Fractal Research Articles

Evolution of Retinal Neuron Fractality When Interfacing with Carbon Nanotube Electrodes.

Exploring how neurons in the mammalian body interact with the artificial interface of implants can be used to learn about fundamental cell behavior and to refine medical applications. For fundamental and applied research, it is crucial to determine the conditions that encourage neurons to maintain their natural behavior during interactions with non-natural interfaces. Our previous investigations quantified the deterioration of neuronal connectivity when their dendrites deviate from their natural fractal geometry. Fractal resonance proposes that neurons will exhibit enhanced connectivity if an implant's electrode geometry is matched to the fractal geometry of the neurons. Here, we use in vitro imaging to quantify the fractal geometry of mouse retinal neurons and show that they change during interaction with the electrode. Our results demonstrate that it is crucial to understand these changes in the fractal properties of neurons for fractal resonance to be effective in the in vivo mammalian system.

Research on the Prediction of Optimal Frequency for Vibration Mixing and Comparison on Initial Performance of Cold-Recycled Asphalt Emulsion Mixture.

The multicomponent cold-recycled asphalt emulsion mixture (CRAEM) has the ability of antireflection cracking between the base and the bottom surface layer, but it has secondary compaction and residual void, which is not conducive to crack resistance and fatigue performance. The application of high-frequency vibration mixing technology can reduce voids and improve crack resistance, but it is limited by the complexity of testing to determine the optimal mixing frequency. The fractal dimension of gradation is deduced by fractal theory, and the prediction model for optimal frequency is proposed. Dry, wet, freeze-thaw splitting tests, and rutting tests were employed to test the early mechanical properties of high-frequency vibration mixing specimens corresponding to different vibration accelerations, and mercury inclusion tests were utilized to compare the void distribution corresponding to the optimal mixing frequency and forced mixing, and to verify the prediction model for optimal frequency. The results indicate that the high-frequency vibration mixing technology is able to benefit the initial cracking resistance (28.1% increase), moisture stability (11.2% increase), and high-temperature stability on the macro level on the optimal frequency. Meanwhile, the void distribution structure can be optimized, reducing the proportion of harmful voids and increasing the proportion of transitional pores on the micro level. However, the freeze-thaw resistance needs to be further studied. This study reduces the number and cost of experiments to determine the optimal frequency, and provides theoretical guidance and technical support for the engineering application of the CRAEM.

Disintegration characteristics and mechanism of red clay improved by steel slag powder

Red clay is prone to softening and disintegration when exposed to water, often triggering severe geological disasters. To enhance its resistance to disintegration, this paper employs a homemade wet-dry cycling disintegration apparatus, combined with XRD, SEM, MIP, and fractal theory, to analyze the effects of different activators on the disintegration characteristics and microscopic mechanisms of SSP-improved soil. Under wet-dry cycling, the disintegration process of SSP-improved soil can be divided into five stages: rapid water absorption, saturation wetting, rapid disintegration, softening and dissolution, and stable non-disintegration. As the activator and SSP content increase, the resistance to disintegration of SSP-improved soil gradually enhances. After the addition of activators, the porosity, pore area, pore length and width, and probability entropy of SSP-improved soil decrease (i.e., 7 % NS < 7 % CSW < 7 % NH < undisturbed soil), while the average shape factor, regional probability distribution index, and fractal dimension increase. This is because activators promote ion exchange, cementation, crystallization, and carbonation reactions in SSP-improved soil, generating a substantial amount of hydration products such as C-S(A)-H, Ca(OH)2 and CaCO3, which fill and encapsulate inter-particle pores, thereby reducing water flow channels within the pores. In contrast, the pore structure between particles in undisturbed soil is simple and well-connected, providing favorable conditions for the migration and dissolution of fine particles and the loss of cementing materials, which accelerates the destruction and disintegration of the soil structure.

