Fragmentation Fractal Dimensions Research Articles

An analysis of impact load and fragmentation dimension to explore energy dissipation patterns in coal crushing

This research delineates the energy dissipation characteristics in coal crushing under impact loads, leveraging the capabilities of Separated Hopkinson Pressure Bar experimental system. A meticulous examination of both burst-prone and non-burst-prone coal samples during destruction processes was undertaken to decipher the dynamic compression mechanical attributes from perspectives of energy and fragmentatio‘s fractal dimensions. Burst-prone coal showcases a more pronounced escalation in fragmentation work in comparison to non-burst-prone samples, thereby illustrating a perceptible strain-rate dependent effect correlating with enhanced strain rates. Additionally, it was observed that incident, reflected, and transmitted energy trajectories for both sample categories follow an approximately linear ascendancy, albeit exhibiting diverse magnitudes. Burst-prone coal manifests a more rapid and focused energy growth compared to its non-burst-prone counterpart. When subjected to impact loads, a notable trend was discerned where the fragmentation’s fractional dimension escalated persistently with both the incident energy and the crushing work, portraying a prominent growth effect. The insights garnered from this study pave the way for distinguishing between impacted and unimpacted coal samples using energy perspectives and fragmentation's fractal dimensions.

Fractal Entropy of Nuclear Medium Probed by KS0 Mesons Produced in AuAu Collisions at RHIC

In this paper, we review our findings concerning fractal entropy of microscopic configurations corresponding to the production of KS0 mesons in AuAu collisions in the z-scaling approach. The entropy is expressed via structural and fragmentation fractal dimensions, and model parameter cAuAu is interpreted as a specific heat of produced medium. These parameters are related to the respective momentum fractions of the colliding nuclei, the momentum fractions of the scattered constituents that fragment into the produced hadrons, and the multiplicity density of negative particles in the central interaction region. The dependence of the entropy on the collision energy over the range of 7.7–200 GeV for most central and most peripheral events is studied as a function of the transverse momentum of the produced KS0 mesons. A non-trivial dependence of the entropy on the collision energy with decreasing transverse momentum is found. This reflects the irregularity of the behavior of the specific heat, cAuAu, and can point to a manifestation of phase transition in nuclear matter.

Experimental Study on Fragmentation Behavior of Specimen with Rockbolts

The effect of rockbolt density on fractured specimens and the distribution law of fragments were studied using the fractal method. The results show that with the increase of rockbolt density, the strength of specimens is improved. Its degree of fragmentation is increased, but the average size of fragments is decreased. The fragmentation and surface fractal dimensions are increased, their difference value is 1 in theory, but the value is slightly less than 1 in practical application due to the uncertain factors such as anisotropy of specimen. There is a positive correlation between the deformation of surrounding rock and its fractal dimension. The surface fractal dimension of surrounding rock can give a quantitative description of its degree of fragmentation and can give a comprehensive reflection to the roof stability. The support mechanism of rockblots can be understood as improving the probability of refragmentation of fragments, making its distribution gradation close to the optimal one with minimum porosity, thus reducing the deformation of surrounding rock.

A fractal evaluation of particle size distributions in an eolian loess-paleosol sequence and the linkage with pedogenesis

The changes of soil particle size distributions (PSDs) can be used as a proxy for soil development intensity and an indicator of soil formation processes. The fragmentation fractal dimensions can describe PSDs of soils in a single parameter. The objective of this research is to use mass-based fragmentation fractal dimension models to describe PSDs and evaluate the relationship between the fragmentation fractal dimension values (DF) and pedogenesis in loess-paleosol. Based on the theory of fragmentation fractal dimension, particle size data from 837 eolian loess samples from the lower part of Chaoyang section uniform loess deposits were analyzed with mass-based models using linear regression and piecewise linear regression. A single power-law exponent can characterize the PSDs across the domain of 0.1–54.4 μm. The trend of fragmentation fractal dimension values is consistent with chemical weathering indices, magnetic susceptibility, and <1 μm clay contents. The DF value of PSDs can be used to assess the degree of particle fragmentation and intensity of pedogenesis for loess-paleosols. A paleosol has a greater DF value than a loess; indicating that the greater the DF value, the greater the effect of energy events that have occurred. Loess particles in paleosols were fragmented to finer particles with an average median grain size (φ = 6.21 ± 0.24, CV = 3.94%) and were poorly sorted to very poorly sorted (So = 1.922 ± 0.128) due to strong pedogenesis under a prominent warm-wet climate. In contrast, particles in loess experienced fewer fragmentations and had a greater average median grain size (φ = 5.89 ± 0.35, CV = 5.89%) due to weak pedogenesis under prominent cold-dry climate. These results confirm the use of fractal characteristics to reflect the loess particle size reduction process. Two domains within the 0.1–54.4 μm domain, F1 and F2, were identified where power-law scaling was applicable. The close transition point from F1 to F2 is 7.64 ± 1.21 μm. The fragmentation fractal dimensions of the two domains decreased in the order: DF1 (close to 3) > DF2 (close to 2). The energy dissipation for fragmenting larger particles from coarse silt (CSI) to fine silt (FSI) is identified in the volume and for fragmenting smaller particles from fine silt (FSI) to clay (CL) it is substantially on the surface. This research indicated that fragmentation fractal model is a good descriptor for the PSDs and DF can be a value quantifying the intensity of loess-paleosol pedogenesis.

