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.

Quantitative topographic analysis of fractal surfaces by scanning tunneling microscopy

1990 ◽  
Vol 5 (10) ◽  
pp. 2244-2254 ◽  
Author(s):  
Morgan W. Mitchell ◽  
Dawn A. Bonnell
Keyword(s):  
Brownian Motion ◽  
Fractal Dimension ◽  
Fracture Surface ◽  
Fourier Analysis ◽  
Scanning Tunneling Microscopy ◽  
Fractal Geometry ◽  
Scanning Tunneling ◽  
Tunneling Microscopy ◽  
Length Scales ◽  
Fractal Models

The applicability of models based on fractal geometry to length scales of nanometers is confirmed by Fourier analysis of scanning tunneling microscopy images of a sputter deposited gold film, a copper fatigue fracture surface, and a single crystal silicon fracture surface. Surfaces are characterized in terms of fractal geometry with a Fourier profile analysis, the calculations yielding fractal dimensions with high precision. Fractal models are shown to apply at length scales to 12 Å, at which point the STM tip geometry influences the information. Directionality and spatial variation of the topographic structures are measured. For the directions investigated, the gold and silicon appeared isotropic, while the copper fracture surface exhibited large differences in structure. The influences of noise in the images and of intrinsic mathematical scatter in the calculations are tested with profiles generated from fractal Brownian motion and the Weierstrass-Mandelbrot function. Accurate estimates of the fractal dimension of surfaces from STM data result only when images consist of at least 1000–2000 points per line and 1/f-type noise has amplitudes two orders of magnitude lower than the image signal. Analysis of computer generated ideal profiles from the Weierstrass-Mandelbrot function and fractional Brownian motion also illustrates that the Fourier analysis is useful only in determining the local fractal dimension. This requirement of high spatial resolution (vertical information density) is met by STM data. The fact that fractal models can be used at lengths as small as nanometers implies that continued topographic structural analyses may be used to study atomistic processes such as those occurring during fracture of elastic solids.

Fractal geometry of collision cascades

1989 ◽  
Vol 4 (1) ◽  
pp. 137-143 ◽  
Author(s):  
François Rossi ◽  
Don M. Parkin ◽  
Michael Nastasi
Keyword(s):  
Monte Carlo ◽  
Fractal Dimension ◽  
Monte Carlo Simulations ◽  
Fractal Geometry ◽  
Critical Energy ◽  
Initial Energy ◽  
Atomic Mass ◽  
Fractal Nature ◽  
Realistic Potential ◽  
Ion Collision

The fractal nature of self-ion collision cascades is first described using an inverse power potential and then by the more realistic potential of Biersack–Ziegler. Based on the model of Cheng et al. and TRIM Monte Carlo simulations, the average cascade fractal dimension is a function of both atomic mass and initial energy. The instantaneous fractal dimension increases as the cascade evolves. A critical energy Ec for producing a dense subcascade is derived and it is shown that Ec agrees well with the onset energy for constant damage efficiency.

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.

Optimal Density Determination of Bouguer Anomaly Using Fractal Analysis (Case of study: Charak area,IRAN)

2005 ◽  
Vol 1 (1) ◽  
pp. 21-24
Author(s):  
Hamid Reza Samadi
Keyword(s):  
Surface Roughness ◽  
Fractal Dimension ◽  
Fractal Analysis ◽  
Fractal Geometry ◽  
Host Rock ◽  
Bouguer Anomaly ◽  
Research Area ◽  
Density Difference ◽  
Optimal Density

In exploration geophysics the main and initial aim is to determine density of under-research goals which have certain density difference with the host rock. Therefore, we state a method in this paper to determine the density of bouguer plate, the so-called variogram method based on fractal geometry. This method is based on minimizing surface roughness of bouguer anomaly. The fractal dimension of surface has been used as surface roughness of bouguer anomaly. Using this method, the optimal density of Charak area insouth of Hormozgan province can be determined which is 2/7 g/cfor the under-research area. This determined density has been used to correct and investigate its results about the isostasy of the studied area and results well-coincided with the geology of the area and dug exploratory holes in the text area

Fractal dimension of the fracture surface of porous ZrO2 – MgO composite

2020 ◽  
pp. 74-82
Author(s):  
A. S. Buyakov ◽  
◽  
Yu. A. Zenkina ◽  
S. P. Buyakova ◽  
S. N. Kulkov ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Fracture Surface

Fractal Dimension of the Fracture Surface of Porous ZrO2–MgO Composite

2020 ◽  
Vol 11 (5) ◽  
pp. 1253-1259
Author(s):  
A. S. Buyakov ◽  
Yu. A. Zenkina ◽  
S. P. Buyakova ◽  
S. N. Kulkov
Keyword(s):  
Fractal Dimension ◽  
Fracture Surface

Characterizing Fractal and Hierarchical Morphologies Beyond the Fractal Dimension

1994 ◽  
Vol 367 ◽  
Author(s):  
Raphael Blumenfeld ◽  
Robin C. Ball
Keyword(s):  
Asymptotic Behavior ◽  
Fractal Dimension ◽  
Finite Size ◽  
Size Analysis ◽  
Corrections To Scaling ◽  
Diffusion Limited Aggregation ◽  
Diffusion Limited ◽  
Markovian Processes ◽  
Branching Patterns ◽  
Admissible Solutions

AbstractWe present a novel correlation scheme to characterize the morphology of fractal and hierarchical patterns beyond traditional scaling. The method consists of analysing correlations between more than two-points in logarithmic coordinates. This technique has several advantages: i) It can be used to quantify the currently vague concept of morphology; ii) It allows to distinguish between different signatures of structures with similar fractal dimension but different morphologies already for relatively small systems; iii) The method is sensitive to oscillations in logarithmic coordinates, which are both admissible solutions for renormalization equations and which appear in many branching patterns (e.g., noise-reduced diffusion-limited-aggregation and bronchial structures); iv) The methods yields information on corrections to scaling from the asymptotic behavior, which is very useful in finite size analysis. Markovian processes are calculated exactly and several structures are analyzed by this method to demonstrate its advantages.

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