Research status and prospects of the fractal analysis of metal material surfaces and interfaces

As a mathematical analysis method,fractal analysis can be used to quantitatively describe irregular shapes with self-similar or self-affine properties.Fractal analysis has been used to characterize the shapes of metal materials at various scales and dimensions.Conventional methods make it difficult to quantitatively describe the relationship between the regular characteristics and properties of metal material surfaces and interfaces.However,fractal analysis can be used to quantitatively describe the shape characteristics of metal materials and to establish the quantitative relationships between the shape characteristics and various properties of metal materials.From the perspective of two-dimensional planes and three-dimensional curved surfaces,this paper reviews the current research status of the fractal analysis of metal precipitate interfaces,metal grain boundary interfaces,metal-deposited film surfaces,metal fracture surfaces,metal machined surfaces,and metal wear surfaces.The relationship between the fractal dimensions and properties of metal material surfaces and interfaces is summarized.Starting from three perspectives of fractal analysis,namely,research scope,image acquisition methods,and calculation methods,this paper identifies the direction of research on fractal analysis of metal material surfaces and interfaces that need to be developed.It is believed that revealing the deep influence mechanism between the fractal dimensions and properties of metal material surfaces and interfaces will be the key research direction of the fractal analysis of metal materials in the future.

Classification of forearm action surface EMG signals based on fractal dimension

Surface electromyogram （EMG） signals were identified by fractal dimension.Two patterns of surface EMG signals were acquired from 30 healthy volunteers＇ right forearm flexor respectively in the process of forearm supination （FS） and forearm pronation （FP）.After the raw action surface EMG （ASEMG） signal was decomposed into several sub-signals with wavelet packet transform （WPT）,five fractal dimensions were respectively calculated from the raw signal and four sub-signals by the method based on fuzzy self-similarity.The results show that calculated from the sub-signal in the band 0 to 125 Hz,the fractal dimensions of FS ASEMG signals and FP ASEMG signals distributed in two different regions,and its error rate based on Bayes decision was no more than 2.26%.Therefore,the fractal dimension is an appropriate feature by which an FS ASEMG signal is distinguished from an FP ASEMG signal.

THEORETICAL VERIFICATION OF THE SCALING RELATION TO DETERMINE SURFACE FRACTAL DIMENSION OF POROUS MEDIA

A simplified procedure is developed to acquire the surface fractal dimension of porous media utilizing the PSD information from mercury porosimetry or other analyses.The self similarity of the inner surface of Sierpinski sponge is analyzed,the result of which demonstrates that the inner surface of Sierpinski sponge is not scale invariant over the whole range of scale transformations.By applying the simplified procedure to analyze and treat the PSD information of Sierpinski sponge over the scale invariant range,it is found that the surface fractal dimension calculated by the scaling relation is in very good agreement with its theoretical value,which virtually provides a theoretical affirmation of the method.

Running-in Test and Fractal Methodology for Worn Surface Topography Characterization

Studying and understanding of the surface topography variation are the basis for analyzing tribological problems,and characterization of worn surface is necessary.Fractal geometry offers a more accurate description for surface roughness that topographic surfaces are statistically self-similar and can be quantitatively evaluated by fractal parameters.The change regularity of worn surface topography is one of the most important aspects of running-in study.However,the existing research normally adopts only one friction matching pair to explore the surface topography change,which interrupts the running-in wear process and makes the experimental result lack authenticity and objectivity.In this paper,to investigate the change regularity of surface topography during the real running-in process,a series of running-in tests by changing friction pairs under the same operating conditions are conducted on UMT-II Universal Multifunction Tester.The surface profile data are acquired by MiaoXAM2.5X-50X Ultrahigh Precision Surface 3D Profiler and analyzed using fractal dimension D,scale coefficient C and characteristic roughness Ra ＊based on root mean square（RMS） method.The characterization effects of the three parameters are discussed and compared.The results obtained show that there exists remarkable fractal feature of surface topography during running-in process,both D and Ra ＊increase gradually,while C decreases slowly as the wear-in process goes on,and all parameters tend to be stable when the wear process steps into the normal wear process.Ra ＊illustrates higher sensitivity for rough surface characterization compared with the other two parameters.In addition,the running-in test carried with a set of identical surface properties is more scientific and reasonable than the traditional one.The proposed research further indicates that the fractal method can quantitatively measure the rough surface,which also provides an evidence for running-in process identification and tribology design.

