Three-dimensional Fractal Dimension Research Articles

Experimental Investigation on the Influence of Strength Grade on the Surface Fractal Dimension of Concrete under Sulfuric Acid Attack

The corrosion of alkaline concrete materials exposed to a sulfuric acid environment is becoming more and more prevalent, and its damage assessment is becoming more and more imperative. This study aims to describe the corroded surfaces of concrete with different strength grades (C30, C50, C80) in sulfuric acid environments in terms of their three-dimensional fractal dimension. Three kinds of concrete with varying strength grades, namely C30, C50, and C80, were immersed in a sulfuric acid solution with pH ≈ 0.85 for four distinct corrosion durations, specifically 0, 28, 56, and 165 days, in accelerated corrosion tests. The 3D laser scanning technique was utilized to capture the 3D coordinates of the surface points of the concrete cylinder before and after corrosion. The fractal dimension of concrete’s uneven surface before and after corrosion was computed via the cube covering method, and the mass loss of the concrete specimen was also obtained. The outcomes demonstrate that the three-dimensional fractal dimension provides a new method for characterizing the degree of corrosion deterioration of concrete samples affected by sulfuric acid via laser scanning technology. From the perspective of the appearance, mass loss, and fractal dimension of a rough surface in the sulfuric acid environment at a pH level of approximately 0.85, the degree of the corrosion deterioration of concrete is ranked from high to low as C80 > C50 > C30. These fractal dimensions of the concrete’s corroded surfaces with various strength grades increase rapidly in the initial period. However, as the corrosion time progresses, the growth rate of the corroded surface fractal dimension gradually decelerates and tends towards stability, which accords with the law of exponential function. The widespread belief is that the higher the strength grade of concrete, the better its durability; however, this pattern varies in sulfuric acid corrosive environments. Therefore, based on this research, it is recommended that in extremely acidic environments (i.e., very low pH), more attention should be paid to high-strength grades of concrete.

Study on the Difference of Fracture Extension in Hydraulic Fracturing of Marine and Continental Shales Based on Optical Scanning Technology

After hydraulic fracturing, the geometric characteristics of rock morphology is a crucial means for evaluating the effectiveness of fracture stimulation in enhancing production. In order to quantitatively analyze the surface undulations of marine and continental shale morphology further following hydraulic fracturing, a method for calculating the three-dimensional fractal dimension of fracture surfaces based on optical scanning technology is proposed. This method involves the acquisition of point cloud data using a 3D surface scanner. The obtained data are subsequently subjected to smoothing processes, followed by the reconstruction of the three-dimensional representation of the fractures. The box-counting dimension algorithm is employed to calculate the fractal characteristics of post-fracture morphology. The research results indicate that marine shale, due to its higher proportion of brittle minerals such as quartz and calcite, predominantly exhibits vertically oriented longitudinal fractures, perpendicular to the minimum horizontal stress. The average initiation pressure is 8% higher compared to continental shale, with an average fractal dimension of 2.24397. In contrast, continental shale, characterized by its high clay content and the development of natural fractures and bedding planes, is more prone to capturing hydraulic fracture features, resulting in predominantly single transverse shear fractures. The average fractal dimension is 2.087065, which is 7% lower than that of marine shale. These research findings offer a certain degree of guidance for the optimization of fracturing process parameters for different types of reservoirs.

