formation occur in the process. It is also seen that at bristle tip velocities of about 7m/s the penetration of abrasives into the foil is dominated by initial contact forces while at about 2m/s it is dominated by bristle stiffness. Long, slender burrs are removed by fatigue fracture under cyclic loading of the bristles. Burr roots are removed in small chunks as cracks formed by abrasive indentation propagate under cyclic loading of the bristles. A model based on Hertzian contact mechanics is proposed to predict the rate of indentation during brushing and is validated using experiments. The surface roughness initially reduces but subsequently increases with further brushing. Change in foil thickness in all cases is ≤1μm. A fracture mechanics model for predicting the deburring rate is also proposed and validated.

Deformable cell–cell and cell–substrate interactions in semi-infinite domain

Leukocyte trafficking in the microvasculature during inflammatory response is known to involve multiple adhesion molecules and is referred to as the leukocyte adhesion cascade (LAC). Surface-bound selectins and their respective ligands are primarily responsible for tethering and rolling of leukocytes over inflamed endothelium. Numerical modeling of this response is challenging due to the nature of cell–cell interactions in Stokes flow (i.e., large domain of influence for each cell over its neighbors). Here, we discuss a novel simulation capable of modeling several steps of the LAC. The new model includes relevant contact and lubrication forces and extends a physics-based model for single particle rolling interactions developed by Hammer and Apte (1992), for multiparticle interactions by King and Hammer (2001a), and for deformable particles by Gee and King (2006). We initially demonstrate the model for cell–cell collisions occurring near a planar substrate, and for cell–substrate adhesive interactions. The adhesion studies provide a new perspective of the contribution of Hertzian contact mechanics toward variations in contact area at the cell–substrate interface. The results confirm that interfacial contact area will increase as a result of the contact formulation and that this mechanism may enhance cell rolling interactions for cells driven toward endothelium by cell–cell collisions. As a result of cell compliance, rolling velocity may decrease significantly, compared to non-compliant cells.

Numerical modeling of force and contact networks in fragmented sea ice

AbstractIn this paper, a molecular-dynamics sea-ice model is used to study contact and force networks in fragmented sea ice, composed of separate floes with power-law size distribution. The momentum equations for individual floes, taking into account floe/floe collisions (with Hertzian contact mechanics), are formulated in a way suitable for a computationally efficient numerical algorithm, allowing simulation of systems of thousands of floes. The simulations are performed for a number of scenarios: pure convergence without wind, through a jamming phase transition; constant wind at a constant ice concentration; and an idealized marginal ice zone. An analysis of the statistical properties of the contact and force networks reveals a highly localized, intermittent character of internal stress in the ice, as well as the role of the size-dependent response of floes to the forcing in formation of spatial patterns of internal stress at lower ice concentrations. The results provide a valuable starting point for formulating improved rheology models for fragmented sea ice.

Quantifying the transverse deformability of double-walled carbon and boron nitride nanotubes using an ultrathin nanomembrane covering scheme

We investigate the characterization of the transverse deformability of double-walled carbon and boron-nitride nanotubes (i.e., DWCNTs and DWBNNTs) using an ultrathin nanomembrane covering scheme. Monolayer graphene oxide sheets (MGOSs) with a sub-nm thickness are used to cover individual double-walled nanotubes on flat substrates. Nanotube cross-section height reduction occurs due to the compression force exerted by the covering membrane, whose morphological conformation is governed by its bending/stretching rigidities and adhesion interaction with the substrate, as well as the radial height and rigidity of the underlying nanotube. The actual transverse deformation of the underlying tube and its effective radial modulus are quantified through interpreting the measured structural morphology of the covering membrane and the nanotube cross-section height reduction using nonlinear structural mechanics and Hertzian contact mechanics theories. The radial deformations in MGOS-covered tubes are found to positively correlate with the nanotube radial rigidity, thus, increasing with the nanotube outer diameter and decreasing with an increase of the number of tube walls. Our results reveal prominent radial strains of about 20% for DWCNTs of 3.55 nm in outer diameter, while about 24% for DWBNNTs of 3.85 nm in outer diameter. Our data about the effective radial moduli of individual DWCNTs and DWBNNTs are in reasonably good agreement with those obtained using atomic force microscopy-based compression methods. Our work shows that the nanomembrane covering scheme is promising as a quantitative technique for studying the radial rigidity of individual tubular nanostructures.

