Fractal Rough Surface Research Articles

Thermo-elastic-plastic deformation of semi-infinite medium under effects of fractal dimension and sliding speed

This study uses elastic–plastic contact theory to analyze the effects of surface roughness and sliding speed on thermo-mechanical deformation and frictional heating of elastic–plastic semi-infinite medium in sliding contact with the fractal rough surface. The evolution of deformation in the semi-infinite medium due to multi-scale surface roughness variation was interpreted in terms of temperature rise, contact pressure; normal and von Mises equivalent stresses using the finite element method. The effect of sliding speed on frictional heating and deformation using time change was also analyzed. This study shows that frictional heating is sensitive to fractal dimension, and sliding speed has a strong effect on frictional heating.

Stress-Dependent Electrical Contact Resistance at Fractal Rough Surfaces

AbstractThe electrical contact resistance between contacting rough surfaces was studied under various compressive stresses. The samples considered here were isotropically roughened aluminium disks with upper and lower surfaces modified through polishing and sand blasting using different sized glass beads. Fractal geometry and roughness descriptors, including root mean square values of roughness and slope, were used to describe the topography of sample surfaces, based on the digitized profiles obtained from interferometry-based profilometry. The electrical contact resistances at the interfaces were obtained by applying a controlled current and measuring the resulting voltage, through the following scenarios: (1) over time for various applied testing currents, the resistance relaxation curves were measured at constant loads; (2) through voltage-current characteristics by means of a logarithmic sweeping current, the influence of the testing current on the electrical response of contacting rough surfaces was ...

Normal contact stiffness of fractal rough surfaces

We used the fractal theory based on a single variable Weierstrass–Mandelbrot function to obtain the normal contact stiffness if rough and smooth isotropic surfaces are pressed against each other. Because in the original fractal theory the distribution of contact area is assumed geometrically, we propose the method in which the actual deformation of asperities and a correction due to asperity coupling (interaction) will be taken into account. This correction is equivalent to an increase of the effective separation by a quantity proportional to the nominal pressure and it has a significant effect on contact stiffness at larger normal loads (low separations). The numerical results demonstrate a nonlinear evolution of the contact stiffness with the normal load in particular in the first stage of loading at low squeezing pressures. We have compared the results of the theoretical contact stiffness using the fractal method with the experimental ultrasonic measurements. Experimental results made on real surfaces agree remarkably well with the theoretical predictions.

Effects of fractal surface roughness and lithology on single and multiphase flow in a single fracture: An experimental investigation

This paper presents qualitative and quantitative analysis of single and multiphase flow in a single fracture based on experimental results and demonstrates relationships between the roughness and fluid movement and distribution.Experiments were conducted on seven perfectly-matching and tightly-closed rough model fractures reproduced from the single fractures of lithologically different seven rock blocks that were jointed artificially through laboratory indirect tensile tests. Transparent upper and opaque lower walls of these models facilitated the visualization of the flow experiments. Rough surfaces of the model fractures were first digitized. Then, using the gathered data in variogram analysis, surface roughness was quantified by fractal dimension. Another roughness quantification parameter was also handled as the ratio between total fracture surface area and planar surface area. Experimental measurements of flow were quantitatively correlated to surface roughness under different normal loading (aperture) conditions. Also, constant rate immiscible displacement experiments were performed to assess the roughness effect represented by seven different lithologies and wettability effect controlled by the material used in manufacturing the fracture samples on the residual saturation development.