Quantitative dispersion characterization of cement particles in hardened cement matrix

Optimizing the cement particle distribution in hardened cement matrix is one of the effective means to utilize cement efficiently, and obtaining information about cement distribution is a prerequisite for this. In view of this, this paper investigates the quantitative dispersion of cement particles in hardened cement paste from an experimental perspective. Cement pastes with different water-binder (W/B) ratios were prepared as the objects. BSE images of these samples were acquired for further analysis, including Delaunay triangulation algorithm, nearest neighbor distance function G(r) and fractal theory. Results showed that when the W/B ratio increases from 0.36 to 0.42, the average cement particle spacing rises from 32.7 μm to 34.9 μm and the proportion of particles with spacing between particle centroids <20 μm decreases from 41.96 % to 25.98 %. Similarly, the increasing W/B ratio causes the nearest neighbor distance function G(r) to shift towards larger distances. When the W/B ratio is 0.34, the median distance between particles (R50) exhibits the most significant change. Additionally, the intricate details of cement particle spacing information can be accurately described by fractal theory. Parameters such as R50 and the proportion of particles with spacing between centroids <20 μm demonstrate strong fits with various fractal parameters. Notably, they exhibit optimal fitting with the spectral width of the generalized fractal spectrum, with an R2 value of 0.99.

Flexural Behavior of Self-Compacting PVA-SHCC Bridge Deck Link Slabs

This paper studied the flexural behavior of bridge deck link slabs made with polyvinyl alcohol–strain-hardening cementitious composites (PVA-SHCC). The tensile and flexural properties of the self-compacting PVA-SHCC with four volume fractions, i.e., 0%, 1%, 1.5%, and 2%, were evaluated first. Next, using the similarity theory, composite models with a geometric similarity ratio of 1:5 were designed to represent the bridge deck with the link slabs. The models considered three materials for link slabs, including concrete, cement mortar, and self-compacting PVA-SHCC, and two different curing ages at 7 and 56 days. Bending tests were performed to investigate the flexural behavior of the models. Based on the fractal theory, the cracking characteristics of the models with different types of link slabs were compared, and the relationship between fractal dimensions and the flexural behavior of the models was studied. Numerical models were built to correlate with the results from the bending tests. It was illustrated that the flexural behavior of the self-compacting PVA-SHCC link slab is better than that of concrete and cement mortar link slabs, where the crack initiation and propagation can be postponed. The results can provide theoretical support and design guidance for the self-compacting PVA-SHCC bridge deck.

Study on the mixed fracture characteristics of concrete-rock Brazilian disks with different fracture angles

The fracture characteristics of cracked straight-through Brazilian discs for concrete-rock composites (CSTBD-CRC) with different fracture inclination angles (Mode I-II) were investigated to evaluate the stability of concrete reinforcement engineering. Acoustic emission (AE) and digital image correlation (DIC) techniques were employed to study crack propagation and fracture modes in CSTBD-CRC Brazil splitting tests, as well as the entropy evolution of AE and DIC. Additionally, 3D scanning technology and fractal theory were utilized to investigate the effect of the fracture mode on the fracture surface morphology. The results indicate that both the fracture energy and peak load of the CSTBD-CRC increase with increasing fracture inclination angle. Moreover, the section fractal dimensions of 0°, 10°, 15° and 25° samples are 2.01879, 2.01528, 2.01186, and 2.01072. Shear fracture mode increases as the dip angle increases and the fracture surface morphology decreases in fractal dimension. Notably, sharp increases in entropy values for both AE and DIC occurred at peak load, suggesting their use as failure criteria for CSTBD-CRC specimens. Furthermore, cracks were found to initiate at the tips of preexisting cracks within the concrete. The displacement in the x direction undergoes the most significant changes on both sides of the crack during crack growth, while the strain in the x direction experiences its greatest variations at the crack location.

Particle shape analysis of calcareous sand based on digital images

Particle geometric is a key parameter that defines the eometric attributes of calcareous sand particles and is intricately related to their mechanical traits, such as compression and shear. The scanning electron microscopy and digital imaging were applied to capture the microscopic properties and geometric projections of calcareous sand. The qualitative analysis, conventional statistical methods and fractal theory were employed to describe the geometric morphology of sand particles. Additionally, we analyzed the structural and physical traits of calcareous sand based on its unique biological genesis. We developed a hypothetical structural-physical model for calcareous sand. Our findings revealed the interwoven reticulation on the surface of calcareous gravel particles, along with an uneven distribution of pores on the external surface. As the particle size increased, the global profile factor decreased and the angularity increased. The critical threshold for the variations in flatness, surface roughness, and circularity was observed at a particle size of 5 mm, with the particle size having a relatively minor effect on these characteristics for particles smaller than 5 mm. The shape of the calcareous sand particles exhibited fractal characteristics, with fractal dimension serving as a measure of surface smoothness, particle breakage, and strength. These experimental results could significantly enhance our understanding of the mechanical behavior of calcareous sand.