The effect of silver nanoparticles on aerobic denitrifying bacteria during biological nitrogen removal: A new perspective based on morphological effects

The morphological effect-dependent biotoxicity of silver nanoparticles (AgNPs) toward aerobic denitrifying bacteria is unclear. In this study, the behavior and biotoxicity of spherical and flaky AgNPs on aerobic denitrifier Pseudomonas stutzeri were investigated based on morphological effects. The three-dimensional fractal dimension of flaky AgNPs with a rough surface and complex structure was 2.73, which was higher than that of spherical AgNPs. Flaky AgNPs more significantly inhibited the growth and nitrogen removal of P. stutzeri than spherical AgNPs, with average inhibition rates of 50.4% and 76.4%, respectively. Flaky AgNPs resulted in higher secretion of extracellular polymeric substances, lower activity of electron transport system and higher concentrations of reactive oxygen species and lactate dehydrogenase in P. stutzeri than spherical AgNPs, indicating that flaky AgNPs displayed stronger inhibitory effects on P. stutzeri. The results of flow cytometry and scanning electron microscopy revealed that flaky AgNPs caused more damage to the cell surface of P. stutzeri, resulting in a higher proportion of cell death. Furthermore, flaky AgNPs more significantly inhibited the expression of napAB and nirS in P. stutzeri than spherical AgNPs, up to 67.7%, resulting in lower nitrate and nitrite reductase activities and worse nitrogen removal efficiency. The results of this study can elucidate the morphology-dependent toxicity of AgNPs and provide new insights into the negative impacts of AgNPs on aerobic denitrification wastewater treatment.

Study of the Microstructure of Coal at Different Temperatures and Quantitative Fractal Characterization.

In order to understand the influence of underground coal fires on coal fractures and pores, mercury intrusion porosimetry (MIP) and scanning electron microscopy (SEM) are combined to study the development of coal pore and fracture under high-temperature treatment and calculate the fractal dimension to analyze the relationship between the development of coal pore and fracture and the fractal dimension. The results show that the volume of pores and fractures of the coal sample (C200) treated at 200 °C (0.1715 mL/g) is greater than that of the coal sample (C400) treated at 400 °C (0.1209 mL/g), and both are greater than the original coal sample (RC) (0.1135 mL/g). The volume increase is mainly due to mesopores and macropores, and the proportions of mesopores and macropores in C200 were 70.15 and 59.97% in C400. The MIP fractal dimension shows a decreasing trend with the increase of temperature, and the connectivity of coal samples improved with the increase of temperature. The changes in volume and three-dimensional fractal dimension of C200 and C400 showed the opposite trend and are related to the different stress of coal matrix at different temperatures. The experimental SEM images confirm that the connectivity of coal fractures and pores improves with the increase of temperature. Based on the SEM experiment, the larger the fractal dimension, the more complex the surface is. The SEM surface fractal dimensions indicate that the surface fractal dimension of C200 is the smallest and that of C400 is the largest, which is consistent with the observations made by SEM. The combination of the two fractal dimensions is used to characterize the self-similarity of coal using the fractal dimension difference. When the temperature increased to 200 °C, the unordered expansion of the coal sample resulted in the largest fractal dimension difference and the lowest self-similarity. When heated to 400 °C, the fractal dimension difference of the coal sample is the smallest, and the microstructure of coal shows a regular groove-like development.

Methodological Study on the Full-Range Pore Structure and Fractal Characteristics of the Tight Reservoirs

Scholars have used the method of splicing various experimental data to evaluate the full-range pore structure of tight reservoirs, but its applicability has not been discussed. When the fractal theory is used to characterize the heterogeneity of tight reservoirs, there is a lack of research on the controlling factors of segmented fractal pore size (the inflection points in the fractal curve) and the relationship between the fractal dimensions of different dimensions. In this paper, mercury intrusion porosimetry (MIP), N2 adsorption (N2-GA), and large-field splicing scanning electron microscopy (MAPs) were conducted to study the pore structure and full-range pore size distribution (PSD) of tight sandstone, and fractal theory was used to evaluate the heterogeneity of the reservoir. Combining the PSD of MIP and N2-GA, two parameters “cosine similarity” and “data similarity” were introduced to characterize the overlapping pore size range of the two experimental methods; then, the PSD of MAPs was used to verify the rationality of the data splicing. The results show that the characterization of the full-range pore structure should not only be based on simple data splicing but should focus on the data similarity of the overlapping pore size range. The study on the segmented fractal pore size of MIP shows that the segmented fractal pore size increases gradually with an increase in the main skeleton mineral content, the decrease in the clay mineral content, and the increase in the pore radius and porosity. For the same sample, the segmented fractal pore size is fixed and does not change with the calculation model of fractal dimension. Comparing the two-dimensional (2D) fractal dimension with the three-dimensional (3D) fractal dimension, calculating the 3D fractal dimension by +1 directly with the 2D fractal dimension is more applicable for the large pores, but it is not applicable for the small pores due to the influence of the extensive development of linear pores.

