Cushioning Performance of Hilbert Fractal Sandwich Packaging Structures under Quasi-Static Compressions

The sandwich structure of cushioning packaging has an important influence on the cushioning performance.Mathematical fractal theory is an important graphic expression.Based on Hilbert fractal theory,a new sandwich structure was designed.The generation mechanism and recurrence formula of theHilbert fractal were expressed by Lin’s language,and the second-orderHilbert sandwich structure was constructed fromthermoplastic polyurethane.The constitutive model of the hyperelastic body was established by using the finite element method.With the unit mass energy absorption as the optimization goal,the fractal sandwich structure was optimized,and the best result was obtained when the order was 2.5 and the unit layer thickness was 0.75 mm.TheHilbert sandwich structure was compared with the rice-shaped sandwich structure commonly used in industry,and the Hilbert fractal structure had better energy absorption.This has practical significance for the development and application of newcushioning packaging structures.

Influence of Acid Treatment on Pore Structure and Fractal Characterization of a Tight Sandstone:A Case Study from Wudun Sag,Dunhuang Basin

In this study,X-ray diffraction,N_(2)adsorption(N_(2)A),and mercury intrusion(MI)experiments were used to investigate the influence of acid treatment on pore structure and fractal characterization of tight sandstones.The results showed that acid treatment generated a certain number of ink-bottle pores in fine sandstone,aggravated the ink-bottle effect in the sandy mudstone,and transformed some smaller pores into larger ones.After the acid treatment,both the pore volume in the range of 2–11 nm and 0.271–8μm for the fine sandstone and the entire pore size range for the sandy mudstone significantly increased.The dissolution of sandstone cement causes the fine sandstone particles to fall off and fill the pores;the porosity increased at first but then decreased with acid treatment time.The fractal dimension obtained using the Frenkel-Halsey-Hill model was positively correlated with acid treatment time.However,the total fractal dimensions obtained by MI tests showed different changes with acid treatment time in fine sandstone and sandy mudstone.These results provide good guiding significance for reservoir acidification stimulation.

Effect of particle gradation on pore structure and seepage law of solution in weathered crust elution-deposited rare earth ores

Both CT and Avizo software were used to explore the effect of particle gradation on the evolution characteristics of pore structure and seepage paths in weathered crust elution-deposited rare earth ores during leaching.The results showed that the pore areas in four kinds of ore samples before leaching were mainly concentrated in 10^(4)–10^(7)μm^(2),whose pore quantities accounted for 96.89%,94.94%,90.48%,and 89.45%,respectively,while the corresponding pore volume only accounted for 30.74%,14.55%,7.58%,and 2.84%of the total pore volume.With the decrease of fractal dimension,the average pore throat length increased,but pore throat quantities,the average pore throat radius and coordination number decreased.Compared with that before leaching,the change degree of pore structure during leaching increased with the fractal dimension decreasing.For example,the reduction rate of the average coordination number of ore samples was 14.36%,21.30%,28.00%,and 32.90%,respectively.Seepage simulation results indicated that seepage paths were uniformly distributed before leaching while the streamline density and seepage velocity increased with the fractal dimension decreasing.Besides,the phenomenon of the streamline interruption gradually reduced during leaching while preferential seepage got more obvious with the decrease of the fractal dimension.

Full-Sized Pore Structure and Fractal Characteristics of Marine-Continental Transitional Shale: A Case Study in Qinshui Basin, North China

