Fragmentation Fractal Dimension Research Articles

TIME‐MASS SCALING IN SOIL TEXTURE ANALYSIS

Popular methods of textural analysis employ the relationship among time, the travel distances, and the radii of particles subject to sedimentation in a viscous liquid. The purpose of this note is to present and test an explicit relationship between time and soil suspension density in the course of the hydrometer analysis procedure applied to particles with the fractal mass‐size distribution. The relationship between the logarithms of mass of particles remaining in the solution and of the observation time is linear with the slope equal to −(3 − D)/2 where D is the fragmentation fractal dimension. This relationship was tested with samples of Typic Argiudoll and 24 soils from Imperial Valley, CA, and gave good approximation of hydrometer data on sedimentation of silt fraction 2 to 50 m. Monitoring soil mass in the solution presents a way of testing the applicability of fractal fragmentation model to particle‐size distributions (PSDs) and estimating the fragmentation fractal dimension.

Hölder spectrum of dry grain volume-size distributions in soil

Fractal parameters have been applied before to describe number and mass size distributions of particles and aggregates through power scaling laws. Exponents such as the fragmentation fractal dimension, D f, represent rough estimates of the scale dependence present in the number or mass accumulated with respect to length scale. However, a deeper and more accurate study of these distributions may be done using multifractal analysis techniques when a large enough amount of data is available. The light scattering method and the multifractal analysis enable the study of these distributions in scales not very often explored. In the present work, attention is focused on the Hölder exponents spectrum, α( q), and its capability to characterize dry grain volume-size distributions in soil. Twenty samples were analyzed by laser diffraction and Hölder spectrum analyses, showing a great variability in Hölder exponent results. Coefficients of determination have been considered to study and characterize scalings achieved. Hölder spectrum analysis revealed to be useful in the characterization of these distributions as they showed suitable scaling properties in general, producing the best fits for the entropy dimension ( D 1). Thus, the combination of laser diffraction and Hölder spectrum analyses is a potential tool for detecting changes in soil grain distributions due to physical or degradation processes. The similarity of these results to those obtained from mathematical multifractal measures suggests that this way of characterizing dry soil grain distributions may be used to simulate empirical data by means of mathematical algorithms based in fractal geometry (Iterated Function Systems and related ones).

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.

Laser diffraction and multifractal analysis for the characterization of dry soil volume-size distributions

Fractal scalings of the type N( x> X)≈ X − D f and M( x< X)≈ X d have been studied in previous works for the cumulative number and mass distributions of soil aggregates and particles of size x greater or less, respectively, than a characteristic size X. The exponents D f (fragmentation fractal dimension) and d represent rough estimates of the scale dependence that takes place in the number or mass accumulated, with respect to length scale. A closer look at the mass of soils (aggregates and primary particles) accumulated across the different regions of the interval of sizes would lead, in general, to detect denser or rarer regions across different length scales. These heterogeneity features are present in most real distributions in nature, displaying a behavior named multifractal. The laser scattering method and the multifractal analysis offer the possibility of studying the volume distribution of soils in scales that are not often explored. The goal of this paper is the application of laser diffraction and multifractal analysis to the characterization of dry volume-size distributions in soils. For each of the 20 soil samples considered for the study, an adequate measure is constructed with data corresponding to laser diffraction analysis of those samples. Multifractal analysis techniques are applied to such measures. Volume-size distribution of soils showed, in general, suitable scaling properties that make multifractal analysis a useful mathematical tool for its characterization by means of the singularity spectrum f( α( q)). Samples showed great variability in their multifractal behavior. An adequate range of the parameter q is selected for the characterization of each sample, in such a way that the R 2 values of the corresponding scaling fits are greater than 0.95. In that range, different features of the singularity spectrum f( α( q)) have been analyzed. The singularity spectrum f( α( q)) reveals to be useful for the characterization of soil volume-size distributions.

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.

An evaluation of a fragmentation fractal dimension technique to determine soil erodibility

The fragmentation fractal dimension, D f, was used to characterise aggregate-size distribution, as an index of soil erodibility, of five soils with contrasting properties from semiarid and temperate areas. The D f was the slope of the regression of aggregate number vs. log of aggregate diameter/diameter of the largest aggregates using scale-variant bulk density and assuming a constant shape for all the aggregates sizes. Six aggregate sizes, ranging from 0.25 to 16 mm, were obtained for each soil. Aggregate bulk density (ABD) ranged from 1.21 to 1.81 mg m −3 for semiarid soils and from 0.83 to 1.55 mg m −3 for temperate soils. Significant correlations (from r=−0.72 to r=−0.98, p<0.05) between ABD and size indicated scale-variance in ABD. D f values ranged from 2.92 to 3.25 for semiarid and from 2.15 to 2.24 for temperate soils. Results from aggregate stability test (using a laboratory rainfall simulator) and soil loss rates (from erosion plots) for each soil were compared with D f to assess the suitability of D f as an erodibility index. Values of D f equal to or less than 2.24 were associated with stable soil, with a predominant percentage of large aggregates (43% to 53% of aggregates 8–16 mm), and negligible soil loss rates. In contrast, D f values greater than or equal to 2.84 were associated with a less stable soil, with a low percentage of large aggregates size (7% to 16% of aggregates 8–16 mm), and notable soil loss rates. The values of D f in describing mass-size distribution were found to be a useful and effective index to discriminate soil erodibility in the range of soil included in this study.

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.

