Hertzian Contact Mechanics Research Articles

Hydromechanical properties of Fontainebleau sandstone: Experimental determination and micromechanical modeling

We measured ultrasonic velocity, hydraulic permeability, and specific storage capacity of eight blocks of Fontainebleau sandstone, a well‐sorted, medium‐grained, almost pure quartz‐sandstone of Oligocene age, covering a range in connected porosity from 3% to 10% and varying significantly in pore geometry for a given porosity. Ultrasonic P‐wave velocity measured on water‐saturated samples covers the full range predicted by variational Hashin‐Shtrikman bounds. Permeability and specific storage were determined by the linear pressurization method at different effective pressures up to 180 MPa and room temperature. The permeability of tested samples varies from 10−13 m2 down to 10−20 m2, depending primarily on porosity and pore geometry, and subordinately on effective pressure. Specific storage capacity always exceeds the contribution of pore fluid compression. The excess corresponds to the contribution of pore deformability controlled by pore geometry and matrix material properties. Since the sandstone samples are composed of a single solid component, quartz, we were able to calculate various poroelastic parameters (Biot‐Willis and Skempton coefficients, drained and undrained bulk moduli) crucial for estimating hydromechanical coupling in fluid‐saturated aggregates. The micromechanical modeling of the effective pressure sensitivity of the hydraulic transport and poroelastic parameters by the contiguity model of Takei (1998) yields good agreement between static and dynamic elastic parameters for saturated samples, partial agreement between the pressure dependence of effective elastic parameters and Hertzian contact mechanics, and an explanation for pressure‐insensitive permeability of samples with the highest porosity.

Elastic contact between self-affine surfaces: comparison of numerical stress and contactcorrelation functions with analytic predictions

Contact between an elastic manifold and a rigid substrate with a self-affine fractalsurface is reinvestigated with Green’s function molecular dynamics. The Fouriertransforms of the stress and contact autocorrelation functions are found to decrease as|q|−μ where q is the wavevector. Upper and lower bounds on the ratio of the two correlationfunctions are used to argue that they have the same scaling exponentμ. Analysis ofnumerical results gives μ = 1+H, where H is the Hurst roughness exponent. This is consistent with Persson’s contact mechanics theory, while asperitymodels give μ = 2(1+H). The effect of increasing the range of interactions from a hard sphere repulsion toexponential decay is analyzed. Results for exponential interactions are accurately describedby recent systematic corrections to Persson’s theory. The relation between thearea of simply connected contact patches and the normal force is also studied.Below a threshold size the contact area and force are consistent with Hertziancontact mechanics, while area and force are linearly related in larger contactpatches.

Experimental Identification of the Nonlinear Parameters of an Industrial Translational Guide for Machine Performance Evaluation

Prediction of machine dynamics at the design stage is a challenge due to lack of adequate methods for identifying and handling the nonlinearities in the machine joints, which appear as the nonlinear restoring force function of relative displacement and relative velocity across the joint. This paper discusses identification of such a nonlinear restoring force function for an industrial translational guide for use with the Nonlinear Receptance Coupling Approach (NLRCA) to evaluate machine dynamic characteristics. Translational guides are among the most commonly used joints in machine tools. Both parametric and nonparametric techniques have been employed to identify the nonlinearities. A novel parametric model based on Hertzian contact mechanics has been derived for the translational guide. A nonparametric method based on two-dimensional Chebyshev polynomials is also used. The models derived from the two techniques, i.e., parametric and nonparametric, are fitted to the experimental data derived from static and dynamic tests to get the restoring force as a function of relative displacement and relative velocity across the joint. The nonlinear representation obtained from both techniques is later converted into the describing function representation which is needed for evaluation of machine dynamic characteristics using the NLRCA. The describing function representations obtained from the two approaches are compared. The design of experiments for evaluating the nonlinearities in such industrial machine tool joints is a challenge, requiring careful alignment and calibration, because they are typically very stiff. This constrains the dynamic experiments to be carried out at high frequencies (e.g. 2000—7000 Hz) where the experimental readings are very sensitive to errors in geometry and calibration.

Generalized Hertzian Model for the Deformation and Cracking of Colloidal Packings Saturated with Liquid

The process of drying colloidal dispersions generally produces particulate solids under stress as a result of capillary or interparticle forces. The derivation of a constitutive relation on the basis of Hertzian contact mechanics between spheres provides a model for quantitatively predicting the conditions under which close-packed colloidal layers form continuous void-free films or homogeneous porous films or crack under tensile stresses.

