Fractal Dimension Research Articles

Rockery morphology based on quantitative analysis of shading

The rockeries of classical Chinese gardens are masterpieces of classical Chinese garden art and form a key element of garden heritage. Consequently, a quantitative study of “rockery shadows” is of considerable importance to the study of “rockery forms.” In this study, we selected the representative North Rockery in Zhanyuan Garden of Nanjing and Ruiyunfeng, Guanyunfeng, and Yulinglong as the objects of the study. This study aims to explore the shadow images of rockeries using image extraction and targeted quantitative analysis methods. Macroscopically, the overall shape of rockery shadows was described using fractal dimensions; microscopically, the aspect ratio, angle, and refinement indexes of each shadow based on each observation angle of the rockery were measured using the PAT-GEOM plug-in in ImageJ software. SPSS Statistics was used for the normal distribution test of the angular distribution data. Consequently, the shadow data of the North Rockery in Zhanyuan Garden and Ruiyunfeng, Guanyunfeng, and Yulinglong, respectively, were analyzed and compared, and four rockery-shadow laws were derived. Finally, the results were applied to the design of the rockery morphological translation based on quantitative analysis of the shadows. The approach presented here will enhance landscape design, support environmental planning, and preserve cultural heritage.

Investigating the compaction and the mechanical behaviors of coal gangue as subgrade filler and constructing highway subgrade in practice

Using coal gangue as subgrade filler (CGSF) can address the accumulated issues of coal mine waste, but also save the constructing costs, which has the important ecological and engineering practical value. The well-graded limit of CGSF was captured based on the fractal model grading equation (FMGE). The large-scale vibration compaction test (LCT) shows that particle grading has a remarkable influence on the compaction of CGSF, with the increase of fine particle content, the dry density of sample first increases and then decreases. On this basis, the optimal range of particle grading was obtained. A series of large-scale triaxial tests (LTT) were conducted to investigate the mechanical behaviors of CGSF. The results shown that (1) the peak deviatoric stress shows a well-quadratic correlation with the fractal dimension. The optimal grading range of CGSF captured by LTT and LCT is basically the same, hence the suggestion of acquiring the optimal grading through LCT was given. (2) Increasing the compaction degree (DC) can significantly improve the strength of CGSF if DC is smaller, while when DC is greater, the strength improved by increasing DC will not be significant. (3) The grading and DC have a significant impact on the cohesive force of CGSF, while have little effect on internal friction angle. In addition, the research results provide good guidance for the constructing of a highway subgrade in practice.

Fundamental cracking performance of asphalt-filler mastics with hydrated lime

This study evaluates the impact of hydrated lime (HL) on the cracking performance at intermediate temperature of asphalt-filler mastics, specifically as a partial replacement for limestone filler (LF). HL, is an active filler known for its multiple benefits in asphalt applications. The Double Edge Notched Tension (DENT) test was used to assess fracture properties, including specific total work of fracture, plastic work of failure, essential work of fracture, and average critical tip opening displacement. The results revealed several key findings: a notable loss of ductility and toughness with increasing HL dosages, a stiffening effect evidenced by increased peak loads and reduced failure deformation, and a decrease in the specific total work of fracture, indicating a loss of fracture energy. However, an enhancement in fracture resistance to plastic deformation is observed with HL addition, although identifying trends at high dosages remains challenging due to complex interactions between HL and LF. Furthermore, using low volumetric concentrations of HL minimizes the adverse impact on fracture performance while leveraging its other beneficial active filler properties. These findings suggest that while HL can improve certain fracture properties, optimal dosages are crucial to avoid adverse effects. Overall, this paper provides an updated and comprehensive study of the fracture of mastics at intermediate temperatures, which adds up to the understanding of asphalt-filler interactions, which can potentially lead to further investigations, such as how the partial replacement of LF with HL affects asphalt mixtures’ performance-related characteristics, particularly the fatigue cracking resistance.