Indentation size effect and shear transformation zone size in a bulk metallic glass in two different structural states

The existence of an indentation size effect (ISE) in the onset of yield in a Zr-based bulk metallic glass (BMG) is investigated by employing spherical-tip nanoindentation experiments. Statistically significant data on the load at which the first pop-in in the displacement occurs were obtained for three different tip radii and in two different structural states (as-cast and structurally relaxed) of the BMG. Hertzian contact mechanics were employed to convert the pop-in loads to the maximum shear stress underneath the indenter. Results establish the existence of an ISE in the BMG of both structural states, with shear yield stress increasing with decreasing tip radius. Structural relaxation was found to increase the yield stress and decrease the variability in the data, indicating “structural homogenization” with annealing. Statistical analysis of the data was employed to estimate the shear transformation zone (STZ) size. Results of this analysis indicate an STZ size of ∼25 atoms, which increases to ∼34 atoms upon annealing. These observations are discussed in terms of internal structure changes that occur during structural relaxation and their interaction with the stressed volumes in spherical indentation of a metallic glass.

Photochemical Synthesis of Polymeric Fiber Coatings and Their Embedding in Matrix Material: Morphology and Nanomechanical Properties at the Fiber–Matrix Interface

In this contribution, we present a three-step pathway to produce a novel fiber coating, study its embedding in epoxy resin and characterize its nanomechanical properties at the interface between fiber and matrix. Inorganic surfaces were sulfhydrylated for subsequent use in thiol-initiated ene photopolymerization. The influence of water on the sulfhydrylation process was studied to find conditions allowing monomolecular deposition. Surface morphology as well as SH-content were evaluated by UV/vis spectroscopy, atomic force microscopy and spectroscopic ellipsometry. Brush-like polymer layers (PS and PMMA) were introduced by UV-light initiated surface polymerization of vinyl monomers. Polymer growth and morphology were studied. After embedding, the nanomechanics of the interfacial region of the fibers was studied. AFM force spectroscopy allowed the mapping of the stiffness distribution at the cross-section of the composite with high spatial resolution. Elastic moduli were determined by Hertzian contact mechanics. The individual phases of the composite material (fiber, interphase, and matrix) can be clearly distinguished based on their mechanical response. The synthesis, morphology, and mechanical properties of an interphase based on a polymeric graft-film swollen with matrix material are shown, and perspectives of these novel coatings for improved matrix-fiber compatibility and interfacial adhesion are discussed.

Load dependence of oxidative wear in metal/ceramic tribocouples in fretting environment

For a large number of non-oxide ceramic/metal as well as for non-oxide particulate reinforced composite/metal tribocouples, oxidative wear is reported to play a major role in the overall damage and material removal process. The designing of new materials with improved resistance to oxidative wear requires understanding of the combined effect of the material properties and operating parameters on the wear resistance. In this paper, an analytical model to predict the dependence of the oxidative wear volume on load and sliding velocity is presented. The model is developed using basic chemical kinetics and Hertzian contact mechanics, and is valid for the case of a stable oxide layer. Exponential dependence of wear volume on operating parameters (load, sliding velocity) is predicted in the proposed model. It has been shown to successfully explain the qualitative experimental load dependence of the wear volume for five ceramic/metal tribocouples (Al2O3/steel, WC–TiC–Co/steel, TiB2–MoSi2/steel, ASP23 steel/alumina and Ti6Al4V/alumina) that exhibit oxidative wear. The activation energy of the oxidation of metallic counter bodies under fretting conditions was also calculated using the model, which turns out to be significantly lower than in the static conditions.

Radial elasticity of multi-walled boron nitride nanotubes

We investigated the radial mechanical properties of multi-walled boron nitride nanotubes (MW-BNNTs) using atomic force microscopy. The employed MW-BNNTs were synthesized using pressurized vapor/condenser (PVC) methods and were dispersed in aqueous solution using ultrasonication methods with the aid of ionic surfactants. Our nanomechanical measurements reveal the elastic deformational behaviors of individual BNNTs with two to four tube walls in their transverse directions. Their effective radial elastic moduli were obtained through interpreting their measured radial deformation profiles using Hertzian contact mechanics models. Our results capture the dependences of the effective radial moduli of MW-BNNTs on both the tube outer diameter and the number of tube layers. The effective radial moduli of double-walled BNNTs are found to be several-fold higher than those of single-walled BNNTs within the same diameter range. Our work contributes directly to a complete understanding of the fundamental structural and mechanical properties of BNNTs and the pursuits of their novel structural and electronics applications.