A numerical study on pile-up in nanoindentation creep

In this paper, we simulate materials of variable hardness, work hardening exponent, χ = ∂1nσ/∂1nε of 0.1?0.3 and a strain rate sensitivity of von Mises stress, vσ = ∂1nσ/∂1nε of 0.08?0.16 to estimate pile-ups and sink-ins during indentation creep and compare the results with literature data. We report that materials pile-up for vΗ/vσ ≥ 1.0 and sink-in for vΗ/vσ < 1.0, vσ is the strain rate sensitivity of the hardness. vΗ/vσ approaches 1 for small Η/Ε*, i.e., soft materials and deformation is dominated by pile-up, whereas when vΗ/vσ approaches 0 for large Η/Ε* corresponds to purely elastic deformation which is apparently dominated by sink-in in a manner prescribed by Hertzian contact mechanics.

Development of synergy model for erosion–corrosion of carbon steel in a slurry pot

A semi-empirical model (model 1) has been developed for synergy of corrosion dominated, non-passivating, erosion–corrosion of carbon steel in a slurry pot, developed from a stripping and recovery model to yield an expression for enhanced corrosion due to erosion damage revealing the underlying oxide–free surface. Dynamic Hertzian contact mechanics is used to model damage during erodent particle impact, as well as the incorporation of squeeze film and collision efficiency effects. The model also accommodates the effect of the erodent deforming the surface leading to increased corrosion activity. Experimental erosion data appear to be consistent with classical erosion behaviour as predicted by the kinetic energy of the erodent model. The experimental erosion–corrosion data show a more complex relationship due to the action of corrosion and its interaction with erosion, ranging from corrosion dominant to an equal influence of erosion and corrosion. Comparisons between modelled and experimental data indicate the nascent surface has a corrosion activity ranging from 30 to 100 times that of the unaffected surface, depending on experimental parameter studied. An empirical approach (model 2) indicates that experimental synergy correlates with a medium synergy level. This model (model 1) represents a step towards modelling the complex interactions of erosion and corrosion.

Nanoplastic deformation of nanoindentation: Crystallographic dependence of displacement bursts

The present paper summarizes the crystallographic dependence of the displacement bursts in nanoindentation using single crystalline aluminum and copper with three kinds of surface indices, namely (0 0 1), (1 1 0) and (1 1 1). It is found that the experimental linear relation between the critical indent load and the burst width of the indent depth at the first displacement burst has significant crystallographic dependence. From the critical indent loads, the critical resolved shear stresses of the dislocation nucleation were estimated to be 3.3 GPa for Al and 3.6 GPa for Cu, using Hertz contact mechanics, which are both close to the ideal values. We explain the nanoplastic mechanics by a comprehensive energy balance model to describe the linear relation between the indent load and the burst width, and by the collective dislocation nucleation model consisting of three-dimensional dislocation loops to evaluate the number of dislocations nucleating. The former model can derive the linear relation qualitatively and link the burst width to the collective dislocation slipping. The latter relates the theoretical critical load and burst width with the function of the punch radius and the distance between dislocations nucleating d D, in which setting d D as about 10 b ( b is the Burgers vector) can fairly represent the Al experimental data.

Nanomechanical properties of individual chondrocytes and their developing growth factor-stimulated pericellular matrix

The nanomechanical properties of individual cartilage cells (chondrocytes) and their aggrecan and collagen-rich pericellular matrix (PCM) were measured via atomic force microscope nanoindentation using probe tips of two length scales (nanosized and micron-sized). The properties of cells freshly isolated from cartilage tissue (devoid of PCM) were compared to cells that were cultured for selected times (up to 28 days) in 3-D alginate gels which enabled PCM assembly and accumulation. Cells were immobilized and kept viable in pyramidal wells microfabricated into an array on silicon chips. Hertzian contact mechanics and finite element analyses were employed to estimate apparent moduli from the force versus depth curves. The effects of culture conditions on the resulting PCM properties were studied by comparing 10% fetal bovine serum to medium containing a combination of insulin growth factor-1 (IGF-1)+osteogenic protein-1 (OP-1). While both systems showed increases in stiffness with time in culture between days 7 and 28, the IGF-1+OP-1 combination resulted in a higher stiffness for the cell-PCM composite by day 28 and a higher apparent modulus of the PCM which is compared to the FBS cultured cells. These studies give insight into the temporal evolution of the nanomechanical properties of the pericellar matrix relevant to the biomechanics and mechanobiology of tissue-engineered constructs for cartilage repair.

