Surface Fractal Dimension Research Articles

Computational model for fractal dimension of anisotropic gear surfaces based on improved structure function method

Computational model for fractal dimension of anisotropic gear surfaces based on improved structure function method

Study on hydrophobic floc structure of coal particles based on fractal simulation

Utilizing the Matlab platform, two-dimensional diffusion-limited aggregation and three-dimensional diffusion-limited cluster aggregation models were established to simulate the hydrophobic flocculation of coal particles. The formation process of flocs aggregated by monomer particles and small groups was simulated from both plane and space perspectives. The gyration radius fractal dimension increased from 1.401 to 1.539 in the DLA model and from 1.623 to 1.729 in the DLCA model, indicating that the floc morphology gradually stabilized and the structure became increasingly regular during the hydrophobic flocculation process. The porosity of DLA increased from 0.762 to 0.851, indicating that the dendritic structure of flocs formed a spatial shielding effect on the planar structure, which prevented the particles from entering the flocs. In contrast, the porosity of DLCA first increased from 0.506 to 0.611 and then decreased, stabilizing at approximately 0.53, indicating that small groups filled the internal voids of flocs in the spatial structure, leading to a gradual increase in floc density. The surface fractal dimension and density of flocs were calculated using the surface fractal and hydrostatic sedimentation-terminal velocity theories, respectively, confirming the conclusions drawn from the simulation analysis.

Spontaneous combustion coal gangue-based composite cement: Compressive performance and environmental benefits under grinding kinetics control

This study aims to tackle the environmental pressure of spontaneous combustion coal gangue by using its active silica-alumina content to create eco-friendly composite cement through a low-energy process. The Divas-Aliavden grinding kinetics equation was established, and powder characteristics were analyzed using particle size distribution, the Rosin-Rammler-Bennett model, and fractal dimension theory. The compressive performance of the coal gangue-based composite cement was tested. A comprehensive evaluation model linked grinding time, powder characteristics, compressive performance, and energy consumption, while also analyzing hydration products and environmental impact. Results showed that initial grinding stages had larger particle sizes and higher energy utilization efficiency. As grinding progressed, energy consumption increased. A grinding time of 35 min achieved optimal balance between energy efficiency and compressive performance, improving particle distribution uniformity and complexity. The compressive performance of the composite cement reached 95 % of ordinary Portland cement after 28 days of curing. The evaluation model indicated that specific surface area, d50, and fractal dimension are key factors influencing energy efficiency, determining energy consumption and particle distribution during grinding. Environmental assessment revealed that substituting 30 % of cement with coal gangue could reduce CO2 emissions by 250 kg CO2/t, achieving a reduction rate of 29.5 %. This study supports the large-scale application of spontaneous combustion coal gangue, contributing to efficient resource utilization and environmental protection.

RECOGNITION OF 3D SURFACE FRACTAL DIMENSION BASED ON CONVOLUTIONAL NEURAL NETWORK

Fractal dimension (FD), as an important parameter for the surface morphology of mechanical machining, can be used to analyze the friction characteristics of contacting surfaces. The accurate and rapid measurement of the FD is of significant importance. To this end, this paper presents a novel approach based on a convolutional neural network (3D-CNN) for the recognition of three-dimensional FDs on machined surfaces. We construct a dataset of anisotropic rough surfaces with different FDs using the Weierstrass–Mandelbrot (WM) fractal function and use a one-factor experimental design to analyze the influence of network parameters (network depth, filter size, filter quantity) on the accuracy of FD identification, finding the optimal combination of neural network parameters. By comparing our 3D-CNN method with three other methods (differential box-counting (DBC), triangular prism surface area (TPSA), and fractal Brownian motion (FBM)), we validated the effectiveness of our proposed method. The experimental results show that the average absolute percentage error of FD calculated by the 3D-CNN method can be controlled within 2%, and the method exhibits small errors throughout the full dynamic range of FDs. We also apply the proposed method to calculate the FD of the vertical milling surface and compare the results with the TPSA and FBM methods. The results show that the FD obtained by our method is close to the other two methods, and can be used to calculate the FD of three-dimensional surface profiles, thereby providing a new approach for the dynamic modeling and parameter identification of contact surfaces.