Based on 10 shale samples collected from 4 wells in Qinshui Basin,we investigate the full-sized pore structure and fractal characteristics of Marine-Continental transitional shale by performing organic geochemistry,mineralogical composition,Nitrogen gas adsorption(N2 adsorption)and Nuclear Magnetic Resonance(NMR)measurements and fractal analysis.Results show that the TOC content of the shale samples is relatively high,with an average value of 2.44wt%,and the thermal evolution is during the mature-over mature stage.The NMR T2 spectrum can be used to characterize the fullsized pore structure characteristics of shale.By combining N2 adsorption pore structure parameters and NMR T2 spectrums,the surface relaxivity of samples are calculated to be between 1.7877 um/s and 5.2272 um/s.On this basis,the T2 spectrums are converted to full-sized pore volume and surface area distribution curves.The statistics show that the pore volume is mainly provided by mesopore,followed by micropore,and the average percentages are 65.04%and 30.83%respectively;the surface area is mainly provided by micropore,followed by mesopore,and the average percentages are 60.8004%and 39.137%respectively;macropore contributes little to pore volume and surface area.The pore structure characteristics of shale have no relationship with TOC,but strong relationships with clay minerals content.NMR fractal dimensions Dmicro and Dmeso have strong positive relationships with the N2 adsorption fractal dimensions D1 and D2 respectively,indicating that Dmicro can be used to characterize the fractal characteristics of pore surface,and Dmeso can be used to characterize the fractal characteristics of pore structure.The shale surface relaxivity is controlled by multiple factors.The increasing of clay mineral content,pore surface area,pore surface fractal dimension and the decreasing of average pore size,will all lead to the decreasing of shale surface relaxivity.

Fractal Characteristics of Fault Structures and Their Use for Mapping Ore-prospecting Potential in the Qitianling Area, Southern Hunan Province, China

Quantitative analyses of the spatial distribution of fault structures can provide a theoretical basis for forecasting prospective ore deposits. Characteristics and complexity of fault structure distribution in the Qitianling area, Southern Hunan Province, China, were quantitatively calculated and appraised by fractal and multifractal methods to evaluate the relation between fault structures and ore-prospecting potential. The results show that the lengths of faults can be modeled as multifractals. Multifractal spectra evidently reflect the characteristics of the scaling of fault structures. The box- counting dimension value （D） of fault structures is equal to 1.656, as indicates complexity of the spatial distribution of faults and favorable structural conditions for the formation of ore deposits. Moreover, the D values of sub-regions were calculated and isopleths of their fractal dimension values were plotted accordingly. Overlay analyses of isopleths of fractal dimension values and distributions of known ore deposits show that areas with the larger fractal dimension values of fault structures have more ore deposits. This spatial coupling relationship between D values and ore deposits can be used to forecast and explore other ore deposits. On the basis of complexity theory for ore-forming systems, three exploration targets with high D values were delineated as prospective ore deposits.

Fractal Dimension of Pore Structure of Combustible Cartridge Cases

A fractal pore structure model of combustible cartridge cases was established by virtue of the fractal geometry. Pore structure information, such as backbone fractal dimension and pore fractal dimension, of four kinds of combustible cartridge case were obtained by mercury intrusion porosimetry (MIP) . The formation mechanism of fractal pore structure of combustible cartridge was studied. The results show that the backbone fractal dimension consists of the component and influenced by the component number and size of components; the pore percolation fractal dimension reflects the pore structures of components; and the fractal dimension of pore structure is positively relative to the tensile strength of combustible cartridge case.

Fractal Structure of Debris Flow

One of the most remarkable characteristics of debris flow is the competence for supporting boulders on the surface of flow, which strongly suggests that there should be some structure in the fluid body. This paper analyzed the grain compositions from various samples of debris flows and then revealed the fractal structure. Specifically, the fractality holds in three domains that can be respectively identified as the slurry, matrix, and the coarse content. Furthermore, the matrix fractal, which distinguishes debris flow from other kinds of flows, involves a hierarchical structure in the sense that it might contain ever increasing grains while the total range of grain size increases. It provides a possible mechanism for the boulder suspension.

Visual Evaluation of the Morphologic Structure of Nonwovens Using Image Analysis and Fractal Geometry

Nonwovens are fiber materials which are based on nonwoven technologies. For the complexity and randomness of nonwovens morphologic structures, it is difficult to express them effectively using classical method. Fractal geometry gives us a new idea and a powerful tool to study on irregularity of geometric objects. Therefore, we studied on the pore size, pore shape, pore size distribution and fiber orientation distribution of real nonwovens using fractal geometry combined with computer image analysis to evaluate nonwovens’ morphologic structures.

