THE DISTRIBUTIONAL DIMENSION OF FRACTALS

In the book [1] H.Triebel introduces the distributional dimension of fractals in an analytical form and proves that： for Г as a non-empty set in R^n with Lebesgue measure ｜Г｜ = 0, one has dimH Г = dimD Г, where dimD Г and dimH Г are the Hausdorff dimension and distributional dimension, respectively. Thus we might say that the distributional dimension is an analytical definition for Hausdorff dimension. Therefore we can study Hausdorff dimension through the distributional dimension analytically. By discussing the distributional dimension, this paper intends to set up a criterion for estimating the upper and lower bounds of Hausdorff dimension analytically. Examples illustrating the criterion are included in the end.

FRACTAL PROPERTIES OF STATISTICALLY SELF-SIMILAR SETS

In this paper, authors compute the Packing dimension of statistically selfsimilar sets and obtaine the dimension and dimension distribution of statistically self-similar measure.

Fractal characteristics of cracks and fragments generated in unloading rockburst tests

True triaxial rockburst experiments with four different unloading rates were performed on four prism specimens of granite sampled from Beishan, China. The damage evolution in the rockburst test was investigated from two aspects including fracture surface crack and fragment characteristics. The scanning electron microscopy was used to observe the micro crack information on fragment surface. Combing binarization and box counting dimensions, the fractal dimensions of cracks were obtained. Meanwhile,the fragments were collected and a sieving experiment was conducted. We weighed the fragments qualities, counted the amount of fragments and measured the fragments length, width and thickness.Utilizing four methods to calculate damage fractal dimensions of fragments, the trend of fractal value changing with unloading rates can be roughly described. It can be concluded from these experiments that the fractal dimension either for crack or for fragment holds a decreasing trend with the decreasing unloading rate, indicating a reduction of damage level.

Distributional Dimension of Fractal Sets in Local Fields

The distributional dimension of fractal sets in R^n has been systematically studied by Triebel by virtue of the theory of function spaces. In this paper, we first discuss some important properties about the B-type spaces and the F-type spaces on local fields, then we give the definition of the distributional dimension dimD in local fields and study the relations between distributional dimension and Hausdorff dimension. Moreover, the analysis expression of the Hausdorff dimension is given. Lastly, we define the Fourier dimension in local fields, and obtain the relations among all the three dimensions. Keywords local field, B-type space, F-type space, distributional dimension, Hausdorff dimension Fourier dimension

Micro-and macro-fractures of coarse granite under true-triaxial unloading conditions

We studied the mechanical behavior of rock under different boundary conditions, based on the fractal characteristics of fractures in terms of microscopic and macroscopic investigations. Three rectangular granite specimens of similar dimensions were tested by a triaxial rock testing machine under uniaxial compression （UC）, confined compression （CC） and true-triaxial unloading conditions （RB） under rock burst boundary conditions. The failure processes of these specimens were investigated via examinations of their fracture behavior on a macro-scale by laser profilometers and on a micro-scale by a scanning electron microscopic （SEM） imaging technique. The SEM images, showing the spailing features of RB frag- ments, are compared with the grain dislocations under UC and CC conditions. Based on a variogram method, two fractal parameters, i.e., fractal dimensions （Dr^d） and the scale dependent fractal parameter Kv, were induced to present the surface roughness of scanning profiles in all directions. The fitted ellipses of Dr^d distribution show that RB surface has the smallest eccentricity, followed by the CC surface, while the UC surface had the largest eccentricity. As a result of this assessment, we conclude that rocks are affected by shear traction in an intermediate stress direction, which will cause fractures generated during rock bursts to twist rather than to tilt as shown in the uniaxial compression and the confined compres- sion tests.

Effect of Fiber Properties on Nonwovens' Pore Structures with Fractal Geometry Analysis

Nonwovens＇ pore structures are very important to their mechanical and physical properties. And the pore structures are influenced by the fiber properties and fibers arrangement in web. In this paper, the fractal geometry, in combination with computer image anaysis, is used to express the irregularity of pore size distribution in nonwovens, and the effect of fiber properties on fractal dimension of pore size distribution is discussed by using simulated images which are composed of nonlinear staple fibers. The results show that the fiber properties, such as crimp, diameter, angular distribution, and especially the number of fibers prominently influence the pore structure.

Collision Statistics of Driven Polydisperse Granular Gases

We present a dynamical model of two-dimensional polydisperse granular gases with fractal size distribution, in which the disks are subject to inelastic mutual collisions and driven by standard white noise. The inhomogeneity of the disk size distribution can be measured by a fractal dimension df. By Monte Carlo simulations, we have mainly investigated the effect of the inhomogeneity on the statistical properties of the system in the same inelasticity case. Some novel results are found that the average energy of the system decays exponentially with a tendency to achieve a stable asymptotic value, and the system finally reaches a nonequilibrium steady state after a long evolution time. Furthermore, the inhomogeneity has great influence on the steady-state statistical properties. With the increase of the fractal dimension df, the distributions of path lengths and free times between collisions deviate more obviously from expected theoretical forms for elastic spheres and have an overpopulation of short distances and time bins. The collision rate increases with df, but it is independent of time. Meanwhile, the velocity distribution deviates more strongly from the Gaussian one, but does not demonstrate any apparent universal behavior.

TWO-DIMENSIONAL CALCULATION FOR WATER SURFACE LINE OF OPEN BEND CHANNEL

This paper presents a differencial equation for two dimensional distribution of water depth and its condition to determine solution, then obtains the analytic solution of a Cauchy problem. By use of this solution the water depth is able to be determined at any point in total bend, which provides a better understanding of depth distribution in bend in comparison with traditional method of one dimensional calculation. The experimental verification shows that the calculated value by this formula well coincides with measured value.

The Morphology and Reciprocation Movement of Honeybee＇s Hairy Tongue for Nectar Uptake

A honeybee uses its brush-like tongue （glossa） to dip nectar and the setae densely distributed on it can increase the amount of trapped nectar observably. The glossa is often simplified as a cylinder covered by uniformly distributed and vertically erected setae during the drinking process, herein variations of the dimensions together with the erection angles of glossal setae are assumed to be negligible. In this paper, a dynamic model for the glossa retraction phase under the specific erection pattern of glossal setae is established, and the energy saving mechanism is extensively studied by comparing four types of erection pat- terns. Then the theoretically-optimal configuration, which satisfies the minimum energy consumption, is achieved from the dynamic model. Using the scanning electron microscope and a specially-designed high-speed camera system, we measure the dimensions of the glossal satae and capture dynamics of the hairy glossa in nectar feeding. It is proven that the erection angle of the glossal setae varies along the tongue axis, which shows a high concordance with our theoretically-optimal configuration. Compared with the hypothetical uniform distribution mode of glossal setae proposed by former researchers, we obtain a 16% increase in energy saving from actual erection pattern.