Fractal Analysis of Fuel Nozzle Surface Morphology Based on the 3D-Sandbox Method.

The dual oil circuit centrifugal fuel nozzle is made of martensitic stainless steel, which has complex morphological characteristics. The surface roughness characteristics of the fuel nozzle directly affect the degree of fuel atomization and the spray cone angle. The surface characterization of the fuel nozzle is investigated by the fractal analysis method. A sequence of images of an unheated treatment fuel nozzle and a heated treatment fuel nozzle are captured by the super-depth digital camera. The 3-D point cloud of the fuel nozzle is acquired by the shape from focus technique, and its three-dimensional (3-D) fractal dimensions are calculated and analyzed by the 3-D sandbox counting method. The proposed method can characterize the surface morphology well, including the standard metal processing surface and the fuel nozzle surface, and the experiments show that the 3-D surface fractal dimension is positively correlated with the surface roughness parameter. The 3-D surface fractal dimensions of the unheated treatment fuel nozzle were 2.6281, 2.8697, and 2.7620, compared with the heated treatment fuel nozzles dimensions of 2.3021, 2.5322, and 2.3327. Thus, the 3-D surface fractal dimension value of the unheated treatment is larger than that of the heated treatment and is sensitive to surface defects. This study indicates that the 3-D sandbox counting fractal dimension method is an effective method to evaluate the fuel nozzle surface and other metal processing surfaces.

Urban spatial form analysis based on the architectural layout -- Taking Zhengzhou City as an example.

The analysis of urban spatial form is the basic research of urban development. Traditional fractal research often focuses on the urban spatial layout, which cannot visually express the specific form, change characteristics and development trend of urban architectural spaces.The urban architectural form is simplified and the basic architectural form templates are extracted, and then, the correlations between architecture form and fractal dimension are built. The results of the case study show that the architectural layout of Zhengzhou City exhibits obvious fractal characteristics, and the combination of the two-dimensional and three-dimensional fractal dimensions is helpful for comprehensively revealing the architectural layout information. Moreover, the fractal dimension of buildings shows that the gradient from the inner to outer ring decreases, similar to the 'annual growth rings' of trees. Obvious differences exist in the fractal dimensions of urban buildings in different directions, reflecting the urban expansion direction. This study promotes the visualization of fractal theory and the expression of fractal theory in spatial gradient, providing theoretical and data reference for urban spatial form optimization.

Three-dimensional fractal dimension and lacunarity features may noninvasively predict TERT promoter mutation status in grade 2 meningiomas.

PurposeThe 2021 World Health Organization classification includes telomerase reverse transcriptase promoter (TERTp) mutation status as a factor for differentiating meningioma grades. Therefore, preoperative prediction of TERTp mutation may assist in clinical decision making. However, no previous study has applied fractal analysis for TERTp mutation status prediction in meningiomas. The purpose of this study was to assess the utility of three-dimensional (3D) fractal analysis for predicting the TERTp mutation status in grade 2 meningiomas.MethodsForty-eight patients with surgically confirmed grade 2 meningiomas (41 TERTp-wildtype and 7 TERTp-mutant) were included. 3D fractal dimension (FD) and lacunarity values were extracted from the fractal analysis. A predictive model combining clinical, conventional, and fractal parameters was built using logistic regression analysis. Receiver operating characteristic curve analysis was used to assess the ability of the model to predict TERTp mutation status.ResultsPatients with TERTp-mutant grade 2 meningiomas were older (P = 0.029) and had higher 3D FD (P = 0.026) and lacunarity (P = 0.004) values than patients with TERTp-wildtype grade 2 meningiomas. On multivariable logistic analysis, higher 3D FD values (odds ratio = 32.50, P = 0.039) and higher 3D lacunarity values (odds ratio = 20.54, P = 0.014) were significant predictors of TERTp mutation status. The area under the curve, accuracy, sensitivity, and specificity of the multivariable model were 0.84 (95% confidence interval 0.71–0.93), 83.3%, 71.4%, and 85.4%, respectively.Conclusion3D FD and lacunarity may be useful imaging biomarkers for predicting TERTp mutation status in grade 2 meningiomas.