Effects of Gel Thickness on Microscopic Indentation Measurements of Gel Modulus

In vitro, animal cells are mostly cultured on a gel substrate. It was recently shown that substrate stiffness affects cellular behaviors in a significant way, including adhesion, differentiation, and migration. Therefore, an accurate method is needed to characterize the modulus of the substrate. In situ microscopic measurements of the gel substrate modulus are based on Hertz contact mechanics, where Young's modulus is derived from the indentation force and displacement measurements. In Hertz theory, the substrate is modeled as a linear elastic half-space with an infinite depth, whereas in practice, the thickness of the substrate, h, can be comparable to the contact radius and other relevant dimensions such as the radius of the indenter or steel ball, R. As a result, measurements based on Hertz theory overestimate the Young's modulus. In this work, we discuss the limitations of Hertz theory and then modify it, taking into consideration the nonlinearity of the material and large deformation using a finite-element method. We present our results in a simple correction factor, ψ, the ratio of the corrected Young's modulus and the Hertz modulus in the parameter regime of δ/ h ≤ min (0.6, R/ h) and 0.3 ≤ R/h ≤ 12.7. The ψ factor depends on two dimensionless parameters, R/h and δ/h (where δ is the indentation depth), both of which are easily accessible to experiments. This correction factor agrees with experimental observations obtained with the use of polyacrylamide gel and a microsphere indentation method in the parameter range of 0.1 ≤ δ/ h ≤ 0.4 and 0.3 ≤ R/ h ≤ 6.2. The effect of adhesion on the use of Hertz theory for small indentation depth is also discussed.

Effect of formulation of alginate beads on their mechanical behavior and stiffness

The aim of this work was to determine the effect of formulation of alginate beads on their mechanical behavior and stiffness when compressed at high speed. The alginate beads were formulated using different types and concentrations of alginate and gelling cations and were produced using an extrusion-dripping method. Single wet beads were compressed at a speed of 40 mm/min, and their elastic limits were investigated, and the corresponding force versus displacement data were obtained. The Young's moduli of the beads were determined from the force versus displacement data using the Hertz's contact mechanics theory. The alginate beads were found to exhibit plastic behavior when they were compressed beyond 50% with the exception of copper–alginate beads for which yield occured at lower deformation. Alginate beads made of higher guluronic acid contents and gelling cations of higher chemical affinity were found to have greater stiffness. Increasing the concentration of alginate and gelling ions also generated a similar effect. At such a compression speed, the values of Young's modulus of the beads were found to be in the range between 250 and 900 kPa depending on the bead formulation.

Solution of the Contact Zone Orientation for Normal Elliptical Hertzian Contact

Many mechanical designs have parts that come into, or lose, contact with each other. When elastic bodies with second order surface geometries come into contact, the contact patch is expected to be approximately flat and to have an elliptical boundary. Classic Hertzian contact mechanics can be used to model such contacts, but since there is no closed-form analytical solution to predict the major and minor axes of the contact zone ellipse, approximate numerical methods have been developed, some of which are very accurate. Predictions of the mutual approach of the bodies and the contact pressure distribution can then be made. Although the shape of the contact ellipse has been modeled and solved for, to date there has been no solution for the orientation of the contact ellipse with respect to either of the contacting bodies. The contact ellipse orientation is needed in order to model the shear stress distributions that occur when sticking friction forces are developed and separate contact zones of sticking and slipping are expected. Using the results of a numerical solution for the conventional contact parameters, this paper presents an analytical solution of the orientation of the contact ellipse, which is shown to depend only on the curvatures and the relative orientation of the contacting bodies. In order to validate the analytical solution, the results are compared with those from ABAQUS™ finite element simulations for cases of identical bodies and bodies with dissimilar curvatures. The predictions of the contact ellipse orientation angles and the major and minor semi-axes agree very well for all cases considered.