Study on the characterization of surface crack structure of coal based on fractal theory

Study on the characterization of surface crack structure of coal based on fractal theory

Linear fractal evolution characteristics of rock crack distributions during loading process

To investigate the fractal characteristics of rock crack distributions during the loading process, discrete element method was used to make rock samples with joints and record the crack propagation. The Box-counting method was used to quantitatively analyze the fractal dimension of the crack distribution at each moment, and the relationship between the crack fractal dimension and strain ratio was established based on fractal theory. The results indicated that the relationship between the fractal dimension of the crack distribution and strain ratio showed a strong linear characteristic. By transforming this linear relationship into a linear function, the slope of the function was found to be linked to the failure patterns of the sample, and a refinement coefficient (damage-fracture reduction factor) was identified from the slope as an effective basis for determining the degree of sample damage and fracture. The damage-fracture reduction factor can be categorized: 0.25–0.5 (spilt and fracture), 0.5–0.9 (synergy between fracture and damage), 0.9–1 (microcrack asymptotic damage). Owing to the linear fractal characteristics, an expression for the damage variables influenced by failure patterns can be established from the geometric aspect. In addition, the linear fractal characteristics of the cracks were verified in other acoustic emission and crack extension experiments.

Sustainable Transportation: Exploring the Node Importance Evolution of Rail Transit Networks during Peak Hours

The scientific and rational assessment of the evolution of node importance in rail transit line networks is important for the sustainability of transportation systems. Based on the complex network theory, this study develops a weighted network model using the Space L method. It first considers the network topology, the mutual influence of neighboring nodes of the transportation system, and the land use intensity in the station influence domain to construct a comprehensive index evaluation system of node importance. It then uses the covariance-weighted principal component analysis algorithm to more comprehensively evaluate the node importance evolution mechanism and analyzes the similarity and difference of the sorting set by adopting three different methods. The interaction mechanism between the distribution of important nodes and the evolution of land use intensity is explored in detail based on the fractal dimension theory. The Xi’an rail transit network is considered an example of qualitative and quantitative analysis. The obtained results show that the importance of nodes varies at different times of the day and the complexity of the morning peak is more prominent. Over time, articulated fragments with significance values greater than 0.5 are formed around the station, which are aligned with the direction of urban development, creating a sustainable mechanism of interaction. As the network’s crucial nodes in the center of gravity increase and the southern network expands, along with the increased intensity of the city’s land utilization, the degree of alignment in evolution becomes increasingly substantial. Different strategies for attaching the network, organized based on the size of Si can lead to the rapid damage of the network (reducing it to 0.2). The identification of crucial nodes highlighted in this paper serves as an effective representation of the functional characteristics of the nodes in transportation networks. The results obtained can provide a reference for the operation and management of metro systems and further promote the sustainable development of transportation networks.

Study on mechanical properties and damage characteristics of acid corrosion in granite based on NMR technology

In acidic environments, rock masses are frequently subjected to severe chemical corrosion, resulting in the initiation of numerous geological engineering disasters. This study aimed to collect physical and mechanical parameters of granite exposed to prolonged acid corrosion and analyze fracture characteristics using acoustic emission (AE) techniques. Additionally, it examined the evolution of pore structure and damage mechanisms through the use of low-field nuclear magnetic resonance (NMR) and fractal theory. The results demonstrate a monotonic decrease in mass, volume, density, P-wave velocity, and S-wave velocity of granite with increasing corrosion time. Particularly notable is the phased reduction observed in uniaxial compressive strength and elastic modulus. The transition from brittle to ductile failure in corroded granite is accompanied by a gradual decrease in internal fracture strength. The trend in the correlation dimension reveals the relationship between the formation time of the main fracture surface and the pore structure. Additionally, total porosity and macropores (D, Da) exhibit significant fractal characteristics. The fractal dimension correlates positively with the damage variable and inversely with uniaxial compressive strength and elastic modulus. This indicates that more severe pore structure damage leads to a higher fractal dimension and lower mechanical performance. Among these, Da demonstrates higher sensitivity in characterizing rock mechanical properties. These findings provide important basis for evaluating the stability of granite geotechnical engineering in acidic environments.