Prediction of Three-Dimensional Fractal Dimension of Hematite Flocs Based on Particle Swarm Optimization Optimized Back Propagation Neural Network

Prediction of Three-Dimensional Fractal Dimension of Hematite Flocs Based on Particle Swarm Optimization Optimized Back Propagation Neural Network

Experimental Study on Grouting Seepage Characteristics of Single-Fractured Rock Masses with Different Inclination Angles under Three-Dimensional Stress

In underground engineering, the fracture dip angle and three-dimensional stress have great influence on the permeability and mechanical properties of fractured rock mass. Firstly, single-fracture samples with similar three-dimensional fractal dimensions and different dip angles were selected by using CT scanning technology. Then, the grouting material seepage tests under different dip angles and different three-dimensional stresses were carried out by self-developed single-fracture stress-seepage coupling true triaxial test system. The experimental data of stress-strain curve, grouting pressure, and grouting material flow rate of single-fracture specimen under different fracture dip angles and three-dimensional stress were obtained. The results show that the three-dimensional fractal dimension can be used as an index to measure the influence of fracture factors on grouting seepage test. The third principal stress σ 3 had a greater influence on the fracture width than the first principal stress σ 1 . The larger the fracture inclination, the lower the strength of the single-fracture specimen. As the third principal stress σ 3 increases, the grouting pressure increased while the grouting flow decreased. With the increase of the fracture inclination angle, the influence of σ 1 on the hydraulic conductivity became larger, while the influence of σ 3 on the hydraulic conductivity became smaller. Moreover, the expression of hydraulic conductivity of single-fracture specimen with different dip angles under three-dimensional stress was obtained by nonlinear fitting of hydraulic conductivity data.

Nano-CT measurement of pore-fracture evolution and diffusion transport induced by fracturing in medium-high rank coal

Fracturing, as a common fracture-making technique, can improve the permeability of coal seams to enhance fluid transport efficiency. To quantitatively evaluate the microscopic characteristics of medium-high rank coal, the loaded pore-fracture system was characterized by computerized tomography (CT) scanning under triaxial loading, followed by the analysis of stress-strain evolution, stress sensitivity and three-dimensional (3D) fractal dimension. Combined with snow algorithm and incompressible steady laminar flow simulation, the heterogeneous distribution of fluid pressure is investigated, focusing on the diffusion effect of gas transport. The results show that the strain of the high-rank coal Chengzhuang (CZ) in the linear elastic stage increases from 0.25% to 1.25%, greater than that of the medium-rank coal Qiyi (QY) from 0.75% to 1.63%, demonstrating a slight lag of the high-rank coal from the linear elastic stage into the yielding stage. The porosity of CZ changes from 1.66% to 13.58% and that of QY varies from 1.74% to 22.28% after fracturing, reflecting that the primary and secondary pores of the medium- and high-rank coals form a complex network structure for fluid transport through continuous connection-expansion. When the strain is between 0.75% and 1.25%, the stress sensitivity coefficient of CZ decreases from 0.13 to 0.02. Moreover, there are many mutation points in the 3D fractal dimension of coal samples after fracturing, mainly due to the generation of new pore-fractures at different locations of the computational domain. For fluid transport, the pressure of QY after fracturing spreads in a wider range than CZ, accompanied by more distribution of high fluid pressure. The diffusion coefficient of the fractured CZ is 350 times higher than that of the original coal under the gas pressure condition of 0.5 MPa, which provides the possibility for more gas to be converted from Knudsen diffusion to transition diffusion or Fick diffusion in the channel.