Insights from tensile fracture properties and full-field strain evolution of deep coral reef limestone under dynamic loads

Coral reef limestone (CRL) commonly undergoes dynamic tension when underground structures of island reefs encounter impacts, explosions, or seismic activities. Given the complexity of biological pores, the dynamic tensile fracture characteristics of CRL are poorly understood. Therefore, the dynamic tensile fracture behaviors of deep CRL were systematically observed by Split Hopkinson Pressure Bar tests and digital image techniques. Comparing to traditional rocks, the macro-pores near failure band would significantly change cracking path. The failure patterns are dominated by loading rate. The strongly dependence of dynamic tensile strength and dynamic crack initiation toughness on loading rate suggests the two indices overcome the effect of CRL macro-pores under dynamic impacts. At low loading rates, tensile fractures predominantly follow intergranular cracks, whereas transgranular cracks dominate at higher rates. The fractal dimension of fracture surface decreases with increasing crack propagation velocity, loading rate, and dynamic crack initiation toughness. Due to the unique marine sedimentary environment, the mechanical heterogeneity in multiple scales distinguishes CRL from terrestrial rock materials. The insights into underlying mechanisms of dynamic tension provide support to optimization of blasting scheme and stability assessments for island underground engineering.

Fractal contact and asperities coalescence of rock joints under normal loading: Insights from pressure-sensitive film measurement

Direct measurement of the real contact area of rock joints under normal loading is crucial for comprehending the subsurface geological processes. However, measuring this phenomenon quantitatively at site-scale or laboratory-scale is challenging. Here, we investigate the evolution mechanism of the real contact area in rock joints by conducting closure tests on artificial and saw-cut sandstone joints under normal stresses up to 50 MPa. Geometrical shapes of contact patches are quantified by the pressure-sensitive film using the adaptive threshold method. An extensive range of contact stress within contact patches is innovatively measured by integrating the results from multi-type pressure-sensitive films. Experimental results demonstrate that the real contact area increases with the increasing normal stress hyperbolically. Such a nonlinear contact evolution behavior can be attributed to the coalescence of adjacent contact patches. The fractal dimension of composite surface governs the geometrical shapes of contact patches and the distribution of contact stress. The relationship between patch areas and bearing loads follows the Hertzian theory when the patches are small, while it gradually becomes linear with the increasing patch size. A power model with exponential cut-off is proposed to predict the size distribution of contact patches. This work can provide new insights for estimating the patch-dependent seismic nucleation length and slip stability of subsurface joints.

Impacts of “Reef Star” coral restoration on multiple metrics of habitat complexity

Coral reefs face threats from climate change and local pressures that lead to reductions in their physical structure, impacting biodiversity by limiting habitat availability. Despite many efforts to actively restore damaged reefs, few projects provide thorough evaluations of their success. This study measured the success of the “Reef Star” method at the Mars Coral Reef Restoration Project in Indonesia in reestablishing the physical structure of reef habitats that were destroyed by blast fishing. We used photogrammetry surveys to measure the physical habitat structure of 17 large sites (1000 m2 each), calculating three complementary measures of small‐ and large‐scale habitat complexity across degraded, restored, and naturally healthy coral reefs. We demonstrate that the restoration efforts have successfully restored small‐scale habitat complexity, as described by surface complexity metrics (3.22 ± 0.27 on restored reefs; 2.85 ± 0.26 on healthy reefs) and fractal dimension (2.27 ± 0.02 on restored reefs; 2.24 ± 0.02 on healthy reefs). This demonstrates the capacity for restored reefs to recover important ecosystem functions that are lost in degradation. However, while restoration has delivered some increases in large‐scale habitat complexity compared to degraded reefs, restored reefs still exhibit lower values of maximum vertical relief than healthy reefs, due to a lack of large physical structures. This lack of available large‐scale habitat might impact fish populations, meaning that restored reefs with limited large‐scale complexity may only support a restricted range of ecosystem functions. Effective reef restoration strategies must use a mixture of different methods that target the recovery of structural complexity at multiple scales.