Compositional analysis for an unbiased measure of soil aggregation

Soil aggregation is an index of soil structure measured by mean weight diameter (MWD) or scaling factors often interpreted as fragmentation fractal dimensions (Df). However, the MWD provides a biased estimate of soil aggregation due to spurious correlations among aggregate-size fractions and scale-dependency. The scale-invariant Df is based on weak assumptions to allow particle counts and sensitive to the selection of the fractal domain, and may frequently exceed a value of 3, implying that Df is a biased estimate of aggregation. Aggregation indices based on mass may be computed without bias using compositional analysis techniques. Our objective was to elaborate compositional indices of soil aggregation and to compare them to MWD and Df using a published dataset describing the effect of 7 cropping systems on aggregation. Six aggregate-size fractions were arranged into a sequence of D-1 balances of building blocks that portray the process of soil aggregation. Isometric log-ratios (ilrs) are scale-invariant and orthogonal log contrasts or balances that possess the Euclidean geometry necessary to compute a distance between any two aggregation states, known as the Aitchison distance (A(x,y)). Close correlations (r>0.98) were observed between MWD, Df, and the ilr when contrasting large and small aggregate sizes. Several unbiased embedded ilrs can characterize the heterogeneous nature of soil aggregates and be related to soil properties or functions. Soil bulk density and penetrater resistance were closely related to A(x,y) with reference to bare fallow. The A(x,y) is easy to implement as unbiased index of soil aggregation using standard sieving methods and may allow comparisons between studies.

A Laboratory Rig and a Scale‐ and Energy‐Controlled Procedure for Tender Soil Fragmentation Test

A laboratory rig was developed for a scale‐ and energy‐controlled tender soil fragmentation test, which required multiple impacts and step‐by‐step decomposition of the soil structure on a macroscale. Soil sample size classes were scaled down by a factor of two, giving ranges of 128, 64, 32, 16, 8, 4, 2, 1, and 0.5 mm, respectively. These size classes were also used as control scales for respective fragmentation stages. Fragmentation tests were conducted to illustrate the possible influences from shape effects, falling weights, displacements, and fragmentation stages, and to investigate the relations among specific fragmentation energy, number of strokes per fragmentation stage, and fragmentation fractal dimension. Specific fragmentation energy was influenced by both the size and the shape of the soil samples. The cylindrical‐ and ball‐shaped soil samples behaved similarly. Thus ball samples were used for a benchmarking test on soil structural stabilities. Subsequent fragmentation on 60‐ and 80‐g ball samples with changed displacements from 5 to 45 cm yielded a decreased number of strokes from around 150 to 10. Whereas, their respective fragmentation fractal dimensions varied insignificantly, ranging from 1.52 to 1.64 for samples of 60‐g mass and 1.55 to 1.72 for 80 g, values much smaller than previously reported. Fragmentation fractal dimension appeared to be a nominal index for remolded soils under laboratory impacting actions. There was a slight increase in specific fragmentation energy as the number of impacts increased. But this increase was aggravated for a single or only few impacts. Thus 20 to 40 strokes was the appropriate stroke number to finish one fragmentation stage. A five‐stage fragmentation on a sample in the 32‐ to 64‐mm size range yielded fragmentation fractal dimensions of 1.29, 1.52, 1.45 and 1.61, for the first four stages, respectively, whereas their specific fragmentation energies increased linearly.

Particle-size distribution in soils: A critical study of the fractal model validation

Particle size distributions (PSDs) have been analyzed in the past with power-law functions relating cumulative number of particles to diameter and mass of particles to diameter, and the exponents interpreted as fragmentation fractal dimensions ( D). More recently, a mass–time scaling model was proposed assuming a fractal mass–size distribution for particles settling in a liquid. The objective of this study was to test the mass–time and the mass–size relations in soil particle distributions settling in water, and to estimate the range of particle diameters where power-law scaling was applicable. Five soil samples from Argentina, with clay contents ranging from 8% to 32%, were used in the experiments. Particle size analysis was done with the Bouyoucos hydrometer method. Results for D ranged from 2.434 to 2.712 with R 2 ranging from 0.922 to 0.998. The power-law scaling was applicable in the range of particle diameters from 1 to 50 μm, to four of the soils, and from 1 to 35 μm to the fifth one. Besides, a previous set of soil PSD from Argentina, with particle diameters ranging from 1 to 40 μm was also studied. The obtained D values were from 2.427 (8% clay) to 2.928 (75% clay), with R 2 from 0.983 to 0.997. Power laws relating mass to size or mass to time of soil particles settling in a liquid were good models to describe particle size distributions in the silt particle range sizes.