Research on Micro-Pore Structure and 3D Visual Characterization of Inter-Salt Shale Based on X-CT Imaging Digital Core Technology

Pore structure is the key factor affecting reservoir accumulation and enrichment behavior. Due to the complex mineral components and pore structure of shale oil reservoirs and strong heterogeneity, it is necessary to explore the micro-pore structure characteristics of inter-salt shale. In this study, in order to qualitatively and quantitatively analyze the pore structure of inter-salt shale reservoirs, as well as evaluate the mineral composition and its spatial distribution characteristics, three shale samples from the 10th cyclothem of the Eq3 (Eq34–10 cyclothem) inter-salt shale were selected to acquire 2D and 3D grayscale images by modular automated processing system (MAPS) and X-ray micro-computed tomography (Micro-CT), respectively. The color map of the inlaid characteristics of mineral aggregates was established by Quantitative Evaluation of Minerals by Scanning Electron Microscopy (QEMSCAN), and different mineral types in the grayscale image were determined. After that, the digital core technology was used to reconstruct the core in 3D, and the maximum sphere method was used to extract the pore network model, so as to realize the quantification of micron pore throats and the 3D visualization of inter-shale samples. Meanwhile, in order to compare the fractal characteristics of the pores of the samples, the two-dimensional and three-dimensional fractal dimensions of the three cores were calculated by combining the digital core technique with fractal theory. The study yielded several notable results: the pore structure of inter-salt shale reservoirs is complex and multi-scale, and the CT scanning digital core technology can effectively realize 3D visualization of rock microstructure without damage. The pore types of rock samples are mainly intergranular pores, interparticle pores, and dissolved pores, and the minerals are mainly dolomite, calcite, and glauberite. The micron pore throat radius of the rock sample is 0.5–13.9 μm, the distribution of coordination number is mainly in the range of 1–4, and the shape of the pore throat is mainly triangular and square. The pore space of inter-salt shale has suitable fractal characteristics, and the three-dimensional fractal dimension of the three cores is in the range of 2.41–2.49. In sum, this work used digital core technology to study the microscopic pore structure of inter-salt shale oil, establishing a basis for further understanding of the seepage characteristics and exploration and development of shale oil.

A New Three-dimensional Fractal Dimension Model to Describe the Complexity of Concrete Pores

A New Three-dimensional Fractal Dimension Model to Describe the Complexity of Concrete Pores

Fractal Design of Indoor and Outdoor Forms of Architectural Space Based on a Three-Dimensional Box Dimension Algorithm

The three-dimensional organic growth features of buildings have been given more and more attention according to the unique view representation regulations of the indoor construction industry, in order to optimize the usage and allocation efficiency of building space. To explore the configuration structure of building function space, the three-dimensional box dimension algorithm is used to measure the fractal dimension of the indoor and outdoor form of building space, and the two-dimensional and three-dimensional fractal characteristics of building space are compared and analyzed; the method of the second derivative is utilized for the automatic recognition of fractal scale-free areas, aiming at the imprecise problem of the artificial judgment of scale-free areas in earlier fractal research of interior and outdoor forms of architectural space. The experimental results show that the second derivative method can effectively and automatically identify the scale-free region. The coefficients of linear fitting are above 0.990, and the indoor and outdoor forms of architectural space have clear fractal characteristics. After applying the method, the three-dimensional fractal dimension grows, indicating that the interior and outdoor form usage of architectural space in this area is generally effective and compact.