Research on image segmentation processing of coal particle flocs combined with clarity detection

ABSTRACT In the image analysis of the hydrophobic flocculation process of coal particles, the recognition accuracy of floc characteristic parameters is affected by bubbles, floc superposition, and floc morphology. An image processing method was studied and designed based on the PyCharm Community platform for clarity recognition and floc segmentation of coal particle floc characteristics. Firstly, the critical value of image clarity was determined to be 1.31 by using the Laplace convolution accounting method after the average gray level correction, and the fuzzy flocs whose clarity was less than the critical value were eliminated. Then, by comparing the segmentation effect of the threshold, edge, and region segmentation algorithm on the flocs, the histogram bimodal method was determined to be the best image processing segmentation method for fine coal floc collection. The fractal theory was used to verify the effect of image processing. The results showed that the surface fractal dimension of the floc image after deblurring and segmentation was maintained near 2.05, indicating that the processed image can obtain more stable and reliable floc characteristic parameters.

Influence of mineral filler characteristics on the filler–asphalt interfacial behavior

The knowledge of mineral filler characteristics and their impacts on filler–asphalt interfacial behavior is not systematic and completed yet. In this paper, eight mineral fillers prepared using the same milling procedure were investigated. The filler–asphalt interfacial behavior was analyzed by differential scanning calorimetric and dynamic shear rheometer tests. Results reveal that in contrast to form factor and specific surface area, particle porosity, particle density, aspect ratio, angularity index, fractal dimension and feature roughness decrease with decreasing particle size. Limestone filler exhibits more regular shape, less significant angularity and richer surface texture than basalt filler. Of all particle characteristics studied, form factor and specific surface area are suggested to be the determining factors affecting filler–asphalt interfacial behavior. Moreover, microscopic morphology analysis provides an insight to interpret the differences explored in mastic interfacial property. The results help employ applicable particle characteristics that fabricate mastic with a stable interface system.

A Comparison of Evaluation Methodologies of the Fractal Dimension of Premixed Turbulent Flames in 2D and 3D Using Direct Numerical Simulation Data

AbstractA Direct Numerical Simulation (DNS) database of statistically planar flames ranging from the wrinkled flamelets to the thin reaction zones regime and DNS data for a Bunsen premixed flame representing the wrinkled flamelets regime have been utilised to evaluate the fractal dimensions of flame surfaces using the filtering dimension method, the box-counting algorithm and the correlation dimension approach. The fractal dimension evaluated based on the fully resolved three-dimensional data has been found to be reasonably approximated by adding unity to the equivalent fractal dimension evaluated based on two-dimensional projections irrespective of the methodology of extracting fractal dimension. This indicates that the flame surface can be approximated as a self-similar fractal surface for the range of Karlovitz and Damköhler numbers considered here. While all methods, provide results identical to each other for benchmark problems, it has been found that the fractal dimension evaluation based on box-counting method provides almost identical results as that obtained using the filtering dimension method for both three and two dimensions, while the fractal dimensions based on the correlation dimension tend to be slightly smaller. The findings of the current analysis have the potential to be used to reliably estimate the actual fractal dimension in 3D based on experimentally obtained 2D binarised reaction progress variable field. The inner cut-off scales estimated based on all three methodologies yield comparable results in terms of order of magnitude with the box-counting method predicting a smaller value of inner cut-off scale in comparison to other methods. The execution times for fractal dimension extraction based on filtering dimension and box-counting methodologies are found to be comparable but the correlation dimension method is found to be considerably faster than the two alternative approaches and provides results consistent with theoretical bounds in all cases.

Digital reconstruction of three‐dimensional contours and its application to microstructural evaluation of postblast rock fissure surfaces