Fractal Prediction Model of Thermal Contact Conductance of Rough Surfaces

The thermal contact conductance problem is an important issue in studying the heat transfer of engineering surfaces, which has been widely studied since last few decades, and for predicting which many theoretical models have been established. However, the models which have been existed are lack of objectivity due to that they are mostly studied based on the statistical methodology characterization for rough surfaces and simple partition for the deformation formats of contact asperity. In this paper, a fractal prediction model is developed for the thermal contact conductance between two rough surfaces based on the rough surface being described by three-dimensional Weierstrass and Mandelbrot fractal function and assuming that there are three kinds of asperity deformation modes： elastic, elastoplastic and fully plastic. Influences of contact load and contact area as well as fractal parameters and material properties on the thermal contact conductance are investigated by using the presented model. The investigation results show that the thermal contact conductance increases with the increasing of the contact load and contact area. The larger the fractal dimension, or the smaller the fractal roughness, the larger the thermal contact conductance is. The thermal contact conductance increases with decreasing the ratio of Young＇s elastic modulus to the microhardness. The results obtained indicate that the proposed model can effectively predict the thermal contact conductance at the interface, which provide certain reference to the further study on the issue of heat transfer between contact surfaces.

Elastoplastic Contact Mechanics Model of Rough Surface Based on Fractal Theory

Because the result of the MB fractal model contradicts with the classical contact mechanics, a revised elastoplastic contact model of a single asperity is developed based on fractal theory. The critical areas of a single asperity are scale dependent, with an increase in the contact load and contact area, a transition from elastic, elastoplastic to full plastic deformation takes place in this order. In considering the size distribution function, analytic expression between the total contact load and the real contact area on the contact surface is obtained. The elastic, elastoplastic and full plastic contact load are obtained by the critical elastic contact area of the biggest asperity and maximun contact area of a single asperity. The results show that a rough surface is firstly in elastic deformation. As the load increases, elastoplastic or full plastic deformation takes place. For constant characteristic length scale G, the slope of load-area relation is proportional to fractal dimension D. For constant fractal dimension D, the slope of load-area relation is inversely proportional to G. For constant D and G, the slope of load-area relation is inversely proportional to property of the material ~b, namely with the same load, the material of rough surface is softer, and the total contact area is larger. The contact mechanics model provides a foundation for study of the friction, wear and seal performance of rough surfaces.

Fractal Characteristics and Fractal Dimension Measurement on Broken Surfaces of Aluminum Electric Porcelain

The characteristics of broken surfaces were r esearched by a scanning electron microscope (SEM) and a reflection microscope, a nd the fractal dimensions of broken surfaces were measured by the Slit Island me thod. The experimental results indicate that the broken surface of aluminum elec tric porcelain is a fractal body in statistics, and the fractal dimensions of br oken surfaces are different with the different amplification multiple value.In a ll of measured fractal dimensions,both of values measured in 100× under reflect ion microscope and in 500× under SEM are maximum, whereas the values measur ed in 63× under reflection microscope and in 2000× under SEM are obviously min imum. The fractal dimensions of broken surfaces are also affected by the degrees of gray comparison and the kinds of measuring methods. The relationships betwee n the fractal dimensions of broken surfaces and porcelain bend strengths are tha t they are in positive correlation on the low multiples and in negative correlat ion on the high multiples.

Effects of AlN hydrolysis on fractal geometry characteristics of residue from secondary aluminium dross using response surface methodology

The effects of aluminium nitride(AlN)hydrolysis on fractal geometry characteristics of residue from secondary aluminium dross were studied using response surface methodology.The results show that the fractal dimensions of the residue can be significantly influenced by the AlN hydrolysis from secondary aluminium dross.The hydrolysis of AlN in the dross was spontaneous under temperatures of303-373K.The actual fractal dimensions of residue were significantly affected by the liquid-solid ratio(p<0.05)and changed from1.16to1.80,which accurately aligned with those from the calculations.Moreover,the fractal dimensions of residue were significantly affected by the interactions between hydrolysis temperature and hydrolysis time,liquid-solid ratio and hydrolysis time,respectively(p<0.01).The minimum fractal dimensions of the residue reached1.15under the optimized conditions,which included a hydrolysis temperature of30℃,liquid-solid ratio of5mL/g and hydrolysis time of10min.The results suggest that response surface methodology can guide in optimizing the conditions of AlN hydrolysis in order to obtain the minimum fractal dimensions of residue for improving the reutilization of the dross.