Fragmentation fractal dimensions of Vertisol samples: influence of sieving time and soil pretreatment

Compaction is a major problem in Vertisols. Physical parameters to quantify the degree of compaction are required. The objectives of this work were to (i) compare the fractal dimensions estimated for soil pre-treated by two different methods and (ii) study relationships between the fragmentation fractal dimension, D f, and the sieving time. Soil samples were collected from the top layer (0–10 cm) of a Vertisol (heavy clay, montmorillonitic). Two methods of soil drying were employed: air-drying, and air-drying, saturation in ethanol, and air-drying again. Fragmentation fractal dimensions were estimated by sieving in a nest of sieves with mesh sizes ranging from 0.25 to 10 mm. The number-based method was used for the estimation. In addition, soil samples were sieved using the following times: 1, 5, 10, 20, 30, 40, 50, 60, 70, and 80 min. The saturation of soil samples in ethanol prior to sieving reduced the resistance of soil to fragmentation. The fragmentation fractal dimension increased as a function of the time of sieving following an exponential model. D f is a valuable parameter to characterize the resistance of Vertisols to fragmentation.

Sensitivity of fractal parameters of soil aggregates to different management practices in a Phaeozem in central Argentina

Changes in soil structure often accompany changes in management practices and may affect the effectiveness of these practices. Parameters are needed to quantify these changes. Our objective was to see if fractal dimensions derived from ‘aggregate bulk density–aggregate size’ and ‘aggregate number–aggregate size’ relationships could be applied to quantify such changes. The study was conducted at the Experimental Farm of the School of Agronomy and Forestry Engineering, National University of La Plata, Argentina. A Vertic Phaeozem soil was sampled at seven locations differing in long-term management practices. The ‘aggregate bulk density–aggregate size’ and ‘number of aggregates–aggregate size’ data were obtained for seven ranges of aggregate sizes. Differences in treatments were reflected by the fragmentation fractal dimension but not the mass fractal dimension. The lowest fragmentation fractal dimensions corresponded to plots under long-term no-tillage and the highest to plots with a history of cultivation of rice ( Oryza sativa L.) under water. The fragmentation fractal dimension reflected the differences in soil management whereas the mass fractal dimension appeared to be insensitive to those differences.

Mass, surface, and fragmentation fractal dimensions of soil fragments produced by tillage

Structural units in freshly tilled soil are mainly fragments derived from tillage. Fractal geometry is potentially useful to model this structure. This research was designed to test the applicability of fractal models for mass– and number–size distributions, and surface roughness of soil fragments produced by tillage; and evaluate relationships among parameters of these fractal models. Fragments were sampled from the soil surface of three soil management experiments on a Normania clay loam: (1) moldboard, chisel, and disc as primary tillage tools, (2) three sequences of primary and secondary tillage, and (3) three crops followed by one tillage sequence. Air-dried samples were sieved to obtain eight fragment–size fractions with average diameters, x m, ranging from 0.4 to 28.0 mm. The size fraction with x m=0.4 mm had different patterns of fragmentation and mass distribution and was not used to estimate fractal parameters. Mass–size relations from size fractions and individual fragments did not show evidence of a fractal distribution of mass; but soil management influenced the estimated mass of fragments of unit diameter, k m. Two-dimensional roughness of fragment surfaces measured on thin sections was fractal with dimensions, D s, between 1.02 and 1.18. Disc tillage produced the highest D s among primary tillage tools. Fragment number–size distributions were fractal with the fragmentation fractal dimension, D f, between 2.14 and 3.19, and sensitive to management effects. The number of fragments of unit diameter, k f, was inversely related to k m, but relationships of D f– k f and D f– D s were not consistent with present fragmentation fractal models. Parameters, D f and k f were valuable for characterizing freshly tilled soils. More research is needed to understand links between D f, k f, and the geometrical configuration of a tilled layer.

A Modified Number‐based Method for Estimating Fragmentation Fractal Dimensions of Soils

AbstractFractal theory has been applied to the characterization of particle‐ and aggregate‐size distributions in soils. We used a number‐based method for estimating fragmentation fractal dimensions from these distributions. This method has several inconsistencies. The objectives of our study were to: (i) propose a modified number‐based method, (ii) evaluate the modified method using published data on particle‐ and aggregate‐size distributions, and (iii) apply the modified method to a large particle‐size distribution data base to analyze the validity of fractal scaling. Assuming scale‐invariant fragmentation to be a valid model of particle‐size distribution within size ranges of fractions, we derived a formula expressing the characteristic grain size as a function of the fractal dimension and limits of the grain size range. Parameters of Turcotte's fractal fragmentation model were found by minimizing the sum of squares of differences between measured and calculated masses of grain fractions. Comparison between original and modified number‐based methods showed that the modified method generally resulted in lower fragmentation fractal dimensions than the original method. The modified method was applied to a data set of particle‐size distributions of 2600 soil samples. In 80% of samples, the fractal scaling was not applicable across the whole range of particle size between 0.002 and 1 mm, since errors of the fractal fragmentation model were statistically significantly larger than measurement errors, and estimates of the fractal dimension were larger than 3. It appears that models more sophisticated than scale‐invariant fragmentation are required to simulate soil particle‐size distributions.