AbstractFracture surface contour study is one of the important requirements for characterization and evaluation of the microstructure of rocks. Based on the improved cube covering method and the 3D contour digital reconstruction model, this study proposes a quantitative microstructure characterization method combining the roughness evaluation index and the 3D fractal dimension to study the change rule of the fracture surface morphology after blasting. This method was applied and validated in the study of the fracture microstructure of the rock after blasting. The results show that the fracture morphology characteristics of the 3D contour digital reconstruction model have good correlation with the changes of the blasting action. The undulation rate of the three‐dimensional surface profile of the rock is more prone to dramatic rise and dramatic fall morphology. In terms of tilting trend, the tilting direction also shows gradual disorder, with the tilting angle increasing correspondingly. All the roughness evaluation indexes of the rock fissure surface after blasting show a linear and gradually increasing trend as the distance to the bursting center increases; the difference between the two‐dimensional roughness evaluation indexes and the three‐dimensional ones of the same micro‐area rock samples also becomes increasingly larger, among which the three‐dimensional fissure roughness coefficient JRC and the surface roughness ratio Rs display better correlation. Compared with the linear fitting formula of the power function relationship, the three‐dimensional fractal dimension of the postblast fissure surface is fitted with the values of JRC and Rs, which renders higher correlation coefficients, and the degree of linear fitting of JRC to the three‐dimensional fractal dimension is higher. The fractal characteristics of the blast‐affected region form a unity with the three‐dimensional roughness evaluation of the fissure surface.

Analysis and Characterization of Micro–Nano Pores in Coal Reservoirs of Different Coal Ranks

Coalbed methane represents a promising source of clean and efficient unconventional energy. The intricate network of micro–nano pores within coal serves as the primary adsorption space for gas, contributing to the complexity of gas migration channels. In this study, based on the box-counting method, three coal samples representing low, medium, and high ranks were subjected to high-precision micro-CT scanning and nano-CT scanning to generate three-dimensional (3D) pore network models using Avizo visualization software. This facilitated the accurate and quantitative characterization of the micro–nano pore structures within coal reservoirs. The results indicated that the face rate distribution range of each sample was large, indicating relatively strong heterogeneity in each sample. The volume fractal dimension of each sample, determined through micro–nano-CT scanning, was around 2.5, while the surface fractal dimension exhibited oscillatory characteristics with moderate uniformity. The pore equivalent radius and throat equivalent radius distributions were unimodal across all the samples, with the micro-CT scanning revealing a concentration primarily within the range of 100–400 μm for the pore equivalent radius and within 200 μm for the throat equivalent radius. Conversely, the nano-CT scanning exhibited concentrations primarily within the range of 500–2500 nm for the pore equivalent radius and within 2000 nm for the throat equivalent radius. The analysis of the 3D reconstruction structures indicated that the middle-rank coal exhibited more developed large–medium pores compared with the low-rank and high-rank coal, while the low-rank and high-rank coal exhibited relatively more micro–small pores. Furthermore, the low-rank coal exhibited the fewest number of pores but the largest average pore equivalent radius and throat radius. Additionally, the middle–high-rank coal exhibited a relatively larger number of pores. Despite the complex topological structures observed in each sample, a significant proportion indicated a coordination number of 0–20, indicating excellent connectivity within the coal samples. This study is conducive to the optimization of coalbed methane surface development blocks and the formulation of reasonable development plans.

Experimental investigation on fractal mechanism and shear failure characteristics of cement mortar at different curing ages

In this paper, the NMR and direct shear tests were conducted successively on cement mortar specimens with different curing ages (7 days, 14 days and 28 days). The porosity, shear strength, shear stiffness and shear energy of cement mortar are measured and analyzed through interpreting the pore distribution patterns and their corresponding fractal dimensions. The cross-sections of shear surface were characterized into point cloud data via 3D laser scanning and quantified by three dimensionless parameters (the root mean square value of the first derivative, root mean square value and joint roughness coefficient). The grey relation analysis (GRA) method and principal component analysis (PCA) is employed to analyze the relationship between the pore structures (i.e., gel pores, transition pores, capillary pores and pores or cracks) and distribution and the fractal dimensions. The results indicate that the shear peak strength is linearly correlated with the joint roughness coefficient. Compared to the fractal characteristics of pore size (T2 spectra) and pore distribution (MRI), we demonstrated that there is a strong correlation between the peak shear strength and the fractal dimension of shear fracture surface, which suggests that the quantitative characterization of internal pores and shear strength variations in cement mortar specimens can be achieved through the numerical values of fractal dimensions.