Implications of the Contact Radius to Line Step (CRLS) Ratio in AFM for Nanotribology Measurements

Investigating the mechanisms of defect generation and growth at surfaces on the nanometer scale typically requires high-resolution tools such as the atomic force microscope (AFM). To accurately assess the kinetics and activation parameters of defect production over a wide range of loads (F(z)), the AFM data should be properly conditioned. Generally, AFM wear trials are performed over an area defined by the length of the slow (L(sscan)) and fast scan axes. The ratio of L(sscan) to image resolution (res, lines per image) becomes an important experimental parameter in AFM wear trials because it defines the magnitude of the line step (LS = L(sscan)/res), the distance the AFM tip steps along the slow scan axis. Comparing the contact radius (a) to the line step (LS) indicates that the overlap of successive scans will result unless the contact radius-line step ratio (CRLS) is < or =(1)/(2). If this relationship is not considered, then the scan history (e.g., contact frequency) associated with a single scan is not equivalent at different loads owing to the scaling of contact radius with load (a proportional variant F(z)(1/3)). Here, we present a model in conjunction with empirical wear tests on muscovite mica to evaluate the effects of scan overlap on surface wear. Using the Hertz contact mechanics definition of a, the CRLS model shows that scan overlap pervades AFM wear trials even under low loads. Such findings indicate that simply counting the number of scans (N(scans)) in an experiment underestimates the full history conveyed to the surface by the tip and translates into an error in the actual extent to which a region on the surface is contacted. Utilizing the CRLS method described here provides an approach to account for image scan history accurately and to predict the extent of surface wear. This general model also has implications for any AFM measurement where one wishes to correlate scan-dependent history to image properties as well as feature resolution in scanned probe lithographies.

Effects of Rolling Friction of the Balancing Balls on the Automatic Ball Balancer for Optical Disk Drives

This study is devoted to evaluating the performance of an automatic ball-type balance system (ABB) installed in optical disk drives (ODDs) with consideration of the rolling friction between the balancing balls and the ball-containing race of the ABB. Research has been conducted to study the performance of the ABB by investigating the nonlinear dynamics of the system; however, the model adopted to describe the rolling friction between the balancing balls and their race was a simple stick-slip type, which does not reflect the realistic contact dynamics, leading to an inaccuracy in predicting ABB performance. In this study, a complete dynamic model of the ABB including a detailed rolling friction model for the balls based on Hertzian contact mechanics and hysteresis loss is established. The method of multiple scales is then applied to formulate a scaled model to find all possible steady-state ball positions and analyze stabilities. It is found that possible steady-state residing positions of the ball inside the race are multiple and form continuous ranges. Numerical simulations and experiments are conducted to verify the theoretical findings, especially for the rolling friction model. The obtained results are used to predict the level of residual vibration, with which the guidelines on dimension design and material choices of the ABB are distilled to achieve desired performance.

The effect of autocorrelation length on the real area of contact and friction behavior of rough surfaces

Autocorrelation length (ACL) is a surface roughness parameter that provides spatial information of surface topography that is not included in amplitude parameters such as root-mean-square roughness. This paper presents a relationship between ACL and the friction behavior of a rough surface. The influence of ACL on the peak distribution of a profile is studied based on Whitehouse and Archard’s classical analysis [Whitehouse and ArchardProc. R. Soc. London, Ser. A 316, 97 (1970)] and their results are extended to compare profiles from different surfaces. The probability density function of peaks and the mean peak height of a profile are given as functions of its ACL. These results are used to estimate the number of contact points when a rough surface comes into contact with a flat surface, and it is shown that the larger the ACL of the rough surface, the less the number of contact points. Based on Hertzian contact mechanics, it is shown that the real area of contact increases with increasing of number of contact points. Since adhesive friction force is proportional to the real area of contact, this suggests that the adhesive friction behavior of a surface will be inversely proportional to its ACL. Results from microscale friction experiments on polished and etched silicon surfaces are presented to verify the analysis.