Heart Shape to Fracture Distance: Characterizing Hydraulic Fracture Propagation before Hits

Summary Estimating the distance from the hydraulic fracture tip to the monitor well can be useful for fracture characterization, well spacing optimization, and preventing parent-child well interference. A heart-shaped signal is referred to as the extensional precursor of a fracture hit recorded by crosswell strain measurements and can serve as a vital tool for such estimation. This study incorporates the 3D displacement discontinuity method (DDM) to understand the impact of fracture geometry and monitor well offset on the heart-shaped signal’s characteristics. Results from numerical simulation and analytical solutions reveal a strong linear correlation between the spatial extent of the heart-shaped signal and the fracture tip distance. This relationship was further developed to predict tip distance using field data from the Hydraulic Fracture Test Site 2 (HFTS2). A reasonable approximation result from field data further validates the methodology. In addition, it is worth noting that the estimation accuracy depends on the ratio between fracture dimension and tip distance. The findings of this study offer a novel approach for real-time monitoring and characterizing hydraulic fracture propagation, which can be further used for well spacing optimization in unconventional and enhanced geothermal system reservoir development, as well as caprock integrity monitoring for carbon sequestration projects.

Development and application of a thermo-hydro-mechanical coupling test system for a single rock fracture

A thermo-hydro-mechanical coupling test system was developed to study the physical and mechanical properties of a single rock fracture under the interaction of multi-physics field. The test system mainly contained a loading system, a shear-flow box and sealing system, a high-temperature environment box and a seepage system. The test system can conduct the normal compression and shear-flow coupling test under complex stress paths within 300 °C. Two load cells were set at both ends of the shear direction to solve problem of inaccurate shear force monitoring. The unique shear-flow box was designed to solve the dynamic sealing problem. To verify the reliability of the test system, the seepage tests during heating–cooling process and a shear-flow coupling test for a single granite fracture under various boundary conditions were carried out. The test system provides research equipment for understanding the interaction mechanism of multi-physical field in deep underground engineering.

Dendritic copper nanostructures for ultrasensitive detection of rhodamine 6G and Congo red

Surface-Enhanced Raman Scattering (SERS) is a fast-responding and non-destructive ultra-sensitive detection technique that can replace traditional chromatographic and spectroscopic analysis techniques. In this experiment, dendritic copper nanostructures were prepared by the solid-state ionics method and the vacuum hot evaporation process with an applied current of I=9μA. Using them as a SERS substrate, Raman detection was performed on the dye molecules Rhodamine 6G (R6G) and Congo Red (CR) to determine SERS enhancement performance. It is found that the growth of the prepared copper nanostructures has a top-end dominant phenomenon. The fractal dimension of the copper nanostructure was calculated using fractal theory, and it was revealed to be 1.560. Scanning electron microscopy (SEM) results showed that a large number of 60-80 nm regularly arranged copper nanoparticles were attached to the surface of the dendritic copper nanostructure. This is beneficial to increase the surface roughness of the substrate and improve the detection level of dye molecules. Subsequently, copper nanostructures can sensitively detect R6G and CR solutions as low as 1×10-13 mol/L and 1×10-7 mol/L, reaching single-molecule levels. The Raman mapping experiment showed that the mean relative standard deviation (RSD) values of R6G and CR were 12.3% and 12.9%, indicating high uniformity and reproducibility of the structure. This structure effectively achieves the fusion of uniformity, repeatability, and sensitivity, and shows broad application prospects in food detection.