Study on Composite Electromagnetic Scattering from 1D Weierstrass Fractal Land Surface with Buried Target Using FDTD

1D band-limited weierstrass fractal rough surface is used to simulate the natural rough land surface, the composite electromagnetic scattering from 1D band-limited weierstrass fractal land surface with buried target is studied using FDTD, the angle distribution curves of the composite bistatic scattering coefficient is obtained by numerical calculation, the variation of the composite bistatic scattering coefficient with fractal Dimension, the root-mean-square of height fluctuation, the target size, the slope angle of target, the buried depth of target is calculated, which is analyzed and discussed in detail at the same time, the characteristics of the composite electromagnetic scattering from fractal land surface with buried plate shape target is obtained

A three-dimensional fusion prediction model for fractal rough surfaces in the sliding contact process

Purpose – The purpose of this manuscript is to analyze the fusion micro-zone generated by typical rough surfaces and investigate the factors of thermal effects on the tribological performance of surface asperities and its results verified by the experiment. Design/methodology/approach – A three-dimensional fractal rough surfaces sliding contact model has been developed, which takes into account temperature rise and distribution. The finite-element method, Green's function method, thermal conduct theory and contact mechanics are used as the solution methods. Findings – The results yield insights into the effects of the sliding velocity, thermal properties of the material, normal load and surface roughness on the temperature rise of the sliding contact surface. It allows the specification of working conductions' properties to reduce fusion. Originality/value – The model is developed and described by using the features of the contact between one flat surface and one rough surface with varied topographies. It can be easily applied for solving the sliding contact problems with different working conditions and specified for designing the surface accuracy in the severe working condition.

Determination of fractal rough surface of polypyrrole film: AFM and electrochemical analysis

Physico-chemical properties of conductive polymers depend on the morphology and characteristic of the synthesized film which emerge from synthesis conditions. For example, conductivity of polypyrrole is related to the fractal dimension of surface. Atomic force microscopy (AFM) is a powerful technique in revealing characteristic of surfaces. In the present research electrochemical techniques and atomic force microscopy are employed to analyze the surface morphology and parameters of surface such as interface width, kurtosis, Hurst exponent, fractal dimension and lateral correlation length for polypyrrole electrodeposited on glassy carbon (GC) electrode and relation between skewness and kurtosis were investigated. Height–height correlation function shows the correlation of heights in different points on the surface and can be calculated by analyzing AFM data and surface profile. Scanning electron microscopy (SEM) analysis carried out to confirm the AFM results. Cyclic voltammetry and electrochemical impedance spectroscopy methods are used to compare the electrochemical capacitance of films with different roughness. AFM images were used to estimate the surface area and results showed that the films synthesized in higher temperatures are rougher and have a larger surface area and according to the results of electrochemical experiments rougher surfaces with higher skews have larger electrochemical capacitance. The study of surface morphology and physico-chemical properties of polypyrrole film can provide suitable methods of synthesis of films with required properties.

Surface roughness effects in the catalytic behavior of vanadia supported on SBA-15

SBA-15 is a template-synthesized mesoporous silica that has found extensive use as a model support for catalytic studies. Thorough structural analyses describe a dual micropore–mesopore structure with a broad distribution of micropore size that we alternatively describe as fractal surface roughness. SBA-15 materials with varying surface roughness were systematically prepared followed by grafting with sub-monolayer coverage of vanadium oxide (VOx). VOx-SBA-15 samples were characterized using nitrogen adsorption, UV–vis spectroscopy, and Raman spectroscopy and tested in the catalytic partial oxidation of methanol to formaldehyde as well as propane to propene. SBA-15 supports with smoother surfaces favor the formation of more polymeric vanadia species at the same surface density loading. Smooth surface catalysts result in a ∼20% lower selectivity of methanol to formaldehyde, and the apparent activation energy on smooth surfaces is ∼25kJ/mol lower than on rough surfaces (75 versus 100kJ/mol, respectively). In contrast to methanol, propane results show a 15% higher selectivity to propene on smooth surfaces. A model of silica hydroxyl distribution is proposed to explain the differences in vanadia speciation and resulting catalytic behavior. These results are significant for our understanding of the nature of vanadium species in partial oxidation catalysts and illustrate the importance of considering differences in support surface morphology in analyzing catalytic behavior.

