A Review on Fractals and Fracture, Part I: Calculating Fractal Dimensions by CAD Model

2011 ◽  
Vol 148-149 ◽  
pp. 818-821
Author(s):  
Asma A. Shariff ◽  
M. Hadi Hafezi
Keyword(s):  
Fractal Dimension ◽  
Fracture Surface ◽  
Fractal Geometry ◽  
Fractal Dimensions ◽  
Experimental Results ◽  
The Other ◽  
Cad Model ◽  
Cloud Of Points

The objective of this paper is to consider the use of fractal geometry as a tool for the study of non-smooth and discontinuous objects for which Euclidean coordinate is not able to fully describe their shapes. We categorized the methods for computing fractal dimension with a discussion into that. We guide readers up to the point they can dig into the literature, but with more advanced methods that researchers are developing. Considerations show that is necessary to understand the numerous theoretical and experimental results concerning searching of the conformality before evaluating the fractal dimension to our own objects. We suggested examining a cloud of points of growth of fracture surface at laboratory using CATIA - Digitized Shape Editor software in order to reconstruct the surface (CAD model). Then, the author carried out measurement/calculation of more accurate fractal dimension which are introduced by [1] in the other paper as Part II.

THE CANTOR SET’S MULTI-FRACTAL SPECTRUM FORMED BY DIFFERENT PROBABILITY FACTORS IN MATHEMATICAL EXPERIMENT

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1750002 ◽  
Author(s):  
XUEZAI PAN ◽  
XUDONG SHANG ◽  
MINGGANG WANG ◽  
ZUO-FEI
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Statistical Physics ◽  
Computer Calculation ◽  
Fractal Dimensions ◽  
The Other ◽  
Mathematical Experiment ◽  
Maximal Width ◽  
Fractal Spectrum ◽  
Vertical Height

With the purpose of researching the changing regularities of the Cantor set’s multi-fractal spectrums and generalized fractal dimensions under different probability factors, from statistical physics, the Cantor set is given a mass distribution, when the mass is given with different probability ratios, the different multi-fractal spectrums and the generalized fractal dimensions will be acquired by computer calculation. The following conclusions can be acquired. On one hand, the maximal width of the multi-fractal spectrum and the maximal vertical height of the generalized fractal dimension will become more and more narrow with getting two probability factors closer and closer. On the other hand, when two probability factors are equal to 1/2, both the multi-fractal spectrum and the generalized fractal dimension focus on the value 0.6309, which is not the value of the physical multi-fractal spectrum and the generalized fractal dimension but the mathematical Hausdorff dimension.

scholarly journals The Solutions to the Uncertainty Problem of Urban Fractal Dimension Calculation

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 453 ◽  
Author(s):  
Chen
Keyword(s):  
Fractal Dimension ◽  
Spatial Analysis ◽  
Fractal Geometry ◽  
Fractal Dimensions ◽  
Dimension Estimation ◽  
Scale Free ◽  
Factors Influencing ◽  
Calculation Results ◽  
Fractal Patterns ◽  
Main Factors

Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of the study area. This phenomenon has been puzzling many researchers. This paper is devoted to discussing the problem of uncertainty of fractal dimension estimation and the potential solutions to it. Using regular fractals as archetypes, we can reveal the causes and effects of the diversity of fractal dimension estimation results by analogy. The main factors influencing fractal dimension values of cities include prefractal structure, multi-scaling fractal patterns, and self-affine fractal growth. The solution to the problem is to substitute the real fractal dimension values with comparable fractal dimensions. The main measures are as follows. First, select a proper method for a special fractal study. Second, define a proper study area for a city according to a study aim, or define comparable study areas for different cities. These suggestions may be helpful for the students who take interest in or have already participated in the studies of fractal cities.

Image Reconstruction and Analysis of Three-Dimensional Fracture Surfaces Based on the Stereo Matching Method

2004 ◽  
Vol 261-263 ◽  
pp. 1593-1598
Author(s):  
M. Tanaka ◽  
Y. Kimura ◽  
A. Kayama ◽  
L. Chouanine ◽  
Reiko Kato ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Image Reconstruction ◽  
Fracture Surface ◽  
Fractal Analysis ◽  
Stereo Matching ◽  
Three Dimensional ◽  
Fractal Dimensions ◽  
Matching Method ◽  
Fracture Surfaces ◽  
Box Counting Method

A computer program of the fractal analysis by the box-counting method was developed for the estimation of the fractal dimension of the three-dimensional fracture surface reconstructed by the stereo matching method. The image reconstruction and fractal analysis were then made on the fracture surfaces of materials created by different mechanisms. There was a correlation between the fractal dimension of the three-dimensional fracture surface and the fractal dimensions evaluated by other methods on ceramics and metals. The effects of microstructures on the fractal dimension were also experimentally discussed.

