The Meso-Analysis of the Rock-Burst Debris of Rock Similar Material Based on SEM

In order to explore the specimen failure characteristics during rock-burst under different gradient stress conditions, in this paper, a novel experimental technique was proposed; a common series of tests under two gradient stress paths were conducted on rock similar material specimens using the true-triaxial gradient and hydraulic-pneumatic combined test apparatus. And plaster was used as the rock similar material. In the experimental process, several rock-burst debris with area sizes of 100 mm2 were collected, and the fractal dimensions of typical detrital section crystal contours were analyzed and calculated using a scanning electron microscopy (SEM) method. The results showed that the specimens’ failure characteristics which had been induced by the two gradient stress processes were various. Also, the mesoscopic morphology of the rock-burst detrital section had effectively reflected its macroscopic failure characteristics. It was found that the fractal dimensions of the crystal contours of the specimen’s detrital section had fractal characteristics, and the box-counting dimension based on the SEM image could quantitatively describe the rock-burst failure characteristics. Furthermore, under the same magnification, the fractal dimensions of the crystal contours of the splitting failures were found to be relatively smaller than those of the shearing failures.

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DIMENSION ANALYSIS OF CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

Fractals ◽

10.1142/s0218348x1730001x ◽

2017 ◽

Vol 25 (01) ◽

pp. 1730001 ◽

Cited By ~ 17

Author(s):

JUN WANG ◽

KUI YAO

Keyword(s):

Hausdorff Dimension ◽

Fractal Dimensions ◽

Continuous Functions ◽

Packing Dimension ◽

Box Dimension ◽

Box Counting ◽

Dimension Analysis ◽

Box Counting Dimension ◽

Self Similar

In this paper, we mainly discuss fractal dimensions of continuous functions with unbounded variation. First, we prove that Hausdorff dimension, Packing dimension and Modified Box-counting dimension of continuous functions containing one UV point are [Formula: see text]. The above conclusion still holds for continuous functions containing finite UV points. More generally, we show the result that Hausdorff dimension of continuous functions containing countable UV points is [Formula: see text] also. Finally, Box dimension of continuous functions containing countable UV points has been proved to be [Formula: see text] when [Formula: see text] is self-similar.

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The linkage between box-counting and geomorphic fractal dimensions in the fractal structure of river networks: the junction angle

Hydrology Research ◽

10.2166/nh.2020.082 ◽

2020 ◽

Vol 51 (6) ◽

pp. 1397-1408

Author(s):

Xianmeng Meng ◽

Pengju Zhang ◽

Jing Li ◽

Chuanming Ma ◽

Dengfeng Liu

Keyword(s):

Fractal Dimension ◽

Fractal Structure ◽

Fractal Dimensions ◽

River Networks ◽

Stream Length ◽

Box Counting ◽

Linear Relationships ◽

Box Counting Dimension ◽

Junction Angle

Abstract In the past, a great deal of research has been conducted to determine the fractal properties of river networks, and there are many kinds of methods calculating their fractal dimensions. In this paper, we compare two most common methods: one is geomorphic fractal dimension obtained from the bifurcation ratio and the stream length ratio, and the other is box-counting method. Firstly, synthetic fractal trees are used to explain the role of the junction angle on the relation between two kinds of fractal dimensions. The obtained relationship curves indicate that box-counting dimension is decreasing with the increase of the junction angle when geomorphic fractal dimension keeps constant. This relationship presents continuous and smooth convex curves with junction angle from 60° to 120° and concave curves from 30° to 45°. Then 70 river networks in China are investigated in terms of their two kinds of fractal dimensions. The results confirm the fractal structure of river networks. Geomorphic fractal dimensions of river networks are larger than box-counting dimensions and there is no obvious relationship between these two kinds of fractal dimensions. Relatively good non-linear relationships between geomorphic fractal dimensions and box-counting dimensions are obtained by considering the role of the junction angle.

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Box-Counting Dimension of Fractal Urban Form

International Journal of Artificial Life Research ◽

10.4018/jalr.2012070104 ◽

2012 ◽

Vol 3 (3) ◽

pp. 41-63 ◽

Cited By ~ 7

Author(s):

Shiguo Jiang ◽

Desheng Liu

Keyword(s):

Fractal Dimension ◽

Urban Form ◽

Fractal Dimensions ◽

Reliable Estimate ◽

Box Dimension ◽

Beijing City ◽

Box Counting ◽

Window Method ◽

Box Counting Dimension ◽

Data Points

The difficulty to obtain a stable estimate of fractal dimension for stochastic fractal (e.g., urban form) is an unsolved issue in fractal analysis. The widely used box-counting method has three main issues: 1) ambiguities in setting up a proper box cover of the object of interest; 2) problems of limited data points for box sizes; 3) difficulty in determining the scaling range. These issues lead to unreliable estimates of fractal dimensions for urban forms, and thus cast doubt on further analysis. This paper presents a detailed discussion of these issues in the case of Beijing City. The authors propose corresponding improved techniques with modified measurement design to address these issues: 1) rectangular grids and boxes setting up a proper box cover of the object; 2) pseudo-geometric sequence of box sizes providing adequate data points to study the properties of the dimension profile; 3) generalized sliding window method helping to determine the scaling range. The authors’ method is tested on a fractal image (the Vicsek prefractal) with known fractal dimension and then applied to real city data. The results show that a reliable estimate of box dimension for urban form can be obtained using their method.