Use of digital image analysis combined with fractal theory to determine particle morphology and surface texture of quartz sands

The particle morphology and surface texture play a major role in influencing mechanical and hydraulic behaviors of sandy soils. This paper presents the use of digital image analysis combined with fractal theory as a tool to quantify the particle morphology and surface texture of two types of quartz sands widely used in the region of Vitória, Espírito Santo, southeast of Brazil. The two investigated sands are sampled from different locations. The purpose of this paper is to present a simple, straightforward,reliable and reproducible methodology that can identify representative sandy soil texture parameters.The test results of the soil samples of the two sands separated by sieving into six size fractions are presented and discussed. The main advantages of the adopted methodology are its simplicity, reliability of the results, and relatively low cost. The results show that sands from the coastal spit(BS) have a greater degree of roundness and a smoother surface texture than river sands(RS). The values obtained in the test are statistically analyzed, and again it is confirmed that the BS sand has a slightly greater degree of sphericity than that of the RS sand. Moreover, the RS sand with rough surface texture has larger specific surface area values than the similar BS sand, which agree with the obtained roughness fractal dimensions. The consistent experimental results demonstrate that image analysis combined with fractal theory is an accurate and efficient method to quantify the differences in particle morphology and surface texture of quartz sands.

Electromagnetic backscattering from one-dimensional drifting fractal sea surface Ⅰ：Wave–current coupled model

To study the electromagnetic backscattering from a one-dimensional drifting fractal sea surface,a fractal sea surface wave–current model is derived,based on the mechanism of wave–current interactions.The numerical results show the effect of the ocean current on the wave.Wave amplitude decreases,wavelength and kurtosis of wave height increase,spectrum intensity decreases and shifts towards lower frequencies when the current occurs parallel to the direction of the ocean wave.By comparison,wave amplitude increases,wavelength and kurtosis of wave height decrease,spectrum intensity increases and shifts towards higher frequencies if the current is in the opposite direction to the direction of ocean wave.The wave–current interaction effect of the ocean current is much stronger than that of the nonlinear wave–wave interaction.The kurtosis of the nonlinear fractal ocean surface is larger than that of linear fractal ocean surface.The effect of the current on skewness of the probability distribution function is negligible.Therefore,the ocean wave spectrum is notably changed by the surface current and the change should be detectable in the electromagnetic backscattering signal.

Effects of fractal surface on rheological behavior and combustion kinetics of modified brown coal water slurries

The paper reports the effects of surface fractal structures on the rheological behavior and combustion kinetics of raw brown coal and three modified coal water slurries （CWSs）. The results show that the fractal structures and physicochemical properties of samples are dependent on various modification processes. The apparent viscosities of the coal water slurries increase with increasing surface fractal dimensions （D）, especially with decreasing shear rates. Fur- thermore, it has been proved that the ignition temperatures and apparent activation energies of modified CWSs are lower than that of raw coal water slurry. Compared with the traditional qualitative analysis of the effect of pore structures on CWSs properties, D can more efficiently indicate the quantificational effect of pore structures on the rheological behavior and combustion kinetics of CWSs.

Synthesis of fractal geometry and CAGD models for multi-scale topography modelling of functional surfaces

In order to support the functional design and simulation and the final fabrication processes for functional surfaces,it is necessary to obtain a multi-scale modelling approach representing both macro geometry and micro details of the surface in one unified model.Based on the fractal geometry theory,a synthesized model is proposed by mathematically combining Weierstrass-Mandelbrot fractal function in micro space and freeform CAGD model in macro space.Key issues of the synthesis,such as algorithms for fractal interpolation of freeform profiles,and visualization optimization for fractal details,are addressed.A prototype of the integration solution is developed based on the platform of AutoCAD's Object ARX,and a few multi-scale modelling examples are used as case studies.With the consistent mathematic model,multi-scale surface geometries can be represented precisely.Moreover,the visualization result of the functional surfaces shows that the visualization optimization strategies developed are efficient.