Assessing the impact of pore‐fracture structures on permeability sensitivity in tectonic coal using computerized tomography scanning

AbstractThe heterogeneity in permeability of coal reservoirs is primarily attributed to the considerable variation in the morphologies and structures of microscopic pore‐fractures, shaped by complex geological processes. This study emphasizes the necessity of understanding the impact and governance of these morphological and structural variations in pore‐fractures across different types of coal bodies on their permeability. Utilizing computerized tomography scanning and three‐dimensional imaging, we examined coal samples from the Datong coalfield in the southeastern Qinshui Basin, Shanxi Province, to characterize the pore‐fracture morphologies and structures distinct to various coal‐body types based on tomographic data. This introduces a methodology for assessing the influence of microscopic pore‐fracture parameters, such as porosity, specific surface area, tortuosity and fractal dimension, on permeability sensitivity. This is achieved through the application of the modified Kozeny–Carman equation and a fractal permeability model. Findings indicate a predominance of slab fractures in raw coal, whereas fragmented coal under weak brittle deformation exhibits small, isolated pore‐fractures with minimal diameter and volume and poor connectivity. In contrast, granular coal subjected to strong brittle deformation features extensive clusters of large pore‐fractures with significant diameter and volume, enhancing connectivity. Moreover, permeability predictions are refined by integrating the modified Kozeny–Carman equation with tomographic data, offering a more precise understanding of the permeability across different coal bodies.

Fractal analysis on the surface topography of Monocrystalline silicon wafers sawn by diamond wire

The direct impact of diamond wire slicing on the cost of subsequent silicon wafer processing underscores the significance of reducing slicing costs for both the integrated circuit and photovoltaic industries. Effectively and reasonably evaluating monocrystalline silicon wafer quality is thus a crucial challenge. In this paper, fractal theory was used to analyze the monocrystalline silicon wafer sawn surfaces combining with surface roughness measurement. The findings reveal the presence of scratches, pits, and prominent saw marks on the sawn surface. Notably, the surface roughness is significantly higher along the feed direction compared to that along the wire direction. Moreover, an increase in the ratio of feed speed to wire speed leads to a rougher surface. The fractal dimension of the sawn surface is inversely proportional to its roughness, indicating that a higher fractal dimension corresponds to a finer surface texture. Moreover, the two dimensional fractal dimension can effectively capture the anisotropic characteristics of monocrystalline silicon wafers. The two dimensional fractal dimension of the wafer surface obtained by removing materials in ductile mode has a strong symmetry. However, the two dimensional fractal dimension of the surface obtained in brittle mode fluctuates obviously and is weakly symmetric. By combining traditional surface roughness analysis with fractal methods, we can gain insights into the material removal mechanisms involved in slicing monocrystalline silicon wafers using diamond wire, which is of great significance for optimizing diamond wire slicing.

3D fractal model with experimental analysis for assessing surface topography in EDM

This work presents a novel three-dimensional fractal model designed specifically for morphological analysis of specimens made by the EDM. Within the 3D fractal framework, an algorithm is developed to estimate fractal parameters such as fractal dimension and periodic length. AISI 316 stainless steel, dielectric media, electrode materials, and powder variations were used in experimental trials to validate the 3D fractal model. Fixed pulse time ratios were used to control the EDM process. The specimen having the lowest fractal dimension, the shortest periodic length, the least amount of surface roughness, and the least amount of ten-point height was discovered to be the one machined using kerosene oil dielectric, brass electrode, graphite powder, and 1.0 pulse ratio time. On the other hand, the specimen machined using copper electrode, graphite powder, transformer oil dielectric, and 1.5 pulse ratio time produced the largest periodic length, maximum surface roughness, fractal dimension, and ten-point height. The developed 3D fractal model evaluates the EDM process well and provides insightful information on how to improve surface properties.

