Papers by Keyword: Fractal Dimension

Abstract: Using a fractal analysis approach to study plant leaf venation and stem sections, we find that plants use very intelligent scaffolding strategies to tune mechanical strength of leaves and stems. Within plant organs, specialized types of tissues with different mechanical properties have evolved. Ideally, the biopolymers cellulose, hemicelluloses and lignin present in plant cell walls confer mechanical rigidity to plant tissues, but our studies reveal that the manner these biopolymers are distributed in the tissue matrix hold the key to the mechanical rigidity of the tissues. We have developed an algorithm to determine fractal dimension of the scaffolding matrix and the well-known box counting algorithm to calculate fractal dimensions of leaf venation in high resolution images of reticulate–veined leaves and optical microscope image of cellulose, hemicellulose, and lignin-stained cross sections of Turbina corymbosa. We found that in leaves with reticulate venation, veins form a scaffolding matrix imparting mechanical rigidity to leaves, and have a fractal dimension close to 1.0 for leaves which have less bending resistance, compared to fractal dimensions close to 1.7 for leaves which have higher bending resistance. Deriving this idea from plants, we use evaporation instability to develop scaffolding matrix with fractal dimensions higher than 1.5 in polymer films. This can form the basis of an efficient strategy to devise thin, stand-alone polymer films with tunable bending stiffness.

Abstract: A numerical model was proposed to investigate the contact behaviour of a solid with a rough surface squeezed against a rigid flat plane. We considered simulated hierarchical surface structures as well as scanned surface data obtained by the profilometry of isotropically roughened specimens. The simulated and treated surfaces were characterised using statistical and fractal parameters. The evolution of contact stiffness under increasing normal compression was analysed through the total truncated area at varying heights, in order to relate contact mechanics to different surface parameters employed for surface characterisation. For a relatively small surface interference, the predicted stress-dependent normal contact stiffness of both scanned and simulated surfaces is in good agreement with experimental observation from nanoindentation tests, revealing a power-law function of the normal load, with the exponent of this relationship closely depending on the fractal dimension of rough surfaces. The numerical results show that the amplitude of a fractal rough surface mainly contributes to the magnitude of the contact stiffness at a given normal load.

Abstract: Forsterite insulating materials were prepared by molten salt method using forsterite powder, pulp waste, NaCl and Na₂CO₃ as raw material. The pore size distribution and pore structural parameters of the sample were studied by nitrogen adsorption-desorption method. The fractal dimension of pore was obtained based on FHH (Frenkel-Halsey-Hill) equation. The results showed that adsorption-desorption isotherm of the sample was identified as type II due to the long-narrow pore with narrow openings. The pore size and fractal dimension decreased with the temperature increased. The maximum pore size was about 2 nm in the sample heated at 1100°C in the condition of different content of molten salt. And optimal thermal insulation performance of the sample were obtained in the sample contained 30 wt% Na₂CO₃ and heat treated at 1100°C.

Abstract: The structure of 4H-SiC and 6H-SiC monocrystals’ nanoscale defects was studied by the small-angle X-ray scattering (SAXS) method. Analysis of dispersion indicatrices has shown the presence of inhomogeneities in the form of linear fractals in silicon carbide crystals of 6H and 4H polytypes, mass fractals in 6H polytype and inhomogeneities partition’s fractal surface in 4H polytype.

Abstract: The shape of particles building the solid phase of soils is an important factor influencing soil behaviour. Three parameters defining the characteristics of particle shape: roundness, angularity and texture are the most commonly analyzed. The most difficult issue is texture determination due to its complex nature. Quantitative evaluation of this parameter creates serious problems, however, is not impossible. A new mathematical tool, such as fractal geometry, may be helpful. Through the use of power law, fractal analysis allows to designate fractal dimension that specifies the complexity of the tested object.