Multi-stage evolution characteristics and particle size effect of sandstone granules subjected to cyclic loads

Broken granules of different sizes and poor bearing capacities produced by repeated mining of coal seams contribute to natural disasters, such as rock strata movement. In this study, mechanical tests were performed on broken sandstone granules to explore their compaction and recrushing characteristics under repeated mining. The change characteristics of the relevant mechanical parameters of sandstone granules with different particle sizes and changes in energy density were examined, and compaction and recrushing mechanisms of sandstone granules under cyclic loading were identified. The results show that an increase in the initial particle size of sandstone granules has a significant effect on the increase in the strain under load. The cumulative dissipation energy corresponding to final crushing was greater for granules with larger initial sizes. The particle size of sandstone was found to be proportional to the fractal dimension after compaction and crushing. The probability of granular particles with larger initial particle sizes being broken into smaller particles is greater, resulting in the porosity attenuating most rapidly during the stress loading process of granular particles under repeated mining. Therefore, the degree of sample breakage was relatively severe. Particles were rearranged after compaction and crushing. Repeated mining results in dynamic evolution of the pore compaction, deformation, and fractur of the rock mass, as well as stable recombination. The study results provide important theoretical support for understanding of the movement mechanism of rock strata in the goaf under repeated mining.

Intensification of flocculation efficiency in multi-stage reactors by optimizing the multi-cone segment configuration

Flocculation is widely used in water treatment, and creating an optimal environment for floc growth is essential to enhancing flocculation efficiency. In this paper, we developed a multi-stage flocculation reactor (MFR) with strong, moderate, and weak mixing zones. Specifically, multi-cone structures with varying diameter ratios (DR) were incorporated in the moderate zone to promote floc growth. Computational simulations revealed that an MFR with a DR of 0.5 generated a uniform and symmetric flow field, reducing zero-velocity zones and forming distinct vortices within cones. This environment facilitated gradual floc growth while minimizing breakage. When applied to treating fracturing flowback fluid, a complex and challenging wastewater, the MFR produced larger and denser flocs, with an average chord length of 180 μm and a fractal dimension of 1.7170. These flocs exhibited high viscoelasticity, with a strength factor of 68.22 and a recovery factor of 47.77, indicating superior shear resistance and recovery ability after breakage. Consequently, the MFR achieved COD and turbidity removal rates of 92.05% and 93.61%, respectively, outperforming a stirred flocculation reactor, which achieved rates of 72.29% and 83.53% for the same parameters. Additionally, the MFR demonstrated excellent pollutant removal efficiency for both mineral processing wastewater and dyeing wastewater. With gradually decreasing energy along the flow direction, the MFR provided appropriate mixing intensities for different stages of floc growth, promoting efficient formation. These findings underscore the MFR’s potential as a highly effective solution for wastewater treatment applications.

A model characterising the fractal supercritical adsorption behaviour of methane in shale: Incorporating principles, methods, and applications

The pore structure of shale is patently heterogeneous, with a conventional adsorption model being difficult to characterise the complex adsorption behavior of methane in shale. Therefore, a model is being proposed to characterise the fractal supercritical adsorption behavior of methane in shale. The principles and derivation of the model have been described in detail. The results showed that the adsorption characteristic parameters relating to different scale pores were affected by temperature and fractal dimension to different degrees. In addition, the relationship between isosteric heat of adsorption (qst) and adsorption capacity of different samples revealed different trends. The absolute values of ΔS of FC-66 and FC-72 decreased linearly. This might have been because of the fact that the adsorption saturation’s inflection point, which caused a fall in the absolute value of the entropy change (ΔS), was not reached. Compared with traditional adsorption models, the proposed model could reflect menthane’s adsorption in shale more accurately. The research results provide theoretical support for the evaluation of deep shale gas reserves, including the optimisation of exploitation strategies.

Radiative Pressure on Fractal Dust Grains in Oxygen-rich AGB Stars

There is still considerable debate on the exact composition of grains formed in the outflows of O-rich, asymptotic giant branch stars. Estimates of the expected condensation distances based on radiative transfer calculations show that iron-free silicates can condense close to the star but typically lack the opacity to drive an outflow unless they are large enough that radiation pressure due to scattering on the grains becomes significant. Iron-containing silicates have a much higher absorption opacity, but due to this stronger absorption, their expected condensation location is well beyond the expected dust formation zone. Recent measurements of the efficiency of SiO condensational growth have shown that this rate is low. The result of this low growth efficiency is that nucleation may persist for longer, giving a larger number of smaller primary particles, leading to an increased likelihood of particle aggregation in these outflows. In this work, we examine how the radiation pressure changes with the possible aggregation of these primary particles into fractal aggregates. Opacity calculations are made using optical properties of both forsterite and astronomical silicate for aggregates containing up to 256 primary particles and for fractal dimensions of 1.8 and 2.8. For primary particles of radius less than ∼0.1 μm, aggregation leads to an enhancement of the radiation pressure over an equivalent cloud of isolated primary particles.