Mapping elasticity moduli of atherosclerotic plaque in situ via atomic force microscopy

Several studies have suggested that evolving mechanical stresses and strains drive atherosclerotic plaque development and vulnerability. Especially, stress distribution in the plaque fibrous capsule is an important determinant for the risk of vulnerable plaque rupture. Knowledge of the stiffness of atherosclerotic plaque components is therefore of critical importance. In this work, force mapping experiments using atomic force microscopy (AFM) were conducted in apolipoprotein E-deficient (ApoE(-/-)) mouse, which represents the most widely used experimental model for studying mechanisms underlying the development of atherosclerotic lesions. To obtain the elastic material properties of fibrous caps and lipidic cores of atherosclerotic plaques, serial cross-sections of aortic arch lesions were probed at different sites. Atherosclerotic plaque sub-structures were subdivided into cellular fibrotic, hypocellular fibrotic and lipidic rich areas according to histological staining. Hertz's contact mechanics were used to determine elasticity (Young's) moduli that were related to the underlying histological plaque structure. Cellular fibrotic regions exhibit a mean Young modulus of 10.4±5.7kPa. Hypocellular fibrous caps were almost six-times stiffer, with average modulus value of 59.4±47.4kPa, locally rising up to ∼250kPa. Lipid rich areas exhibit a rather large range of Young's moduli, with average value of 5.5±3.5kPa. Such precise quantification of plaque stiffness heterogeneity will allow investigators to have prospectively a better monitoring of atherosclerotic disease evolution, including arterial wall remodeling and plaque rupture, in response to mechanical constraints imposed by vascular shear stress and blood pressure.

Measuring Viscoelasticity of Soft Samples Using Atomic Force Microscopy

Relaxation indentation experiments using atomic force microscopy (AFM) are used to obtain viscoelastic material properties of soft samples. The quasilinear viscoelastic (QLV) model formulated by Fung (1972, "Stress Strain History Relations of Soft Tissues in Simple Elongation," in Biomechanics, Its Foundation and Objectives, Prentice-Hall, Englewood Cliffs, NJ, pp. 181-207) for uniaxial compression data was modified for the indentation test data in this study. Hertz contact mechanics was used for the instantaneous deformation, and a reduced relaxation function based on continuous spectrum is used for the time-dependent part in the model. The modified QLV indentation model presents a novel method to obtain viscoelastic properties from indentation data independent of relaxation times of the test. The major objective of the present study is to develop the QLV indentation model and implement the model on AFM indentation data for 1% agarose gel and a viscoelastic polymer using spherical indenter.

Modeling Mixed-Lubrication of a Shoe-Floor Interface Applied to a Pin-on-Disk Apparatus

While slip-and-fall accidents are a serious health concern, few attempts have been made to tribologically model the shoe-floor-contaminant interface. To this end, modeling techniques are introduced here for shoe and floor materials operating in mixed lubrication. The proposed analytical model results are compared with experimental data in order to assess the validity of the developed model. Coefficient of friction (COF) values are generated using a pin-on-disk apparatus across a range of sliding speeds with two different shoe materials operating in the mixed-lubrication regime. The model solves for the contact condition using Hertzian contact mechanics theory and the hydrodynamic pressure condition using the Reynolds equation. The amount of contact deformation is adjusted iteratively such that the summed force from the fluid and contacting asperities is equivalent to the total normal force. The model predicts friction values based on the proportion of the load supported by the fluid versus the proportion of the load supported by contacting asperities. The model-generated COF-velocity plots showed close agreement with experimental values for both shoe materials studied. In addition, the model predicts that as the speed between the surfaces increases, the hydrodynamic lift increases. This in turn decreases the contact area and the load borne by the contacting surfaces. Hence, the model presented serves as an initial step towards developing shoe-floor-contaminant friction models.

Evaluation of a semi-empirical model in predicting erosion–corrosion

The phenomenon of erosion–corrosion has been studied extensively by various investigators but no accurate model has been developed to predict the interactions between erosion and corrosion. This is mainly attributed to the complexity of the interactions that generate either a synergistic or antagonistic wear effect for a particular material in a certain environment. A semi-empirical model has recently been developed at the University of Southampton which incorporates dynamic Hertzian contact mechanics to model the damage during particle impact and accommodates the effect of erodent deforming the surface leading to an increased corrosion activity. The model was found to have good agreement with erosion–corrosion rates of carbon steel. The aim of this paper is to evaluate the robustness of this semi-empirical model by testing it on a passive metal. UNS S31603 was chosen due to its inherent passivity to corrosion. A slurry pot erosion tester was used as the test rig to perform the experiments. It was found that this passive metal produces high synergistic levels when exposed to erosion–corrosion in 0.3 M HCl with variation in erodent concentrations and flow velocities. SEM and surface profilometry show typical ductile material behaviour with cutting mechanism and deformation mechanism occurring simultaneously. A wear map is presented and it is observed that the increase in velocity and sand concentration causes the material to shift from a corrosion–erosion dominated region to an erosion–corrosion dominated region. This paper will also evaluate the semi-empirical model and discuss its applicability in predicting erosion–corrosion.