Abstract: Many physical effects, such as dc conductivity and percolation, depend on the morphology of the silicate structure and its relationship to adsorbed water. These effects play an important role in numerous technological applications, in geology, oil-extracting industry, and other practical fields. In this study, all the samples: natural montmorillonite, kaolinite, and сlinoptilolite with different exchangeable cations in their structures, – were stored in ambient air humidity. The investigation was carried by using two separate techniques, namely Dielectric Spectroscopy and a fractal analysis of electron micrographs. The aims of this work were to analyze the complex relaxation behavior of the relaxation process in temperature range –70°C ÷ +70°C and to determine the fractal dimensions of silicates from the dielectric response at percolation. Dielectric measurements in the frequency range of 1 Hz ÷ 1 MHz were performed using a BDS 80 Dielectric Spectrometer based on an Alpha Impedance Analyzer (Novocontrol). The micrographs were analyzed using a special Matlab based program. The analysis of aspects of the dielectric relaxation spectra related to percolation was used for the determination of the numerical characteristics of geometric heterogeneity of natural silicates. The percolation temperatures of the studied samples were determined. The percolation phenomenon in the silicates is related to the transfer of the electric excitation within the developed network of open pores due to the migration of protons and ions along the surface of connected pores on the outer surfaces of the granules. The analysis of these processes allows one to extract the fractal dimensions associated with the migration of charge carriers within the porous medium. Fractal dimensions of the silicates calculated in two ways: from dielectric spectroscopy study and from fractal analysis of the micrographs, – are in good agreement with each other. It was demonstrated that conventional method of the spatial fractal dimension determination using fractal analysis of electron micrographs leads to overestimation in the case of spatial fractal bounded by a surface fractal. The dielectric spectroscopy method is free from such overestimation.

Abstract: In order to study the relationship between the fractal dimension of rock fracture surface and temperature, uniaxial compression experiments on the granite rock and scanning electron microscopy experiments on the rock fracture surface were carried out after different temperatures. The fractal dimension was calculated by using scanning electron microscopy images. The results are as following: (1) The cracks show intergranular damage like rock candy, rivers and so on at the temperature from 25°C to 200°C. The fractal dimension are volatile, the fractal dimension reach the maximum at 200 °C, it can be inferred that the granite rock burst the most energy consumption at 200 °C; (2) From 200°C to 800°C, the cracks transform from stepped cleavage, intergranular fracture gradually to grain crushing, transgranular cracking damage with the rise of temperature. When the temperature is above 800 °C, the cracks transform from the transgranular cracks, shear slip bands gradually to dimples and microporous points. The cracks of rock surface transform from irregular crack structure to homogeneous pore structure gradually at high temperature, the weakening of rock heterogeneity is the basic reason for the decrease of fractal dimention of rupture surface, the fractal dimension is reduced from 1.634 at 200 °C to 1.595 at 1200 °C, the margin of reduction is 2.39%. At the same time, the energy consumption of rock rupture decreases with the rise of temperature, rock transforms from brittle to plastic gradually.

Abstract: Combined with the characteristics of forest image, the paper puts forward a new forest fire image segmentation method based on YCBCR and local fractal dimension. First of all, convert the image in RGB color space into YCBCR. Secondly, original triangle prism model algorithm which calculates the square of block matrix is replaced with four pyramid model. This algorithm is easy to be realized, reduce the amount of calculation model, and speed up the image segmentation speed. Segmentation effect significantly is improved, and the simulation experimental results show that this algorithm can be very good for the segmentation image of the flame extracted from complex background. It has obvious, accurate and fast effect and provides a good foundation for the future image analysis and recognition processing.

Abstract: Use scissors and cutting pliers to produce some striation marks. The data collection apparatus is used to collect the surface data of such marks produced by scissors and cutting pliers, and then get the profile curve that is vertical to the surface of striation marks. In application of fractal theory, the fractal dimension of such profile curve is then calculated, and further studies are made on its fractal characteristics. As an exploration on tool types and individual identification, this is aimed to provide a new theory and approach to examination identification of striation tool marks.

Abstract: The spindle or spherical cobalt oxalate powders were prepared by conventional precipitation in the presence of pulsed electromagnetic field (PEMF). The morphology and phase structure were characterized by scanning electron microscopy (SEM) and X-ray powder diffraction (XRD). The aspect ratio and surface fractal dimension of cobalt oxalate aggregate were calculated using Image tool software and Differential Box Counting (DBC) algorithm. The results indicate that the products prepared from ammonium oxalate and cobalt chloride give rise to the formation of β-CoC₂O₄∙2H₂O. With the increase of pulsed time, the morphology of cobalt oxalate sample varies periodically from rod-like to spheroidic. For the sample treated for 60 s, the degree of sphericity is the highest. The surface fractal dimension is 2.096 and the corresponding fractal form is tight type.