Revealing the transformation of spatial structure of greater Cairo: Insights from satellite imagery and geospatial metrics

Greater Cairo (GC) is the seventh-largest metropolitan city globally. In recent decades, GC has witnessed massive urban expansion, which has yet to be empirically measured or characterized. This research seeks to explore the patterns of urban growth and changes in urban form in GC from 1973 to 2021 using remote sensing and geospatial metrics. Six Landsat images in 1973, 1984, 1992, 2003, 2013, and 2021 were used to explore urban growth in GC. Urban Land Density Function (ULDF), Landscape Expansion Index (LEI), and Fractal Dimension Index (FDI), were utilized to analyze various aspects of urban expansion. Additionally, the ratio of Land Consumption Rate (LCR) to Population Growth Rate (PGR) was calculated to assess the correlation between urban expansion and population changes. Our results reveal a substantial addition of 203 thousand hectares of urban areas over the last half-century. The ULDF highlights GC’s transformation from a condensed to a scattered urban landscape, particularly with the establishment of the New Administrative Capital in the vicinity. Edge expansion emerged as the principal type of urban growth throughout the research duration. The consistent increase in FDI from 1.44 in 1973 to 1.75 in 2021 signifies the fragmentation of the urban landscape. Moreover, the progressive rise in LCR from 0.023 in 1984 to 0.050 ​ha per person in 2021 suggests a steady expansion of low-density urban areas. Notably, the LCR-to-PGR ratio escalated from 0.878 in 1984 to 2.428 in 2021, indicating that urban expansion surpassed population growth. These results will help further understand the characteristics of GC’s urban expansion and may offer a scientific foundation for policymakers to reconsider the current horizontal expansion policy in GC.

Evolution of pore structure and cement composition during weathering of conglomerate at Mogao Grottoes in alkaline and arid regions, NW China

Numerous conglomerate grottoes have been excavated and preserved in the arid and alkaline regions of NW China. After millennia of weathering, these conglomerate grottoes have suffered from various weathering issues (such as seepage and collapse) that are internally governed by the distribution of pore networks and the degree of cementation. However, studies on conglomerate weathering have mostly been reported in humid and tropical climates, where intensive chemical weathering occurs. Additionally, characterizations of pore evolution during conglomerate weathering have been restricted due to limitations in detecting pore size and pore type using a single pore characterization method. To fill this gap, this study jointly used mercury intrusion porosimetry (MIP) and computed tomography (CT) to characterize the pore evolution process during conglomerate weathering at Mogao Grottoes, a world heritage site in NW China. Mineralogical, hydrological, mechanical and microscopic characterization were also conducted to understand the changes in compositions of bulk samples and cements during weathering. Results revealed that the majority of pores in conglomerate are concentrated in macropores > 5 μm, followed by mesopores and micropores. During weathering, volume fraction of macropores significantly increases and that of micropores decreases, with median/average pore diameters experiencing an 8- to 10-fold increase. Permeability, porosity and fractal dimension exhibited strong positive linear correlations with median/average pore diameters. Throughout weathering, the abundances of calcite and clay minerals show notable decrease, particularly in cements, resulting in an enrichment of feldspars. Frequently occurring sand-carrying wind activities and salt weathering in arid regions are inferred to be responsible for the major cement loss and porosity generation through physical processes. Our combined MIP-CT method for pore network characterization is also applicable to other lithologies with a wide range of pore sizes. The proposed mechanism of pore evolution and cementation failure provides a scientific base for protecting stone heritages in arid zones.