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IMAGE PYRAMIDS FOR CALCULATION OF THE BOX COUNTING DIMENSION

Fractals ◽

10.1142/s0218348x12500260 ◽

2012 ◽

Vol 20 (03n04) ◽

pp. 281-293 ◽

Cited By ~ 7

Author(s):

H. AHAMMER ◽

M. MAYRHOFER-REINHARTSHUBER

Keyword(s):

Digital Images ◽

Fractal Dimensions ◽

Counting Method ◽

Image Pyramids ◽

Box Counting ◽

Aspect Ratios ◽

Box Counting Dimension ◽

Method Effects ◽

Box Counting Method ◽

Processing Techniques

The fractal dimensions of real world objects are commonly investigated using digital images. Unfortunately, these images are unable to represent an infinitesimal range of scales. In addition, a proper evaluation of the applied methods that encompass the image processing techniques is often missing. Several mathematical well-defined fractals with theoretically known fractal dimensions, represented by digital images, were investigated in this work. The very popular Box counting method was compared to a new image pyramid approach as well as to the Minkowski dilation method. Effects from noise and altered aspect ratios were also considered. The new Pyramid method is quite identical to the Box counting method, but it is easier to implement. Additionally, the calculation times are much shorter and memory requirements are almost comparable.

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Multifractal behaviour of river networks

Hydrology and Earth System Sciences ◽

10.5194/hess-4-105-2000 ◽

2000 ◽

Vol 4 (1) ◽

pp. 105-112 ◽

Cited By ~ 30

Author(s):

S. G. De Bartolo ◽

S. Gabriele ◽

R. Gaudio

Keyword(s):

Fractal Dimensions ◽

Multifractal Spectrum ◽

River Networks ◽

Legendre Transform ◽

Box Counting ◽

Box Counting Dimension ◽

Mathematical Problems ◽

Fractal Spectrum ◽

Mass Exponents ◽

Specific Number

Abstract. A numerical multifractal analysis was performed for five river networks extracted from Calabrian natural basins represented on 1:25000 topographic sheets. The spectrum of generalised fractal dimensions, D(q), and the sequence of mass exponents, τ(q), were obtained using an efficient generalised box-counting algorithm. The multi-fractal spectrum, f(α), was deduced with a Legendre transform. Results show that the nature of the river networks analysed is multifractal, with support dimensions, D(0), ranging between 1.76 and 1.89. The importance of the specific number of digitised points is underlined, in order to accurately define, the geometry of river networks through a direct generalised box-counting measure that is not influenced by their topology. The algorithm was also applied to a square portion of the Trionto river network to investigate border effects. Results confirm the multifractal behaviour, but with D(0) = 2. Finally, some open mathematical problems related to the assessment of the box-counting dimension are discussed. Keywords: River networks; measures; multifractal spectrum

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Fractal Researches on the Spatial Structure of the Urban System in Sheyang Economic Zone

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.689.467 ◽

2013 ◽

Vol 689 ◽

pp. 467-472

Author(s):

Shao Hong Ren ◽

Lei Su ◽

Jing Hai Zhu

Keyword(s):

Spatial Structure ◽

Central City ◽

Fractal Dimensions ◽

Urban System ◽

Liaoning Province ◽

Space Distribution ◽

Urban Systems ◽

Box Counting ◽

Box Counting Dimension ◽

Economic Zone

This paper used three kinds of fractal dimensions to characterize spatial structure of the urban system in Shenyang Economic Zone, namely aggregation dimension, spatial correlation dimension, and box-counting dimension. It was found that the spatial structure of the Shenyang Economic Zone urban system has obvious fractal properties. In analysing the results of our research the following conclusions emerged: The space distribution of the Shenyang Economic Zone urban system is centripetal gathered around the central city of Shenyang, the accessibility of the transport network between cities is good, and the spatial shape is more compact than other urban systems of Liaoning Province. The authors believe that the structure of the Shenyang Economic Zone urban system is more effective than the structure of Liaoning Province urban system, and put forward rationalization proposals to optimize the urban system of the Shenyang Economic Zone.

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Experimental Investigation on the Damage Localization of Limestone under Triaxial Compression Condition

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.168-170.1524 ◽

2010 ◽

Vol 168-170 ◽

pp. 1524-1530

Author(s):

Hai Qing Yang ◽

Xiao Ping Zhou

Keyword(s):

Computerized Tomography ◽

Damage Evolution ◽

Axial Stress ◽

Triaxial Compression ◽

Fractal Dimensions ◽

Rock Failure ◽

Damage Localization ◽

Deformation Localization ◽

Box Counting ◽

Box Counting Dimension

The phenomena of Damage localization, which characterized as the rock mass suddenly enter into the deformation localization stage after a period of uniform deformation, is the beginning of rock failure. Damage localization is also the precursor of rock failure. Utilizing the image enhancement and segmentation technology, the rule of damage evolution of limestone under triaxial compression is analyzed. Moreover, The computerized tomography images are analyzed by the method of digital image processing, which includes threshold partition and edge detection, and then the relationship between computerized tomography image and damage evolution is discussed. Meanwhile, the dependence of fractal dimensions of rock section on the axial stress is determined by the method of box-counting dimension. It is concluded that the fractal dimension may reach a minimum value at the point of damage localization, and after that damage become more severe.

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Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽

10.3390/gels7020046 ◽

2021 ◽

Vol 7 (2) ◽

pp. 46

Author(s):

Pedram Nasr ◽

Hannah Leung ◽

France-Isabelle Auzanneau ◽

Michael A. Rogers

Keyword(s):

Fractal Dimension ◽

Crystallization Temperature ◽

Cayley Tree ◽

Fractal Dimensions ◽

Self Similarity ◽

Avrami Model ◽

Cluster Aggregation ◽

Box Counting ◽

Self Assembled ◽

Limited Diffusion

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

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