Validation of the Cooray‐Rubinstein (C‐R) formula for a rough ground surface by using three‐dimensional (3‐D) FDTD

AbstractIn this paper, we have extended the Cooray‐Rubinstein (C‐R) approximate formula into the fractal rough ground surface and then validate its accuracy by using three‐dimensional (3‐D) finite‐difference time‐domain (FDTD) method at distances of 50 m and 100 m from the lightning channel. The results show that the extended C‐R formula has an accepted accuracy for predicting the lightning‐radiated horizontal electric field above the fractal rough and conducting ground, and its accuracy increases a little better with the higher of the earth conductivity. For instance, when the conductivity of the rough ground is 0.1 S/m, the error of the peak value predicted by the extended C‐R formula is less than about 2.3%, while its error is less than about 6.7% for the conductivity of 0.01 S/m. The rough ground has much effect on the lightning horizontal field, and the initial peak value of the horizontal field obviously decreases with the increase of the root‐mean‐square height of the rough ground at early times (within several microseconds of the beginning of return stroke).

A complete contact model of a fractal rough surface

This study suggested a revision to the contact model of a fractal rough surface, by extending the modified asperity contact model developed by Morag and Etsion, into a complete contact model of a fractal surface. According to the modified asperity model, the critical area of a single asperity was scale dependent and that the asperity's plastic to elastic mode transition agreed with classical contact mechanics. The total load, area and stiffness of a fractal rough surface were studied, and obtained by summing over the contact force, area and stiffness of the asperities at all length scales. The results revealed that the contact area depended linearly on the contact load and that the contact stiffness increased with increasing contact load. The share of plastic contact area decreased as the contact load increased. The impact of the fractal roughness parameter G and fractal dimension D on the contact stiffness was also discussed, and the results showed that the rougher the surface became, corresponding to a smaller value of D and a larger value of G, the smaller the contact stiffness was.

Thermal contact conductance of elastically deforming nominally flat surfaces using fractal geometry

Purpose – This work aims at constructing a continuous mathematical, linear elastic, model for the thermal contact conductance (TCC) of two rough surfaces in contact. Design/methodology/approach – The rough surfaces, known to be physical fractal, are modelled using a deterministic Cantor structure. Such structure shows several levels of imperfections and including, therefore, several scales in the constriction of the flux lines. The proposed model will study the effect of the deformation (approach) of the two rough surfaces on the TCC as a function of the remotely applied load. Findings – An asymptotic power law, derived using approximate iterative relations, is used to express the area of contact and, consequently, the thermal conductance as a function of the applied load. The model is valid only when the approach of the two surface in contact is of the order of the surface roughness. The results obtained using this model, which admits closed form solution, are displayed graphically for selected values of the system parameters; the fractal surface roughness and various material properties. The obtained results showed good agreement with published experimental results both in trend and the numerical values. Originality/value – The model obtained provides further insight into the effect that surface texture has on the heat conductance process. The proposed model could be used to conduct an analytical investigation of the thermal conductance of rough surfaces in contact. This model, although simple (composed of springs), nevertheless works well.

Non-contact adhesion to self-affine surfaces: A theoretical model

Abstract Strength of adhesion between materials is known to be strongly influenced by interface irregularities. In this work, I devise a perturbative approach to describe the effect of self-affine roughness on non-contact adhesive interactions. The hierarchy of the obtained analytical solutions is the following. First, analytical formulae are deduced to describe roughness corrections to the van der Waals interaction energies between a hemi-space adherend, bounded by a self-affine surface, and a point-like adherent. Second, the problem of two hemi-spaces, one of which has a planar surface, and the other is bounded by a self-affine surface, is solved analytically. In the latter case, a numerical analysis is performed to delineate the behavior of the roughness corrections as a function of the parameters, characterizing self-affine fractal surface roughness. The problem of two hemi-spaces, both bounded by self-affine fractal surfaces, is also addressed in this work. The modelʼs predictions are compared with previously reported theoretical results and available experimental data.