Crystallite coalescence during film growth based on improved contact mechanics adhesion models

Intrinsic tensile stress, which can lead to problems in deposited thin films such as cracking, peeling, and delamination, often develops during the early stages of thin film growth. Many attempts have been made to estimate the tensile stress during crystallite coalescence, both experimentally and analytically. Most recently, using a combination of Hertzian contact mechanics and elasticity theory, Freund and Chason applied the Johnson-Kendall-Roberts (JKR) theory to account for adhesion between crystallites under specific conditions. Other existing contact mechanics models that naturally account for adhesion include the improved Derjaguin-Muller-Toporotov and Maugis-Dugdale theories. The objective of this study is to provide useful analytical and numerical techniques based on these contact mechanics theories for a wide range of conditions that accurately approximate the intrinsic tensile stress that develops during crystallite coalescence. As an analytical method, the Maugis-Dugdale model is proposed as a more general alternative to the JKR model. Parameters such as the contact radius and “net” adhesive force are computed as a function of the relative separation between two adjacent crystallites in a thin gold film. Another useful parameter known as the “jump-to-contact” separation is also calculated by the Maugis-Dugdale and JKR models. For comparison to the analytical models, a finite element method is used to simulate the crystallite coalescence problem. The numerical technique is based on a nonlinear surface interaction element developed to approximate van der Waals adhesion, and allows for full-field analysis of stress and displacement in crystallites. Two different boundary conditions are used, for which corresponding contact radius and tensile stress are computed and compared to the analytical results. As a further study, the length scale effect is also investigated by varying the radius of individual crystallites from 20nm to 300nm. It is concluded that in order to estimate the average tensile stress accurately using analytical models, the radius of individual crystallites must be large compared to the contact radius. For small length scales, the finite element approach is more appropriate.

Spherical Load Indentation in Submicron NiTiCu Shape Memory Thin Films

ABSTRACTNanoindentation with spherical tipped indenters provides a powerful technique to explore surface and thin film mechanical properties through the application of Hertzian contact mechanics. The full range of mechanical response can be obtained from elastic, through the yield point, to permanent deformation. In this study spherical indentation has been used for probing MBE-grown NiTiCu alloy thin films into superelasticity or stress-induced martensitic transformation. By this way, obstacles typically occurring related to the fabrication of freestanding films (film thickness &lt; 1 μm) are avoided. The indentation measurements were performed starting from the parent austenite state. Notably, for loads as small as 0.5 mN, deformation appears to be completely reversible. As loading is increased (up to 5 mN) the indent becomes irreversible following local plastic deformation within the tip-specimen contact area. Using finite-element simulations the indentation data were converted into a stress-strain diagram aimed at simulating uniaxial tension load. Therefrom, the superelastic strain is estimated to be around 3%.

Elastic analysis for liquid-bridging induced contact

In the present paper, micro-scale contact induced by liquid bridging is studied numerically via FEM with consideration of coupling effect between the liquid bridging and the elastic deformation of the half space. The variations of the liquid-bridging force and contact area as functions of a dimensionless parameter χ (measurement of relative stiffness of the contact bodies) and the filling angle ψ (measurement of liquid volume) are obtained and discussed. The FEM numerical solution is compared with two approximate analytical solutions based on Hertzian contact mechanics. It is found that the dimensionless liquid bridging force and contact area obtained via the approximate solutions are much smaller than the values obtained via FEM under certain conditions. The error inhabited in the approximate solution is due to the coupling effect between the liquid-bridging force and the elastic deformation of the half space.

Determination of Elastic Moduli of Thin Layers of Soft Material Using the Atomic Force Microscope

We address three problems that limit the use of the atomic force microscope when measuring elastic moduli of soft materials at microscopic scales. The first concerns the use of sharp cantilever tips, which typically induce local strains that far exceed the linear material regime. We show that this problem can be alleviated by using microspheres as probes, and we establish the criteria for their use. The second relates to the common use of the Hertz contact mechanics model, which leads to significant errors when applied to thin samples. We develop novel, simple to use corrections to apply for such cases. Samples that are either bonded or not bonded to a rigid substrate are considered. The third problem concerns the difficulty in establishing when contact occurs on a soft material. We obtain error estimates for the elastic modulus resulting from such uncertainty and discuss the sensitivity of the estimation methods to error in contact point. The theoretical and experimental results are compared to macroscopic measurements on poly(vinyl-alcohol) gels.