Fractal Surface Synthesis Based on Two Dimensional Discrete Fourier Transform

The discrete Fourier transform（DFT） is used for fractional Brownian motion（FBM） surface synthesis in tribology（i.e., contact, sliding, and sealing, etc）. However, the relationship between fractal parameters（fractal dimension and scale factor） and traditional parameters, the influence of fractal parameters on surface appearance, have not been deeply discussed yet. These lead to some kind of difficulty to ensure the synthesized surfaces with ideal fractal characteristic, required traditional parameters and geometric appearance. A quantitative relationship between fractal parameters and the root mean square deviation of surface （Sq） is derived based on the energy conservation property between the space and frequency domain of DFT. Under the stability assumption, the power spectrum of a FBM surface is composed of concentric circles strictly, a series of FBM surfaces with prescribed Sq could be synthesized with given fractal dimension, scale factor, and sampling numbers, but the ten-point height（Sz）, the skewness（Ssk） and the kurtosis（Sku） are still in random, where the probability distributions of Sz and Ssk are approximately normal distribution. Furthermore, by iterative searching, a surface with desired Abbott-Firestone curve could be obtained among those surfaces. An intuitive explanation for the influence of fractal dimension and scale factor on surface appearance is obtained by discussing the effects on the ratio of energy between high and low frequency components. Based on the relationship between Sq and surface energy, a filtering method of surface with controllable Sq is proposed. The proposed research ensures the synthesized surfaces possess ideal FBM properties with prescribed Sq, offers a method for selecting desired Abbott-Firestone curve of synthesized fractal surfaces, and makes it possible to control the Sq of surfaces after filtering.

An ultrathin wide-band planar metamaterial absorber based on a fractal frequency selective surface and resistive film

We propose an ultrathin wide-band metamaterial absorber （MA） based on a Minkowski （MIK） fractal frequency selective surface and resistive film. This absorber consists of a periodic arrangement of dielectric substrates sandwiched with an MIK fractal loop structure electric resonator and a resistive film. The finite element method is used to simulate and analyze the absorption of the MA. Compared with the MA-backed copper film, the designed MA-backed resistive film exhibits an absorption of 90% at a frequency region of 2 GHz-20 GHz. The power loss density distribution of the MA is further illustrated to explain the mechanism of the proposed MA. Simulated absorptions at different incidence cases indicate that this absorber is polarization-insensitive and wide-angled. Finally, further simulated results indicate that the surface resistance of the resistive film and the dielectric constant of the substrate can affect the absorbing property of the MA. This absorber may be used in many military fields.

Fractal characteristics of surface crack evolution in the process of gas-containing coal extrusion

In this paper, simulated experiment device of coal and gas outburst was employed to perform the experiment on gas-containing coal extrusion. In the experiment, coal surface cracks were observed with a high-speed camera and then the images were processed by sketch. Based on the above description, the paper studied the fractal dimension values from different positions of coal surface as well as their changing laws with time. The results show that there is a growing parabola trend of crack dimension value in the process of coal extrusion. Accordingly, we drew the conclusion that extruded coal crack evolution is a process of fractal dimension value increase. On the basis of fractal dimension values taken from different parts of coal masses, a fractal dimension of the contour map was drawn. Thus, it is clear that the contour map involves different crack fractal dimension values from different positions. To be specific, where there are complicated force and violent movement in coal mass, there are higher fractal dimension values, i.e., the further the middle of observation surface is from the exit of coal mass, and the lower the fractal dimension value is. In line with fractal geometry and energy theory of coal and gas outburst, this study presents the relation between fractal dimension and energy in the process of extruding. In conclusion, the evolution of crack fractal dimension value can signify that of energy, which has laid a solid foundation for the quantification research on the mechanism of gas-containing coal extrusion.

The autocorrelation of speckles in deep Fresnel diffraction region and characterizations of random self-affine fractal surfaces

This paper studies the correlation properties of the speckles in the deep Fresnel diffraction region produced by the scattering of rough self-affine fractal surfaces. The autocorrelation function of the speckle intensities is formulated by the combination of the light scattering theory of Kirchhoff approximation and the principles of speckle statistics. We propose a method for extracting the three surface parameters, i.e. the roughness w, the lateral correlation length ξ and the roughness exponent α, from the autocorrelation functions of speckles. This method is verified by simulating the speckle intensities and calculating the speckle autocorrelation function. We also find the phenomenon that for rough surfaces with α= 1, the structure of the speckles resembles that of the surface heights, which results from the effect of the peak and the valley parts of the surface, acting as micro-lenses converging and diverging the light waves.