Effect of fractal dimension on mechanical behaviour and energy absorption of Menger sponge-inspired fractal structures

Effect of fractal dimension on mechanical behaviour and energy absorption of Menger sponge-inspired fractal structures

Fabrication and characterization of foam ceramics from recycled waste concrete powder and waste glass powder

Incorporating multi-solid wastes to prepare foam ceramics has attracted widespread attention due to its excellent characteristics and environmental protection. In this study, foam ceramics were prepared by sintering method using recycled waste concrete powder (RWCP) and waste glass power (WGP) as the major material, Na2CO3 as foaming agent and borax as fluxing agent. The effect of sintering temperature on the physical properties and pore structure of foam ceramics were studied. The phase transformation and reaction mechanism were also discussed. The results reveal that when the RWCP content was 40%, the WGP content was 60% and the sintering temperature was 890 °C, the foam ceramics showed the best overall performance. The compressive strength, bulk density, water absorption and porosity were 2.39MPa, 0.83g/cm3, 8.27% and 66.68%, respectively. At this time, the pore size distribution was optimal, and mesopore occupied the vast majority. The fractal dimension of pore reached the minimum of 1.09. In addition, The XRD results showed that WGP and RWCP gradually depolymerized and reconstructed under the action of flux. Then the foam ceramics with mullite, pyroxene and wollastonite as the main phases were formed. The experimental results showed that it was feasible to prepare foam ceramics using RWCP. This study can provide a new train of thought for the reuse of RWCP.

Fractional quantum mechanics meets quantum gravity phenomenology

This letter extends previous findings on the modified Schrödinger evolution inspired by quantum gravity phenomenology. By establishing a connection between this approach and fractional quantum mechanics, we provide insights into a potential deep infrared regime of quantum gravity, characterized by the emergence of fractal dimensions, similar to behaviors observed in the deep ultraviolet regime. Additionally, we explore the experimental investigations of this regime using Bose-Einstein condensates. Notably, our analysis reveals a direct implication of this analogy: general experiments probing fractional quantum mechanics may serve as equivalent models of quantum gravity. We identify instances of nonlocal behavior in such systems, suggesting an analogous phenomenon of nonlocality in quantum gravity.

Study on the pore structure of eco-regenerated mortar using corn cob based on nuclear magnetic resonance

This study aims to investigate the influence of corn cob particles on pore structure of mortar. By using NMR technology, the pore parameters of fast-hardening sulfoaluminate cement mortar (SAC) and ordinary Portland cement mortar (PO) were measured and compared. The influence of corncob particles on the mechanical properties of eco-mortar was evaluated by uniaxial compression test. The results indicate that medium and large pores of both SAC and PO increases with corn cob particle, as well as the porosity. The fractal dimensions of harmful and multiple harmful pores are negatively correlated with the volume fraction of corn cob particles. The addition of corn cob particles reduced the overall fractal dimension of the mortar and decreased the complexity of internal pores. The probability density function of grayscale values transitions from being “thin and tall” to “short and wide”, with an increasing proportion of high grayscale values. This leads to an increase in the proportion of large pores within the mortar, resulting in a continuous deterioration of pore structure. The addition of corn cob particles greatly reduces the strength of SAC and PO, but the peak stress of SAC is higher than that of PO. When corn cob particle content is 30wt% and 50wt%, the peak stress of SAC is increased by 23.37% and 14.25%, respectively, compared with the peak stress of PO.

Fractal Modelling of Heterogeneous Catalytic Materials and Processes.

This review considers the use of fractal concepts to improve the development, fabrication, and characterisation of catalytic materials and supports. First, the theory of fractals is discussed, as well as how it can be used to better describe often highly complex catalytic materials and enhance structural characterisation via a variety of different methods, including gas sorption, mercury porosimetry, NMR, and several imaging modalities. The review then surveys various synthesis and fabrication methods that can be used to create catalytic materials that are fractals or possess fractal character. It then goes on to consider how the fractal properties of catalysts affect their performance, especially their overall activity, selectivity for desired products, and resistance to deactivation. Finally, this review describes how the optimum fractal catalyst material for a given reaction system can be designed on a computer.