Multi-scale investigation on staged deterioration mechanism of sliding-zone soils induced by reservoir fluctuations

Water level fluctuations in the reservoir deteriorate soils and rocks on the bank landslides by drying-wetting (D-W) cycles, which results in a significant decrease in mechanical properties. A comprehensive understanding of deterioration mechanism of sliding-zone soils is of great significance for interpreting the deformation behavior of landslides. However, quantitative investigation on the deterioration characteristics of soils considering the structural evolution under D-W cycles is still limited. Here, we carry out a series of laboratory tests to characterize the multi-scale deterioration of sliding-zone soils and reveal the mechanism of shear strength decay under D-W cycles. Firstly, we describe the micropores into five grades by scanning electron microscope and observe a critical change in porosity after the first three cycles. We categorize the mesoscale cracks into five classes using digital photography and observe a stepwise increase in crack area ratio. Secondly, we propose a shear strength decay model based on fractal theory which is verified by the results of consolidated undrained triaxial tests. Cohesion and friction angle of sliding-zone soils are found to show different decay patterns resulting from the staged evolution of structure. Then, structural deterioration processes including cementation destruction, pores expansion, aggregations decomposition, and clusters assembly are considered to occur to decay the shear strength differently. Finally, a three-stage deterioration mechanism associated with four structural deterioration processes is revealed, which helps to better interpret the intrinsic mechanism of shear strength decay. These findings provide the theoretical basis for the further accurate evaluation of reservoir landslides stability under water level fluctuations.

An improved mathematical model of gas–water two-phase flows in fractured horizontal wells accounting for physical contact behaviors of fractures with total factor characteristics

In water-bearing gas reservoirs, water existence affects gas production performances due to two-phase flows occurring in matrix and fracture systems. Under the background of staged multi-cluster fractured horizontal wells, this work focuses on an improved gas–water two-phase flow model accounting for physical contact behaviors of fractures with total factor characteristics. Combining a point-convergence method with fractal theory and applying Laplace transform and Stehfest numerical inversion, analytical solutions of this proposed model are solved to validate against the field production data and numerical simulation results. Subsequently, the analytical nephograms of formation pressure and water saturation at different stages are given for the first time, throughout which formation pressure and water saturation distributions at different locations of the fractured reservoir embedded with segmented multi-cluster tree-shaped fracture networks are visually exhibited. The results demonstrate that draining area evolution is limited to each segment at early stage and mid-stage, whereas these multi-cluster segments are really integrated into the whole draining area at late stage. Thus, the moving boundary of fracture-controlled unit is also delineated based on water saturation nephograms and expands with continuous production from 65 to 180 m in the y axis direction, contributing to the effective fracturing area for a high productivity. Furthermore, the share of free gas and adsorbed gas are 83% and 17% at early stage, while free gas and adsorbed gas account for 57% and 43% of total gas at the late stage. Those findings contribute to high-efficient and sustainable development of unconventional gas resources.

Regulating pore structure of diatomite with alkali dissolution and its influence on humidity control performance

Alkali dissolution is an effective method to regulate the pore structure of porous mineral material. The main chemical composition of diatomite is amorphous SiO2, which can be dissolved in alkali dissolution. The diatomite samples with alkali dissolution treatment were systematacially characterized by the particle size analysis, low temperature nitrogen adsorption, MIP, fractal theory, SEM, TEM, XRD, FTIR and surface hydroxyl density to analyze the pore structure and surface properties. And the humidity control performance of diatomite was tested under different temperatures and relative humidities. The relationship among pore structure, surface properties and humidity control performance were analyzed. The results show that alkali dissolution can regulate the mesoporous and macroporous of diatomite, and some new microporous can generate at high alkali dosage. The specific surface area, mesoporous pore volume, proportion of macroporous volume, surface roughness, heterogeneity of pore structure and number of hydroxyl groups on the surface of diatomite are important factors which determining the humidity control performance. The humidity control performance of diatomite is positively correlated with the specific surface area, mesoporous volume, surface roughness, heterogeneity of pore structure and number of hydroxyl groups on the surface, while is negatively correlated with the proportion of macroporous volume.