True triaxial experimental study on fluid flow in single fracture with different dip angles under three-dimensional stress at different depths

The shape of ore body in deep underground engineering is complex, and its fracture development degree is affected by water abundance, fracture zone, fault and other factors. With the deepening of depth, the vertical and horizontal stress changes significantly, which has a great impact on the fracture opening. Therefore, the fracture dip angle and its three-dimensional stress state are the main factors affecting the fluid flow of fractures in a large area of ore body and its hanging and footwall rock mass. In this study, firstly, the three-dimensional fractal dimension of a single-fracture specimen was obtained by using CT (Computer Tomography) scanning technology and the single-fracture specimen with a similar three-dimensional fractal dimension was selected for testing. Then, the self-developed single fracture stress seepage coupling true triaxial test system is applied to test the seepage of single fracture samples with six different fracture dip angles under different stress states of underground depth. The experimental results show that the fluid flow and strain of fractured rock mass are comprehensively affected by the fracture dip angle and the stress state of the underground depth, and the influence degree of the fracture dip angle changes with the deepening of the underground depth. The above conclusions can be mutually verified in the aspects of stress-strain, water pressure and permeability coefficient. Furthermore, the relationship between the minimum injectable water pressure (MIWP) of fractured rock mass with different dip angles and underground depth is obtained, the expression of seepage coefficient of single fracture specimen with different dip angles and underground depth under three-dimensional stress is summarized, and the relationship between eh and em of single fracture specimen with different fracture dip angles is determined. This study provides a basic reference for the design scheme of hydraulic conductivity of fractured zone or fractured rock mass with different underground depths.

Effect of raindrop splashes on topsoil structure and infiltration characteristics

Raindrop splashing causes pore blockage in topsoil and reduces its infiltration capacity. In this study, simulated rainfall and infiltration tests were used to analyze the relationship between differences in the topsoil structure and the infiltration capacity of Lou soil on the Loess Plateau under different raindrop splashing conditions. The results show that, compared with soil existing under no raindrop splashing, the structure of the disturbed topsoil was altered, and the number of aggregates (A) and the three-dimensional fractal dimension (FD) degree were significantly increased (P < 0.05). At the same time, the pore volume (VP) and macroporosity (PL) were significantly reduced (P < 0.05). Aggregates splashed by 4.05 mm diameter raindrops were the most severely disturbed, and the pore clogging rate (CR) reached as high as 46.40%. After the broken aggregates blocked the pores, a dense and low permeable crust was formed on the topsoil, and it decreased the cumulative amount of infiltration and the infiltration rate. With increasing splash frequencies and raindrop diameters, the topsoil infiltration rate decreased gradually, and the soil infiltration reduction effect (IR) increased significantly. Three splashes later, the IR effect reached 94.21%. The study reveals the processes and mechanisms driving changes in soil infiltration characteristics from the perspective of changes in topsoil structure caused by raindrop splashing.

Investigation of Tangential Contact Damping of Rough Surfaces From the Perspective of Viscous Damping Mechanism

AbstractThe contact damping between rough surfaces has an important influence on the wear, vibration, contact fatigue, and energy dissipation between interfaces. In this paper, based on contact theory, a tangential damping mathematical model of rough surfaces is established from the point of view of viscous contact damping energy dissipation mechanism of asperities and considering the fractal characteristics of three-dimensional topography of rough surfaces. Through the combination of micro-contact modeling and macro dynamic testing of composite beams, the analysis results show that there are important evolution rules between tangential damping and surface fractal parameters and material parameters. The nonlinear relations between them are as follows: tangential contact damping is positively correlated with normal load, load ratio, and maximum contact area of asperity, and negatively correlated with fractal roughness; tangential contact damping increases first and then decreases with the increase of three-dimensional fractal dimension. The results of computational and experimental modal analysis show that the established mathematical model is feasible for predicting tangential damping. The study of tangential contact damping between surfaces can lay a foundation for improving the performance of assembly interfaces.