Experimental analysis of pore structure and fractal characteristics of soft and hard coals with same coalifcation

Accurate and quantitative investigation of the physical structure and fractal geometry of coal has important theoretical and practical signifcance for coal bed methane(CBM)development and the prevention of dynamic disasters such as coal and gas outbursts.This study investigates the pore structure and fractal characteristics of soft and hard coals using nitrogen and carbon dioxide(N_(2)/CO_(2))adsorption.Coal samples from Pingdingshan Mine in Henan province of China were collected and pulverized to the required size(0.20–0.25 mm).N_(2)/CO_(2)adsorption tests were performed to evaluate the specifc surface area(SSA),pore size distribution(PSD),and pore volume(PV)using Braunuer-Emmett-Teller(BET),Barrett-Joyner-Halenda(BJH),and Density Functional Theory(DFT).The pore structure was characterized based on the theory of fractal dimensions.The results unveiled that the strength of coal has a signifcant infuence on pore structure and fractal dimensions.There are signifcant diferences in SSA and PV between both coals.The BJH-PV and BET-SSA obtained by N_(2)-adsorption for soft coal are 0.029–0.032 cm^(3)/g and 3.523–4.783 m^(2)/g.While the values of PV and SSA obtained by CO_(2)-adsorption are 0.037–0.039 cm^(3)/g and 106.016–111.870 m^(2)/g.Soft coal shows greater SSA and PV than hard coal,which is consistent with the adsorption capacity(VL).The fractal dimensions of soft and hard coal are respectively diferent.The Ding coal exhibits larger D1 and smaller D_(2),and the reverse for the Wu coal seam is observed.The greater the value of D1(complexity of pore surface)of soft coal is,the larger the pore surface roughness and gas adsorption capacity is.The results enable us to conclude that the characterization of pores and fractal dimensions of soft and hard coals is diferent,tending to diferent adsorption/desorption characteristics.In this regard,the results provide a reference for formulating corresponding coal and gas outburst prevention and control measures.

Pore structure characterization based on NMR experiment: A case from the Shanxi Formation tight sandstones in the Daning-Jixian area, eastern Ordos Basin

The study of pore structure requires consideration of important factors including pore throat size,pore radius composition,and pore-throat configuration.As the nuclear magnetic resonance(NMR)experimental results contain rich information about pore structures and fluid occurrence states,this study investigated the pore structures of the tight sandstone reservoirs of the Shanxi Formation in the Daning-Jixian area,eastern Ordos Basin.Firstly,by making the inverse cumulative curve of the NMR T2 spectrum coincide with the capillary pressure curves which were obtained by the mercury injection capillary pressure(MICP)technique,this study derived a conversion coefficient that can be used to convert the NMR T2 spectrum into the pore throat radius distribution curves based on the NMR experimental results.Subsequently,we determined the pore radius intervals corresponding to irreducible water distribution using the NMR-derived pore radius distribution curves.Finally,the NMR T2 distribution curves based on the fractal theory were analyzed and the relationships between fractal dimensions and parameters,including permeability,porosity,reservoir quality index(RQI),flow zone indicator(FZI),irreducible water saturation,RT35,and RT50,were also discussed.The NMR-derived pore throat radius distribution curves of the study area are mainly unimodal,with some curves showing slightly bimodal distributions.The irreducible water mainly occurs in small pores with a pore radius less than 100 nm.As the permeability decreases,the contribution rate of small pores to the irreducible water gradually increases.The NMR-based fractal dimensions of pores show a two-segment distribution.Small pores have small fractal dimensions and are evenly distributed,while large pores have large fractal dimensions and complex pore structures.The fractal dimension of large pores(Dmax)is poorly correlated with porosity but strongly correlated with FZI,RQI,RT35,and RT50.These results indicate that large pores are the main pore zones that determine the seepage capacity of the reservoirs.Additionally,there is a certain correlation between Dmax and the irreducible water saturation.