A nonlinear seepage theory model is developed using nuclear magnetic experiment and fractal theory

A nonlinear seepage theory model is developed using nuclear magnetic experiment and fractal theory

Cave morphometric analysis: A review

Morphometric analysis is the quantification of shapes, which makes irregular shapes found in nature analyzable and comparable. Cave morphometry has been used for the genetic classification of caves, the digital reconstruction of their conduits, the decoding of their paleoenvironment, and other research purposes. Ratios and indices that have been derived from Euclidean geometry and application of fractal geometry onto karst features and topological parameters are the basic methodologies that have been used for shape quantification. This paper reviews the literature that focuses on methodologies used for morphometric analyses and the applications that these methodologies have found.

One-Dimensional Compression Fractal Theory and Experimental Verification of Coarse-Grained Soil

One-Dimensional Compression Fractal Theory and Experimental Verification of Coarse-Grained Soil

Universal superdiffusion of random walks in media with embedded fractal networks of low diffusivity.

Diffusion in composite media with high contrasts between diffusion coefficients in fractal sets of inclusions and in their embedding matrices is modeled by lattice random walks (RWs) with probabilities p<1 of hops from fractal sites and 1 from matrix sites. Superdiffusion is predicted in time intervals that depend on p and with diffusion exponents that depend on the dimensions of matrix (E) and fractal (D_{F}) as ν=1/(2+D_{F}-E). This contrasts with the nonuniversal subdiffusion of RWs confined to fractal media. Simulations with four fractals show the anomaly at several time decades for p≲10^{-3} and the crossover to the asymptotic normal diffusion. These results show that superdiffusion can be observed in isotropic RWs with finite moments of hop length distributions and allow the estimation of the dimension of the inclusion set from the diffusion exponent. However, displacements within single trajectories have normal scaling, which shows transient ergodicity breaking.

Surface roughness measurement using microscopic vision and deep learning

Due to the self-affine property of the grinding surface, the sample images with different roughness captured by the micron-scale camera exhibit certain similarities. This similarity affects the prediction accuracy of the deep learning model. In this paper, we propose an illumination method that can mitigate the impact of self-affinity using the two-scale fractal theory as a foundation. This is followed by the establishment of a machine vision detection method that integrates a neural network and correlation function. Initially, a neural network is employed to categorize and forecast the microscopic image of the workpiece surface, thereby determining its roughness category. Subsequently, the corresponding correlation function is determined in accordance with the established roughness category. Finally, the surface roughness of the workpiece was calculated based on the correlation function. The experimental results demonstrate that images obtained using this lighting method exhibit significantly enhanced accuracy in neural network classification. In comparison to traditional lighting methods, the accuracy of this method on the micrometer scale has been found to have significantly increased from approximately 50% to over 95%. Concurrently, the mean squared error (MSE) of the surface roughness calculated by the proposed method does not exceed 0.003, and the mean relative error (MRE) does not exceed 5%. The two-scale fractal geometry offers a novel approach to image processing and machine learning, with significant potential for advancement.

Iʿjāz of the Qurʿān for Using the Words Gold, Silver and Not Money

The production of knowledge on iʿjāz in economy is strongly influenced by the literature on scientific miracles of the Qurʿān. The interpretation of the verses of the Qurʿān pertaining to maʿāsh comes up against the binary logic aimed at constructing a scientific identity by summoning key concepts from mainstream economic thought. This article is based on philology as an art of reading well to shed light on the iʿjāz of the Qurʿān through the use of the words gold and silver and not that of money. It is structured around multidisciplinary approach and a qualitative analysis through the meanings in reference to the literature on the secrets of eloquence and evidence of the inimitability of the Qurʿān. The results showed that the iʿjāz in the use of the words gold and silver invites to live in the world of rizq, as an umwelt or 風土 fūdo. This mode of existence provides resilience against voluntary servitude to the system of monetary creation ex nihilo by the interest loan granted by commercial banks. Such particular way is a thematic interpretation of the verses relating to the means to acquit the rights of the Creator and those of the creatures. It converges like Hénon attractor as a chaotic fractal set with research in history, anthropology and numismatics which does not use the word money.