Fractal Characterization of Soil Aggregation and Fragmentation as Influenced by Tillage Treatment

AbstractParameters are needed to quantify tillage‐induced changes in soil structure. Our objective was to see if fractal parameters from ped mass‐ and number‐size relations could be applied to quantify such changes. Soil from long‐term conventional (moldboard plow, MP, plus two passes with a disk harrow, DH) and no‐till (NT) plots on a Maury silt loam (fine, mixed, mesic Typic Paleudalf) was air dried and sieved into six size classes ranging from 0.5–1 to 16–31.5 mm. The mass percentage and weight of 100 peds in each size class were used to calculate the mean mass per ped and number of peds as a function of scale. Fractal equations were then fitted to both data sets (R2 always ≥ 0.99). The resulting parameter estimates were Dm (mass fractal dimension) and km (mass of an aggregate of unit length) for the mass‐size relation, and Dt (fragmentation fractal dimension) and kf (number of fragments of unit length) for the number‐size distribution. Significant tillage effects were observed for Dm, Df, and kf. The Dm increased from 2.88 for NT to ≥ 2.94 for MP and DH, suggesting differential compaction of large aggregates by plowing and disking. The Df was always less than the Dm, as predicted by theory. Soil fragmentation was least for NT (Df = 2.02) and greatest for MP (Df = 2.55); for DH, the Df = 2.15 suggesting that disking broke up large fragments and/or coalesced small fragments following plowing. The kf, which was shown to be nonlinearly related to the Df, was least for MP and greatest for NT. Similar studies on other soils are needed to test the general applicability of these fractal relations to soil structure.

A fractal description of the progeny of double impact single particle breakage

This paper outlines the results of the double impact failure of individual cores of rock, the progeny of which were described by a fragmentation fractal dimension. This, swing pendulum generated, data was compared to that previously established from the single impact breakage of rock cores, and the fragmentation fractal was found to be a suitable tool by which the two mechanisms could be assessed. The swing pendulum data was seen to adequately comply with an empiric based fractal model of fragmentation derived from the above mentioned single impact tests. Although an additional consideration was necessary to account for the velocity of the impact event, this indicated that the model may have wider application and be relevant to the impact failure of rock in general. However, ultrafast load cell testing of selected rocks illustrated that a previously postulated function for describing the relation of fragmentation fractal and size of the parent particle did not hold. The extrapolated fragmentation fractal significantly under estimated that directly obtained from the progeny of the ultrafast load cell experiments.

A fractal description of the progeny of single impact single particle breakage

An initial empiric fractal based model for the breakage of rock has been outlined, which was based on the single impact failure of single cores of rock. This incorporates the variance of specific impact energy, and to some degree the starting size, upon the resultant breakage products. The important parameter of interest is the fragmentation fractal dimension which represents the power-law size frequency distribution of the progeny, and is in essence a measure of the degree of fineness of the population. This quantity possesses a theoretical basis and offers an alternative approach for studying comminution events. As the application of fractal geometry has become more widespread the associated terminology has been widely adopted, but it can sometimes represent a pointless shift in vocabulary. This is particularly the case with power-law relationships that do not directly relate to a physical structure. In the situation described here the notation is justified as the fragmentation fractal can be derived directly from fractal theory, and is fundamentally associated with the coastline or boundary fractal dimension of the particles within the distribution.

Soil aggregates as mass fractals

Soil aggregates have a fractal mass. That is, they are porous and, as they are studied in greater detail, more pores may be observed. Mass fractals have scale-dependent bulk density. Larger objects, or soil aggregates, have a smaller bulk density. Bulk density in soil studies is sometimes assumed to be constant. If this was the case, soil aggregates would not be mass fractals, and their porosity would not change with scale. The fact that soil aggregates are mass fractals places restrictions on the estimation of the fragmentation fractal dimension (Df) of soil. The mass fractal dimension of soil (Dm) may be calculated from bulk density-aggregate size data. Linear and nonlinear methods of estimating Dm were compared and were shown to give similar results. The Dm is shown to influence porosity and the saturated water content. Fractal theory, in particular Dm, has implications for the calculation of the pore-size distribution and the moisture characteristic. By equating Campbell's (1985) Version of the Brooks-Corey water retention function, � Proportional �(-1 / b)and an equivalent form to the Brooks-Corey relation given by Crawford (1994), � Proportional �(Dm - d) it is suggested that D-m = d - 1/b, where d is the embedding dimension.

Surface area and size distributions of soil particles

Small-angle X-ray scattering was employed to show that the surface of soil particles is rough and scales as A ∝ r D s where A is the surface area of a given size fraction of radius r and D s is the surface fractal dimension ( D s = 2.4 ± 0.1). This relation has been confirmed by independent surface-area measurements on fractionated soil samples using nitrogen-gas adsorption and Methylene Blue adsorption from solution. These results bear an interesting relationship to recent size-distribution measurements of soil particles. The number of particles per unit volume with a radius larger than r has been shown to follow a power law N( r) ∝ r −D where the exponent D is the fragmentation fractal dimension ( D = 2.8 ± 0.1). The power law is typically valid between two cut-off radii r 1⪡ r⪡ r 2 with values around r 1 ≈ 10–100 nm and r 2 ≈ 10–5000 μm. The specific surface area of the unfractionated soil sample depends critically upon the position of the lower cut-off r 1 and can be accurately estimated from size-distribution data and the knowledge of D s. These features can be related to a class of fragmented fractals which are characterized by the two fractal dimensions D and D s. These fractal dimensions obey the inequalities 2< D s< D<3 and 2 D s– D⩽2 which are also satisfied by the present experimental estimates.