CONSTRUCTING CROSSOVER-FRACTALS USING INTRINSIC MODE FUNCTIONS

2010 ◽  
Vol 02 (04) ◽  
pp. 509-520 ◽  
Author(s):  
SY-SANG LIAW ◽  
FENG-YUAN CHIU
Keyword(s):  
Fractal Dimension ◽  
Real Time ◽  
Empirical Mode Decomposition ◽  
Fractal Dimensions ◽  
The Other ◽  
Partial Sums ◽  
Intrinsic Mode Functions ◽  
Displacement Method ◽  
Mode Decomposition ◽  
Time Sequences

Real nonstationary time sequences are in general not monofractals. That is, they cannot be characterized by a single value of fractal dimension. It has been shown that many real-time sequences are crossover-fractals: sequences with two fractal dimensions — one for the short and the other for long ranges. Here, we use the empirical mode decomposition (EMD) to decompose monofractals into several intrinsic mode functions (IMFs) and then use partial sums of the IMFs decomposed from two monofractals to construct crossover-fractals. The scale-dependent fractal dimensions of these crossover-fractals are checked by the inverse random midpoint displacement method (IRMD).

scholarly journals Fractal Dimension of Fracture Surface in Rock Material after High Temperature

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Z. Z. Zhang
Keyword(s):  
Fractal Dimension ◽  
Electron Microscope ◽  
High Temperature ◽  
Fracture Surface ◽  
Scanning Electron Microscope ◽  
Fractal Dimensions ◽  
Threshold Value ◽  
Fracture Surfaces ◽  
Increasing Temperature ◽  
Scanning Electron

Experiments on granite specimens after different high temperature under uniaxial compression were conducted and the fracture surfaces were observed by scanning electron microscope (SEM). The fractal dimensions of the fracture surfaces with increasing temperature were calculated, respectively. The fractal dimension of fracture surface is between 1.44 and 1.63. Its value approximately goes up exponentially with the increase of temperature. There is a quadratic polynomial relationship between the rockburst tendency and fractal dimension of fracture surface; namely, a fractal dimension threshold can be obtained. Below the threshold value, a positive correlativity shows between rockburst tendency and fractal dimension; when the fractal dimension is greater than the threshold value, it shows an inverse correlativity.

Detection of Wear Condition of Micro Milling Cutters Based on Length Fractal Dimension

2014 ◽  
Vol 577 ◽  
pp. 697-700
Author(s):  
Zhi Qiang Wang ◽  
Gui Ying Zhang ◽  
Bin Liu
Keyword(s):  
Fractal Dimension ◽  
Time Domain ◽  
Fractal Dimensions ◽  
Experimental Results ◽  
New Method ◽  
Micro Milling ◽  
Vibration Signals ◽  
Reference Samples ◽  
Wear Condition

In this paper, a new method to realize online wear detection of micro-milling cutters based on length fractal dimension is proposed. On the basis of expression derivation of length fractal dimension, experiments are conducted. First, several cutters with different wear condition are chosen as reference samples. Their multi-section vibration signals in time-domain are collected and the clustering domain δ of each sample are obtained based on length fractal dimensions. Then, the vibration signals of tested cutters are monitored and analysed in time domain, thus their length fractal dimension are abstracted. Comparing the length fractal dimension of tested cutters with the clustering domain δ of reference samples, the wear condition of tested cutters are detected. The experimental results show that the length fractal dimension of each tested cutter falls in the clustering domain corresponding to the actual wear condition.

scholarly journals Surface Morphology: Theoretical Model and Correlation with Experimental Results

2019 ◽  
Vol 9 (1) ◽  
pp. 3308-3313
Keyword(s):  
Fractal Dimension ◽  
Surface Morphology ◽  
Fractal Geometry ◽  
Experimental Results ◽  
Dispersed Phase ◽  
Dynamic Parameters ◽  
Dispersed System ◽  
Roughness Exponent ◽  
Theoretical Results ◽  
Solid Type

In this work, we propose a model for the description of the surface morphogenesis of a dispersed system of the solid-solid type. To obtain the model, stochastic formalism based on the master equation and the principles of fractal geometry was applied, so that the surface morphology is characterized by the fractal dimension and the roughness exponent, which are expressed as a function of the composition of the dispersed system and the dynamic parameters associated with surface formation. Theoretical results obtained were compared with experimental results, finding that the variable that shows a significant effect on the morphology of the surface of the solid-solid dispersed system is the specific surface area of the particles of the dispersed phase found in the surface, as predict theoretically.