A novel box-counting method for quantitative fractal analysis of threedimensional pore characteristics in sandstone

Fractal theory offers a powerful tool for the precise description and quantification of the complex pore structures in reservoir rocks,crucial for understanding the storage and migration characteristics of media within these rocks.Faced with the challenge of calculating the three-dimensional fractal dimensions of rock porosity,this study proposes an innovative computational process that directly calculates the three-dimensional fractal dimensions from a geometric perspective.By employing a composite denoising approach that integrates Fourier transform(FT)and wavelet transform(WT),coupled with multimodal pore extraction techniques such as threshold segmentation,top-hat transformation,and membrane enhancement,we successfully crafted accurate digital rock models.The improved box-counting method was then applied to analyze the voxel data of these digital rocks,accurately calculating the fractal dimensions of the rock pore distribution.Further numerical simulations of permeability experiments were conducted to explore the physical correlations between the rock pore fractal dimensions,porosity,and absolute permeability.The results reveal that rocks with higher fractal dimensions exhibit more complex pore connectivity pathways and a wider,more uneven pore distribution,suggesting that the ideal rock samples should possess lower fractal dimensions and higher effective porosity rates to achieve optimal fluid transmission properties.The methodology and conclusions of this study provide new tools and insights for the quantitative analysis of complex pores in rocks and contribute to the exploration of the fractal transport properties of media within rocks.

SOME FRACTAL STRUCTURES IN RAPIDLY SOLIDIFIED Al-Mn ALLOY

The fractal structures formed in rapidly quenched Al_(86)Mn_(14)alloy have been observed on SEM.Their fractal dimensions are from 1.46 to 1.97.The principal phase examined by TEM and X-ray diffractometer is icosahedral quasicrystalline one with a small amount of Al phase. It is believed that these fractal structures are formed by many aggregated particles during the rapid-quenching process.

CRYSTALLIZATION AND FRACTAL STRUCTURE OF MOLYBDENA SINGLE CRYSTALS

When a molybdenum sheet was heated properly at 900℃ in an oxidizing atmosphere, Mo_5O_(14) single crystals will grow.The size of the most coarse single crystal is up to 15×1.1 mm and 310 nm thick as measured by Rutherford backscattering spectrometry.Some fractal structure consisting of many molybdena whiskers were observed after heating at 750℃.An approach to explain the crystal growth and fractal structure has been discussed.

Fractals of Protein Backbone Structures

Ths paper，based on the principles of geometric self-similarity of fractal theory and some research results of rotein chemistry，improved the method of comput－ing protein fractal dimensions，and computed fractal dimensions of some protein back bone，secondary and assumed folding structures．The relationship between protein back-bone strucrural fractal dimensions and its spatial structures was investigated．The results indicated that protein backbone fractal dimensions not only have a close relation with protein secondary structure，but also with its folding．In addition，the folding of protein Polypeptide chains in 3-D space may be similar to the other macromolecular chain be haviour described by the self-avoiding walks（SAW）model．

FRACTAL ANALYSIS APPLIED TO SPATIAL STRUCTURE OF CHINA'S VEGETATION

Based on the fractal theory, the spatial structure of China＇s vegetation has been analyzed quantitatively in this paper. Some conclusions are drawn as the following. 1） The relationships between size and frequency of patch area and patch shape index exist objectively for China＇s vegetation. 2） The relationships between perimeter and area exist objectively for China＇s vegetation. 3） The fractal dimension of evergreen needleleaf forests on mountains in subtropical and tropical zones is the largest, while the smallest for deciduous broadleafand evergreen needleleaf mixed forests in temperate zone, reflecting the most complex spatial structure for evergreen needleleaf forests on mountains in subtropical and tropical zones and the simplest for deciduous broadleaf and evergreen needleleaf mixed forests in temperate zone. 4） The fractal dimensions of China＇s vegetation types tend to decrease from thc subtropics to both sides. 5） The stability of spatial structure of deciduous broadleaf and evergreen needleleaf mixed forests in temperate zone is the largest, while the smallest for double-cropping rice, or double-cropping rice and temperate-like grain, and tropical evergreen economic tree plantations and orchards, reflecting the steadiest for deciduous broadleaf and evergreen needleleaf mixed forests in temperate zone and the most unstable for double-cropping rice, or double-cropping rice and temperate-like grain, and tropical evergreen economic tree plantations and orchards in spatial structure. 6） The stability of spatial structure of China＇s vegetation tends to decrease from the temperate zone to both sides, it is significantly pertinent to understand the formation, evolution, dynamics and complexity rule of ecosystem of vegetation.