Plastic strain localization in Bouligand structures

The Bouligand structure represents helicoidal stacking of aligned fibers; such a structure is widely observed in biological composites. Despite the progress in characterization of toughening caused by Bouligand arrangement of fibers, the inelastic deformation mechanisms of this structure remain elusive. In this study, we carry out calculations for plastic deformation of Bouligand structure, crossed-lamellar structure and the single lamellar structure. It is found that the single lamellar structure and crossed-lamellar structure can undergo necking, while in the Bouligand structure, plastic strain localization bands develop, which is accompanied by plastic rotating of fibers, lamellar twisting and lamellar delamination. Compared with crossed-lamellar structure, the Bouligand structure exhibits lower plastic energy dissipation. However, the Bouligand pattern can activate delamination of lamellae, generating high level of damage energy dissipation. The plastic deformation of Bouligand structure depends on the fracture properties of interface between adjacent lamellae. It is identified that the plastic dissipation of Bouligand structure increases with increasing cohesive strength of lamellar interface, and dominant shear bands emerge in the case of weak lamellar interface. The high strength of lamellar interface plays a role in promoting twisting of lamellae. We have further revealed the effect of the thickness of individual lamella on plastic deformation of the Bouligand structure. The thick lamella is capable of suppressing plastic strain localization in Bouligand structure, thereby giving rise to high plastic dissipation. The findings of this study shed new light on the development of bioinspired Bouligand-type materials.

Effect of pre-shock on the expanding fracture behavior of 1045 steel cylindrical shell under internal explosive loading

Explosively driven expansion fractures in metallic shells will be preceded by a compression shock wave, which inevitably changes the mechanical and microstructural state of the shell materials and thus affects their subsequent dynamic deformation and expansion fracture processes. To understand the influence of pre-shock pressure on the dynamic expansion fracture behavior of metallic shells, the incident shock waves with varying peak pressure were generated in a 1045 steel cylindrical shell by sweeping detonation wave loading. The pre-shock and subsequent expansion fracture processes were continuously diagnosed by combining a high-speed framing camera and the arrayed Photon Doppler Velocimetry (PDV) measurements to obtain the fracture strains at different axial positions (corresponding to different pre-shock pressures). The results clearly show that the pre-shock pressure during detonation loading has a significant effect on the expansion fracture properties of the 1045 steel cylinder; the fracture strain remarkably decreases with the increase of pre-shock pressure from ∼19 GPa to ∼20 GPa and then changes gently in the range of ∼20 GPa to ∼22 GPa. Metallurgical analyses reveal increased densities of microstructural defects, including dislocations and deformation twins, when a shock wave passes through, resulting in a reduced capacity for further defect storage in the shocked metals, ultimately leading to a decrease in the fracture strain during the subsequent expansion deformation process.

Specimen reconstitution method for the assessment of fracture properties of reactor pressure vessel steels

Routine testing of the fracture properties of irradiated surveillance specimens is a critical method for evaluating irradiation embrittlement in reactor pressure vessel (RPV) steels. Given the limited space in irradiation surveillance tubes, specimen reconstitution technology is expected to provide a viable solution to the shortage of irradiated specimens. This study investigates the specimen reconstitution method for 16MND5 steel 0.5 T CT specimens used in the RPV. By studying the local mechanical properties of the residual specimen and performing finite element analysis of the crack tip plastic zone, the appropriate size for specimen reconstitution is determined. The electron beam welding method is used to join the residual specimen and the base material, and the J-integral formula for the reconstituted specimen is determined based on the finite element analysis, taking into account the influence of the weld seam and the degradation of the mechanical properties of the residual specimen. The feasibility of the specimen reconstitution method is validated through fracture tests on base material specimens and reconstituted specimens at different temperature.