Nonintrusive investigation of large Al-kaolin fractal aggregates with slow settling velocities

Although a combination of aggregate characteristics dictate particle settling, it is commonly assumed that large particles have higher terminal velocities. This simplifying assumption often leads to overprediction of large aggregate settling velocities which in turn negatively impacts on estimates of sedimentation clarification efficiency. Despite its importance, little attention has been given to large aggregates with slow-settling velocities. This paper addresses this gap by investigating slow-settling velocities of large, heterodisperse and multi-shape Al-kaolin aggregates using non-intrusive methods. A particle image velocimetry technique (PIV) was applied to track aggregate velocity and a non-intrusive image technique was used to determine aggregate characteristics, including size (df), three-dimensional fractal dimension (Df), density (ρf), aggregate velocity (Vexp) and Reynolds number (Re). Results showed no strict dependence of settling velocity on large aggregate size, shape and density, as Al-kaolin aggregates with the same size exhibited different settling velocities. A comparison of the results with the well-known Stokes’ law for velocity modified by a shape factor showed that the settling velocities measured here can vary from 2 to 14 fold lower than the predicted values for perfect sphere-shape aggregates with the same density and size. Furthermore, results have also shown large Al-kaolin aggregate's drag coefficient (Cd) to be around 56/Re, for average fractal aggregate sphericity of around 0.58.

Self-similar propagation of spherically expanding flames in lean hydrogen–air mixtures

The self-similar propagation of expanding spherical flames in lean hydrogen–air mixtures at elevated pressure was experimentally investigated using a dual-chamber apparatus with the quartz windows that enabled observing a flame radius of up to 110 mm. The flame images of the fully developed cellular structure were recorded, and the largest values of r/rc and Pe for the dual-chamber apparatus were estimated. The acceleration exponent, α, corresponding to the fractal dimension in lean hydrogen–air mixtures at elevated pressure was evaluated by plotting the experimental flame radius as a function of time. In this study, the experimental values of α (α = 1.25–1.43) occur within a wide range of r/rc (r/rc = 2–15). The observed trend of α to increase with a decrease in equivalence ratio, φ, and an increase in initial pressure, Pi, is owing to the effects of the diffusional–thermal and Darrieus–Landau instabilities. The value of α increased with an increase in r/rc and saturated at α = 1.43 at r/rc > 10, and the saturated values of the hydrogen–air flame are within α = 1.4–1.5. The results demonstrated that the flame propagation in lean hydrogen–air mixtures was classified as laminar (r/rc < 1), acceleration (r/rc > 1), transition (1 < r/rc < 10), and self-similar (r/rc > 10) regimes. Consequently, the practical three-dimensional fractal dimensions for accidental hydrogen–air explosions are D3 = 2.24–2.33.

Three-Dimensional Fractal Characterization of Concrete Surface Subjected to Sulfuric Acid Attacks

Sulfuric acid corrosion on concrete structures is more prevalent and its damage evaluation becomes gradually imperative. This study attempts to characterize the sulfuric acid corroded surfaces of concrete in terms of three-dimensional fractal dimension. Accelerated sulfuric acid corrosion tests were conducted using concrete cylinders with six different corrosion durations, and the 3D coordinates of points on the surface of cylinders were captured using 3D laser scanning technique. The fractal dimensions were calculated using the cubic covering method, and the corrosion depth and mass loss of concrete were obtained correspondingly. The results indicated that fractal dimension can be considered as an indicator to evaluate the corrosion deterioration. The surface fractal dimension was positively correlated with the corrosion duration by a power function, where the value increased dramatically in the early stage, and gradually slowed down to maintain constant. Therefore, empirical functions to evaluate the corrosion depth and mass loss of concrete after sulfuric acid attacks were proposed. The parameters to establish the equivalent accelerated relation between experimental and natural conditions were also recommended under different degradation degrees.