Sphractal: Estimating the Fractal Dimension of Surfaces Computed from Precise Atomic Coordinates via Box‐Counting Algorithm (Adv. Theory Simul. 6/2024)

Sphractal: Estimating the Fractal Dimension of Surfaces Computed from Precise Atomic Coordinates via Box‐Counting Algorithm (Adv. Theory Simul. 6/2024)

A New R35 and Fractal Joint Rock Typing Method Using MICP Analysis: A Case Study in Middle East Iraq

Reservoir characterization in carbonate formations plays a crucial role in understanding the complex pore structures and permeability properties. While absolute pore-throat radius (APTR) and R35 have been widely accepted in pore-scale rock typing, they fall short of providing detailed pore-throat distribution (PTD) information. To address this limitation and enhance PTD indication during rock classification, we introduce a novel approach—the R35 and fractal joint rock typing method—incorporating a new parameter, Dn (fractal dimension). This method is developed based on the analysis of mercury injection capillary pressure (MICP) data obtained from 20 carbonate samples in the Middle East, specifically Iraq. We delve into the discussion of APTR and R35 methods, employing both PTD and thin-section images to gain comprehensive insights into rock typing. Our approach incorporates a whole curve fractal-based model to determine the fractal dimension, Dn. The analysis reveals that Dn in each R35 rock type exhibits a decreasing trend with higher Hg intrusion peaks and wider pore-throat radius, providing valuable information on the distribution of pores within the rock samples. Furthermore, we extend our analysis to include the surface fractal dimension, Ds, obtained through two-dimensional (2D) fractal analysis on typical pores in binary thin-section images. The consistent decreasing trend of Dn aligns with the findings from Ds analysis, underscoring the effectiveness of parameter Dn in capturing the intricate pore structures within carbonate formations. Our results suggest that parameter Dn has the potential to serve as a standalone criterion in rock typing, eliminating the need for pretyping by R35 or APTR. The versatility of Dn as an indicator of pore distribution and complexity underscores its significance in advancing our understanding of carbonate reservoirs. As a recommendation, further exploration through additional fractal-based analyses is encouraged to solidify the role of parameter Dn in becoming a pivotal factor in the evolving field of rock typing.

Fracture behaviors of sustainable recycled aggregate concrete under compression-shear loading: Laboratory test, numerical simulation, and environmental safety analysis

A series of compression-shear tests were carried out to investigate the shear failure of sustainable recycled aggregate concrete, and shear failure surfaces were scanned by a 3D laser scanner. The effect of recycled coarse aggregate (RCA) replacement rates and normal stresses on the shear strength, failure mode, and fracture morphology were analyzed. Meanwhile, discrete element method was used to study shear behaviors at a mesoscale. Finally, the carbon emission was calculated to evaluate the effect of RCA on the environment. The result shows that the shear strength, cohesion c, and internal friction angle φ all decrease as the RCA replacement rate increases. The shear damage of RAC can be aggravated by increasing the RCA replacement rate. Fractal dimension and joint roughness coefficient of shear failure surface tend to increase with increasing RCA replacement rate and normal stress. In addition, more microcracks can be found in coarse aggregate with the increase in RCA replacement rate, and the increasing normal stress can cause more microcracks. Given the nonlinear development of stress-displacement, a modified nonlinear constitutive model is proposed. Finally, the total carbon emissions of recycled aggregate concrete are calculated considering production, transportation, and construction, the carbon emission of 100% sustainable recycled aggregate concrete is 10.2% less than that of 100% natural aggregate concrete.

Study on coal pulverization characteristics and gas desorption mechanism based on impact crushing experiment

The coal's particle size distribution properties after pulverization and the gas desorption behavior driven by pulverization are of profound meaning to the study of coal and gas outburst mechanism. In this paper, based on the impact crushing experiment, the tectonic coal and primary coal are crushed under different impact energy conditions. After screening the broken coal, the particle size distribution law is analyzed, and the characterization function suitable for the particle size distribution of coal particles after crushing is determined. The relationship between crushing work and new surface area and fractal dimension of coal body is discussed. The consequences indicated that the mass proportion of tectonic coal below 0.074 mm particle size is much huger than that of raw coal. G-S, R–R, and fractal distribution model describe the best particle size distribution of the two coals in the scope of 0.074∼4 mm. The new surface area added increases with the crushing work, and the tectonic coal is 1.34–1.96 times that of the raw coal. The fractal dimension diminishes first and then increases with the crushing work ratio. In addition, the gas desorption amount of coal particles with different particle sizes after coal pulverization was measured, and a dynamic model suitable for coal pulverization-driven gas desorption was established, and the experimental results were verified. The research results of this paper can provide experimental and theoretical basis for the analysis of energy dissipation in coal and gas outburst.