On the fractal nature of crack branching in MgF2

1998 ◽  
Vol 13 (11) ◽  
pp. 3153-3159 ◽  
Author(s):  
J. J. Mecholsky ◽  
Richard Linhart ◽  
Brian D. Kwitkin ◽  
Roy W. Rice
Keyword(s):  
Fractal Dimension ◽  
Fracture Surface ◽  
Fractal Geometry ◽  
Magnesium Fluoride ◽  
Crack Branching ◽  
Far Field ◽  
Fractal Nature ◽  
Branching Patterns ◽  
Basic Studies ◽  
Far Field Stress

Nineteen disks of IR window grade, hot pressed magnesium fluoride (˜0% porosity, grain size ˜1 μm) previously loaded in ring-on-ring flexure tests were used to analyze the crack branching patterns. Fractal geometry was used to determine the crack branching fractal dimension which was named the crack branching coefficient or CBC. The failure stress was proportional to the CBC and the number of pieces generated during the fracture. Thus, the number of pieces was proportional to the crack branching coefficient. The crack branching coefficient is distinct from the fractal dimension obtained from the onset of mist and hackle on the fracture surface. The fractal dimension of the fracture surface is, in most cases for brittle materials, a constant and related to the crack tip stress field. The crack branching fractal dimension is a function of the stress at fracture and the far-field stress distribution, or in other words, related to both the type and magnitude of loading. The findings in this work have strong implications for many commercial processes such as comminution, attrition, grinding, and basic studies in crack branching.

scholarly journals Comparative analysis of cranial suture complexity in the genus Caiman (Crocodylia, alligatoridae)

2000 ◽  
Vol 60 (4) ◽  
pp. 689-694 ◽  
Author(s):  
L. R. MONTEIRO ◽  
L. G. LESSA
Keyword(s):  
Fractal Dimension ◽  
Significant Interaction ◽  
Fractal Dimensions ◽  
The Other ◽  
Regional Differentiation ◽  
Cranial Sutures ◽  
Food Items ◽  
Facial Skull ◽  
Species Effect ◽  
Functional Stress

The variation in degrees of interdigitation (complexity) in cranial sutures among species of Caiman in different skull regions was studied by fractal analysis. Our findings show that there is a small species effect in the fractal dimension of cranial sutures, but most variation is accounted for by regional differentiation within the skull. There is also a significant interaction between species and cranial regions. The braincase sutures show higher fractal dimension than the facial skull sutures for all three species. The fractal dimension of nasal-maxilla suture is larger in Caiman latirostris than in the other species. The braincase sutures show higher fractal dimensions in C. sclerops than in the other species. The results suggest that different regions of the skull in caimans are under differential functional stress and the braincase sutures must counteract stronger disarticulation forces than the facial sutures. The larger fractal dimension shown by C. latirostris in facial sutures has probably a functional basis also. Caiman latirostris is known to have preferences for harder food items than the other species.

The fractal facets of turbulence

1986 ◽  
Vol 173 ◽  
pp. 357-386 ◽  
Author(s):  
K. R. Sreenivasan ◽  
C. Meneveau
Keyword(s):  
Fractal Dimension ◽  
Turbulent Flows ◽  
Fractal Dimensions ◽  
Dissipative Structures ◽  
The Other ◽  
Turbulent Shear ◽  
Constant Property ◽  
Turbulent Environment ◽  
Developed Turbulence ◽  
Self Similar

Speculations abound that several facets of fully developed turbulent flows are fractals. Although the earlier leading work of Mandelbrot (1974, 1975) suggests that these speculations, initiated largely by himself, are plausible, no effort has yet been made to put them on firmer ground by, resorting to actual measurements in turbulent shear flows. This work is an attempt at filling this gap. In particular, we examine the following questions: (a) Is the turbulent/non-turbulent interface a self-similar fractal, and (if so) what is its fractal dimension ? Does this quantity differ from one class of flows to another? (b) Are constant-property surfaces (such as the iso-velocity and iso-concentration surfaces) in fully developed flows fractals? What are their fractal dimensions? (c) Do dissipative structures in fully developed turbulence form a fractal set? What is the fractal dimension of this set? Answers to these questions (and others to be less fully discussed here) are interesting because they bring the theory of fractals closer to application to turbulence and shed new light on some classical problems in turbulence - for example, the growth of material lines in a turbulent environment. The other feature of this work is that it tries to quantify the seemingly complicated geometric aspects of turbulent flows, a feature that has not received its proper share of attention. The overwhelming conclusion of this work is that several aspects of turbulence can be described roughly by fractals, and that their fractal dimensions can be measured. However, it is not clear how (or whether), given the dimensions for several of its facets, one can solve (up to a useful accuracy) the inverse problem of reconstructing the original set (that is, the turbulent flow itself).

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