One Method for Estimating Normal Contact Stiffness with <i>R<sub>a</sub> </i>and Adhesion Rate

Based on the fractal contact theory, the model about the relation between normal contact stiffness and material properties as well as fractal parameters is established. Then by making use of the relation between fractal parameters and surface roughness, a new method for estimating the normal contact stiffness is put forward. The numerical results indicate that the contact stiffness increases as surface roughness decreases or adhesion rate increases. When the adhesion rate is constant, contact stiffness changes a little while Ra≥6.3μm, but the change is relatively bigger when Ra≤6.3μm. Considering both the cost of surface machining and the influence of contact stiffness on the whole system, the surface roughness is suggested to be Ra=6.3μm. The results provide theoretical supports for vibration isolation analysis with gaskets and the process of low-noise construction.

Computational homogenization of rubber friction on rough rigid surfaces

The main component of rubber friction is known to be of hysteretic nature, i.e. it is due to viscoelastic energy dissipation taking place in the bulk of the material as a result of the pulsating forces induced by the surface asperities whenever rubber slides on a rough substrate, such as in the case of a car tire on a road surface. This implies that the observed macroscopic friction depends upon the constitutive behavior of the rubber and the characteristics of the rough surface profile. In contrast to analytical models, numerical approaches can fully account for geometric and material non-linearities arising in the rubber behavior, especially at small scales. However, explicit numerical modeling of rough surface features spanning a wide range of significant length scales would result prohibitively expensive, which motivates the need for a computational multiscale framework. As shown by previous related research, fractal surface profiles can be decomposed into a finite number of sinusoidal terms, so that a central ingredient of a multiscale approach becomes the homogenization of rubber friction on a sinusoidal surface. This work proposes a computational homogenization procedure where a macroscale coefficient of friction for rubber is derived from the solution of a microscale boundary-value problem. The latter considers contact of a representative volume element (RVE) with a sinusoidal rigid surface, which is assumed to represent the smallest length scale of a fractal rough surface. The numerical model is developed within the isogeometric framework and features a mortar formulation for the unilateral contact problem in the discretized setting. Numerical aspects related to the choice of the RVE, the setup of the test parameters and the convergence rate of different discretizations are discussed. Physically relevant observations concern the role of the macroscopic applied pressure and sliding velocity on the homogenized friction coefficient. Some comparisons with analytical results as well as dimensional analysis considerations are further reported.

Fractal approach to surface roughness of TiO2/WO3 coatings formed by plasma electrolytic oxidation process

In this study, we have shown that atomic force microscopy is a powerful technique to study the fractal parameters of TiO2/WO3 coatings prepared by plasma electrolytic oxidation (PEO) process. Since the surface roughness of obtained oxide coatings affects their physical properties, an accurate description of roughness parameters is highly desirable. The surface roughness, described by root mean squared and arithmetic average values, is analyzed considering the scans of a series of atomic force micrographs. The results show that the oxide coatings exhibit lower surface roughness in initial stage of PEO process. Also, the surfaces of TiO2/WO3 coatings exhibit fractal behavior. Positive correlation between the fractal dimension and surface roughness of the surfaces of TiO2/WO3 coatings in initial stage of PEO process was found.

Temperature Dependence of Coercivity Behavior in Fe Films on Fractal Rough Ceramics Surfaces

In this paper we study the anomalous magnetic properties in Fe films on fractal rough α-Al2O3 ceramics surfaces. It is found that the temperature dependence of the coercivity Hc of the Fe films exhibits a peak around a critical temperature Tcrit = 10 K: firstly, Hc increases with the temperature T; then it decreases rapidly as T further increases. The peak of the coercivity goes up with the decrease of the film thickness. It is estimated that, for T < Tcrit, the anomalous coercivity behavior may be resulted from the coupling effect of the spin-glass like states and ferromagnetic phases, for T > Tcrit, the observed coercivity behavior is mainly dominated by the effect of the non-uniform size distribution of the single-domain particles or magnetic clusters. The study shows that the enhancement Hc of the Fe films is strong related to the roughness of the film surfaces induced by the fractal ceramics surfaces. The results are further confirmed by the thermoremanence and vibrating sample magnetometer measurements.