Quantitative imaging of nanoscale mechanical properties using hybrid nanoindentation and force modulation

In this article, we present a quantitative stiffness imaging technique and demonstrate its use to directly map the dynamic mechanical properties of materials with nanometer-scale lateral resolution. For the experiments, we use a “hybrid” nanoindenter, coupling depth-sensing nanoindentation with scanning probe imaging capabilities. Force modulation electronics have been added, enhancing instrument sensitivity and enabling measurements of time dependent materials properties (e.g., loss modulus and damping coefficient) not readily obtained with quasi-static indentation techniques. Tip–sample interaction stiffness images are acquired by superimposing a sinusoidal force (∼1 μN) onto the quasi-static imaging force (1.5–2 μN), and recording the displacement amplitude and phase as the surface is scanned. Combining a dynamic model of the indenter (having known mass, damping coefficient, spring stiffness, resonance frequency, and modulation frequency) with the response of the tip–surface interaction, creates maps of complex stiffness. We demonstrate the use of this approach to obtain quantitative storage and loss stiffness images of a fiber-epoxy composite, as well as directly determine the loss and storage moduli from the images using Hertzian contact mechanics. Moduli differences as small as 20% were resolved in the images at loads two orders of magnitude lower than with indentation, and were consistent with measurements made using conventional quasi-static depth-sensing indentation techniques.

Interfacial Force Microscopy and its Application in Metal Matrix Composites

AbstractMetal matrix composites (MMCs) combine the properties of metal and ceramic or intermetallic materials. Common examples of metal matrix composites are Cu-Al2O3, SiCw-Al, Al-Al2O3, Al-B4C, and Ni-NiAl3. Mechanical or thermal properties, such as strain-stress behavior, or thermal expansion coefficient can be tailored by changing the content of the reinforcing phase. The most common techniques of measuring mechanical properties of composite materials rely on macroscopic approach. During the past fifteen years, a significant effort has been made to develop various techniques of measuring mechanical properties on a microscopic level. These techniques include atomic force microscope (AFM) and depth sensing indentation techniques, based on Hertzian contact mechanics. However, it is still a challenge to measure reliably and quantitatively the Young's modulus and Poisson's ratio of individual phases as well as properties at the interfaces. This presentation will focus on fundamental aspects of measuring of mechanical properties of metal matrix composites at nano-scale using Interfacial Force Microscopy (IFM). The IFM is a scanning probe microscope, which utilizes a unique self-balancing capacitance force sensor. Force-displacement curves obtained with the IFM are analyzed using Hertzian contact mechanics to extract the Young's moduli of the individual phases and interface region with high spatial resolution. The properties of Cu-Al2O3, Al-SiCp composites will be discussed in detail. Furthermore, a comparison of experimental data with mechanical properties calculated from first principles will be discussed.

Time domain simulation of railway noise

The most difficult problem concerned with railway noise generation is the process occurring in the contact region during rolling. The following computation efficient time domain model determines the vertical rail deflection by use of Green’s function of a periodically supported beam. Nonlinear Hertzian contact mechanics are used, less realistic, however, than a discretized contact region allowing for rough surfaces. The wheel is as yet a rigid mass. More advanced contact and wheel models can readily be included later. Calculation results include the following observations. The contact force peaks sharply precisely above the support points, followed by a short loss of rail–wheel contact, also for smooth rails. Vibrations of the rail pinned–pinned mode are impossible to eliminate by removing surface roughnesses, because the excitation is above all caused by forward wheel velocity together with discrete supports. The response amplitude increases with forward velocity. Optimum pad stiffness depends on corrugation amplitude. These new findings should be considered when trying to noise optimize track constructions.

Relationships between acoustic emission signals and physical phenomena during indentation

A commercial piezoelectric acoustic emission transducer has been used in conjunction with nanoindentation techniques to study the relationship between acoustic emission signals and discrete physical events to identify the type and strength of an event. Indentations into tungsten and iron single crystals have been used to study dislocation generation and passive film failure. In addition, indentations made into a thin nitride film on sapphire have been used to cause film delaminations. Parameters such as signal rise time and frequency for a piezoelectric sensor are related to sample geometry, and not to the type of event which caused the acoustic emission signal. As a possible calibration for acoustic emission sensors, the most meaningful parameter is the acoustic emission energy, which has been shown to scale with the elastic energy released during the event. The measured values of elastic energy released correspond very closely to those calculated using Hertzian contact mechanics.

Evaluation of the Effective Radius of Spherical Indenters for Ultra-Microindentation

To determine the mechanical properties of materials, using spherical ultra-microindentation techniques, the geometry of the part of the indenter in contact with the surface must be well known. However, crystallographic anisotropy ensures that nominally spherical diamond indenters with very small tip radii will not be truly spherical. This paper examines the shape of typical indenters, with nominal tip radii of 5 μm and 10 μm. Atomic force microscopy techniques are used to determine the general form and then using a technique based on Hertzian contact mechanics in combination with load/penetration data obtained by indentation, to determine the effective radius of the part of the indenter in actual contact with the surface.