The fractal structure in the ionization dynamics of Rydberg lithium atoms in a static electric field

The ionization rate of Rydberg lithium atoms in a static electric field is examined within semiclassical theory which involves scattering effects off the core. By semiclassical analysis, this ionization process can be considered as the promoted valence electrons escaping through the Stark saddle point into the ionization channels. The resulting escape spectrum of the ejected electrons demonstrates a remarkable irregular electron pulse train in time-dependence and a complicated nesting structure with respect to the initial launching angles. Based on the Poincaré} map and homoclinic tangle approach, the chaotic behaviour along with its corresponding fractal self-similar structure of the ionization spectra are analysed in detail. Our work is significant for understanding the quantum-classical correspondence.

Fractal Analysis on the Spatial Structure of Land Use Patterns in a Non-Point Source Polluted Area in Southern China

Based on the TM remote sensing interpretation of land use data of a non-point source polluted area in southern China in 2015, supported by ArcGIS and the fractal theory, this paper established the perimeter-area fractal equation of the spatial structure of land use types, quantitatively calculated the fractal dimension and stability index of various land use types in this non-point source polluted area in 2015, and analyzed the spatial structure characteristics of land use types. The results showed that the fractal dimension values of the different land use types in the research area ranged 1.141–1.657, and the whole fractal dimension of the research area was 1.206. To be specific, the fractal dimension values of unused land, grass land and rural residential land were high, and the fractal dimension values of dry land, paddy filed and waters were low, and the fractal dimension values of urban land, industrial traffic construction land and forest land were moderate. The area of the land use types whose stability index values were 0.2–0.4 accounted for 89.5% of total area, suggesting that the stability of the spatial structure of land use types of Beibei District in 2015 was not high generally. The rural residential land and urban land of the research area should be planned and managed more properly from now on.

A Fractal Space-Time Structure of Nonequilibrium Living Systems Is Essential Property and Fractal Property of Dynamics Systems on Cosmic Scales Is Foundation for the Origin of the Living Systems

The experimental data represented in the bibliography are theoretically analyzed to construct an adequate model for dynamics of an open nonequilibrium living system. It is shown that the viable microorganisms are capable of forming the fractal structure, whose dimensionality is certainly non-integral. In addition, we have attempted to provide a generalized description of the properties of living and nonliving matter (in the addition to that described in work [1]). Relevant published data were used to demonstrate a fractal structure of the space in the vicinity of centrally gravitating bodies with satellites revolving around them along closed trajectories and serving as a kind of testers of the neighboring space. A local violation of its discontinuity is likely to be a necessary (yet not sufficient!) dynamic characteristic of the spatiotemporal continuum for self-organization of molecules into a living, i.e., self-replicating, system.

Fractal Characteristics of Population Canopy Structure of the Mangrove,Bruguiera gymnorrhiza (L.) Lamk

The fractal characteristics of the canopy structure of B. gymnorrhiza population are investigated by fractal dimension analysis in the National Shankou Mangrove Nature Reserve. The 3-year-old branches have box dimensions between 1.22 and 1.55, showing the complexity degree of branching structure and the ability of occupying and utilizing ecological space. It may be considered that fractal dimension provides a useful index for the study of light utilization efficiencies and growth processes of B. gymnorrhiza. Calculated by using the two-surface method, the fractal dimensions for the crown pattern of individuals with ages of 20 to 50 years range from 2.21 to 2.54, indicating that the filling degree of foliage to a tree crown is relatively low and B. gymnorrhiza has the property of a sun plant. Along with the increase of ages of individuals, the filling degree of foliage to a tree crown changes from high to low, and so does the fractal dimension. The box dimensions obtained from the grayscale curves of population canopy are between 1.47 and 1.61. The greater the box dimension, the more loosely organized the canopy spatial structure, and the more the light spots. The canopy structural information and complexity of a population can be effectively captured by box dimensions obtained from canopy grayscale curves.