Electromagnetic Scattering from One-Dimension Fractal Surface

In this paper,the surfaces are modeled by multiscaled band-limited continuous fractal function and fractional Brownian motion function.The relationship between the scattering behavior and geometrical characteristics of the fractal surface is analysed in detail based on the perturbation solution and Kirchhoff approximation.Finally,some discussions are given about scattering from fractal surface.

Implication of Surface Fractal Analysis to Evaluate the Relative Sensitivity of Topography to Active Tectonics,Zagros Mountains, Iran

Fractal geometry is increasingly becoming a useful tool for modeling and quantifying the complex patterns of natural phenomena. The Earth's topography is one of these phenomena that have fractal characteristics. This paper investigates the relative sensitivity of topography to active tectonics using ASTER Global Digital Elevation Model. The covering divider method was used for direct extraction of surface fractal dimension(D surf) to estimate the roughness-surface of topography with aid of geographic information system(GIS)techniques. This evaluation let us highlight the role of the geomorphic and tectonic processes on the spatial variability of fractal properties of natural landforms.Geomorphic zones can be delineated using fractal dimension mapping in which variability of surface fractal dimension reflects the roughness of the landform surface and is a measure of topography texture. Obtained results showed this method can be a quick and easy way to assess the distribution of land surface deformation in different tectonic settings. The loose alluvial deposits and irregularities derived by tectonic activity have high fractal dimensions whereas the competent formations and higher wavelength folded surfaces have lower fractal dimensions.According to the obtained results, the Kazerun Fault Zone has a crucial role in the separation of the Zagros Mountain Ranges into the different lithological,geomorphological and structural zones.

Fractal Model for Estimating the Roughness of Rock Fracture Surfaces

The joint roughness coefficient (JRC) is an important geometric parameter in rock mechanics, seismo-geology, engineering geology, and structural geology. In recent years, the fractol geometry has been widely applied in researching JRC In this paper, a simpler fractal model for estimating JRC was developed, whick can be applied in simulating rock discontinuity profiles, aud the rougher the surface, the greater the fractal dimension D The epirical equation of JRC and D was established according to the model’ The JRC values of six joint profiles were estimated according to Eq (10),which are ratker approximate to the typical JRC values; so the method may be applied to estimate the roughness of the fractured surfaces of natural rocks For a give profile the estimates of D and JRC can be obtained from Eqs (5) and (l0) with only one parameter e’ (the average angle of the profile fluctuation) being needed.

Fractal Measures of Drainage Network to Investigate Surface Deformation from Remote Sensing Data: A Paradigm from Hindukush (NE-Afghanistan)

This approach represents the relative susceptibility of the topography of the earth to active deformation by means of geometrical distinctiveness of the river networks. This investigation employs the fractal analysis of drainage system extracted from ASTER Global Digital Elevation Model (GDEM-30m resolution). The objective is to mark active structures and to pinpoint the areas robustly influenced by neotectonics. This approach was examined in the Hindukush, NE-Afghanistan. This region is frequently affected by deadly earthquakes and the modern fault activities and deformation are driven by the collision between the northward-moving Indian subcontinent and Eurasia. This attempt is based on the fact that drainage system is strained to linearize due to neotectonic deformation. Hence, the low fractal dimensions of the Kabul, Panjsher, Laghman, Andarab, Alingar and Kocha Rivers are credited to active tectonics. A comprehensive textural examination is conducted to probe the linearization, heterogeneity and connectivity of the drainage patterns. The aspects for these natural textures are computed by using the fractal dimension (FD), lacunarity (LA) and succolarity (SA) approach. All these methods are naturally interrelated, i.e. objects with similar FD can be further differentiated with LA and/or SA analysis. The maps of FD, LA and SA values are generated by using a sliding window of 50 arc seconds by 50 arc seconds (50" × 50"). Afterwards, the maps are interpreted in terms of regional susceptibility to neotectonics. This method is useful to pinpoint numerous zones where the drainage system is highly controlled by Hindukush active structures. In the North-Northeast of the Kabul block, we recognized active tectonic blocks. The region comprising, Kabul, Panjsher, Andrab, Alingar and Badakhshan is more susceptible to damaging events. This investigation concludes that the fractal analysis of the river networks is a bonus tool to localize areas vulnerable to deadly incidents influencing the Earth’s topography and consequently intimidate human lives.