Fractal characteristics and microstructure evolution of magnetron sputtering Cu thin films

How to describe surface morphology characteristic and microstructure evolution are the hottest researches of current thin film researches. But in traditional characterization of surface morphology, the roughness parameters are scale related. And the microstructure evolution of thin film during post-treatment is usually not considered in detail. To give a better understanding of the roughness of thin films topography, fractal method is carried out. In addition, microstructure evolution of thin films is analyzed based on the crystallography and energy theory. Cu thin films are deposited on Si (100) substrates by magnetron sputtering, and then annealed at different temperatures. Surface topography is characterized by atomic force microscope (AFM). Triangular prism surface area (TPSA) algorithm is used to calculate the fractal dimension of the AFM images. Apparent scale effect exists between the surface morphology roughness and film thickness. Relationship between the fractal dimension and roughness is analyzed by linear regression method and linear relationship exists between fractal dimension and surface roughness root mean square (RMS). Fractal dimension can be characterized as a scale independence parameter to represent the complex degree and roughness level of surface. With the increase of annealing temperature, surface roughness and fractal dimension decrease. But when the annealing temperature exceeds the recrystallization temperature, due to the agglomeration and coalescence of Cu grain, surface roughness and fractal dimension increase. Scale effect and changing regularity of grain growth and shape evolution for different film thickness under different annealing temperatures are analyzed. Based on minimum total free energy, regularity of grain growth and changing is proposed. The proposed research has some theory significance and applicative value of Cu interconnect process and development of MEMS.

Study on Reconstruction of Fractal Rough Surfaces

The geometric topography of the engineering surfaces play key roles in many issue of engineering and science including the analysis of friction, wear, lubrication, sealing and contact conductivity, especially microcosmic surface contact, thus the characterization of surface is one of the most important topics of tribology. Using the technology of CAD/CAE, a three-dimensional solid model of the fractal surfaces was established and meshed to provide the computer models for microcosmic contact analysis between two rough surfaces in this paper.

Investigation of the Heterogeneous Nucleation on Fractal Surfaces

Classical theory of heterogeneous nucleation has been developed with an implied hypothesis of smooth substrate surfaces; however, morphologies of any real substrate surfaces are generally complicated and demonstrate fractal characteristics. In this paper, the wettability between the embryo and the fractal substrate surface was discussed, and heterogeneous nucleation behaviors were theoretically analyzed. The result shows that the roughness factor of a fractal surface varies with the scale of the embryo. As a result, the fractal character of the substrate surface has important effects on heterogeneous nucleation behaviors. It has been shown that the energy barrier for heterogeneous nucleation of a non-wetting phase on a fractal rough surface increases with increasing fractal dimensions, and both the critical nucleus radius and the nucleation energy barrier decrease with increasing fractal dimensions for heterogeneous nucleation of a wetting phase on the fractal rough surface. For a non-wetting system, the critical nucleus radius shows a slight shift with changes of the intrinsic wetting angle, but for a wetting system, the critical nucleus radius shows an obvious change with decreasing intrinsic wetting angle, thus imposes a stronger effect on the heterogeneous nucleation behaviors.

Investigation of the dry normal contact between fractal rough surfaces using the reduction method, comparison to 3D simulations

The reduction method was applied to solving the elastic contact problem of fractal rough surfaces having different fractal dimensions. These surfaces are of particular interest, as contact spots influence each other on large and small scales, depending on the power spectrum. Results were compared to previous boundary-element method simulations that had been conducted with 3D surfaces, taking into account all interdependencies between contact spots. Excellent agreement in the contact stiffness was found. Even though no interactions are allowed in the reduced model, the same power-law behavior can be observed as in the 3D case, due to the transformation of the power spectrum. Theoretical considerations about the power-law for different fractal dimensions are given.