Fracture fractal model

Fractal models for soil structure and rock fractures are becoming increasingly popular (e.g., Sahimi, 1993 Baveye et al., 1998). The primary appeal of these models is their ability to parsimoniously parameterize complex structures. Scale symmetry or scale invariance, in which an object is at least statistically the same after magnification, is a fundamental property of fractals and can also be observed in numerous natural phenomena. Thus, it is logical that some investigators have examined theoretical transport in known prefractals. 

It is difficult to describe the complex forms and status of natural fracture surfaces with ordinary functions. The fractal theory can be used to describe extremely irregular geometric patterns, and many researches have indicated that it is reasonable to simulate rough fracture surfaces with fractals theory. At present, the self-affine fractal model is regarded as the best fractal model to simulate fracture surface. In the following, random Brown function method (Weierstrass- Mandelbrot function) is used to simulate a fractal fracture surface. 

The electron microscopy data confirm the made conclusion. In Fig. 8.2, the microphotograph of stable crack boundary in PASF sample (solvent -chloroform) is adduced. As one can see, the fracture surface has microroughnesses at any rate of two levels ( 1 mcm and 20 mn) that allows to apply fractal models for PASF samples fracture process [4, 5]. 

Hence, the adduced above results have shown that polyethylenes samples fracture criterion in tests on cracking under stress in active mediums is polar liquid reaching of sample median plane. The stability to cracking is described correctly within the frameworks of fractal model of transport in polymers. The stability to cracking extreme growth cause is structural changes, which are due to high-disperse mixture Fe/FeO introduction and characterized by dimension D. ... 

T-H-M-C processes are significantly affected by subsurface heterogeneity, which results in scale-dependence of the related parameters. To handle this scale-dependent behavior, we need to characterize this heterogeneity and consider its effects at different scales. In this study, we demonstrate that the measured permeability data from Sellafield site, UK, are very well described by fractional Levy motion (fLm), a stochastic fractal. This finding has important implications for modeling large-scale coupled processes in heterogeneous fractured rocks. 

Yamamoto, H., Kojima K. Tosaka, H. 1993. Fractal clustering of rock fractures and its modeling using cascade process. In Pinto da Cunha (ed.). Scale Effects in Rock Masses 93, Balkema, Rotterdam. 

Four fracture sets are used and their orientation show near-random patterns due to their low Fisher constants. The size of the model is as 5 m x 5 m (Figure 2). The fracture trace lengths are characterized by a fractal scaling law as follows... 

Forming, storming, norming, performing model of decision making, 2210 Forrester Research, 781 Forward-reserve allocation (wetfehouse operation), 2093 Forward selection, 2289 Fourth generation R D, 148 FOV (field of view), 2505 Fox Meyer, 949-950 Fractals, 404 Fractures, 1169... 

These phE fluctuations may shed light on the dynamics of possible chaotic processes responsible for the fractal nature of many fracture surfaces. Measurements of the fractal dimensions of various fracture surfaces have been reported by several authors.(65-6 ) since the underlying structure of many chaotic systems is fractal, this suggests that fracture is to some degree chaotic, that is, deterministic, yet aperiodic and unpredictable. Some simple models of the fracture process have been proposed which display chaos or result in fractal... 

Thus, the stated above results demonstrated, that fractal analysis application for polymers fracture process description allowed to give more general fracture concept, than a dilation one. Let us note, that the dilaton model equations are still applicable in this more general case, at any rate formally. The fractal concept of polymers fracture includes dilaton theory as an individual case for nonfractal (Euclidean) parts of chains between topological fixation points, characterized by the excited states delocalization. The offered concept allows to revise the main factors role in nonoriented polymers fracture process. Local anharmonicity ofintraand intermolecular bonds, local mechanical overloads on bonds and chains molecular mobility are such factors in the first place [9, 10]. 

Polymer mechanical properties are one from the most important ones, since even for polymers of different special-purpose function a definite level of these properties always requires [20]. Besides, in Ref [48] it has been shown, that in epoxy polymers curing process formation of chemical network with its nodes different density results to final polymer molecular characteristics change, namely, characteristic ratio C, which is a polymer chain statistical flexibility indicator [23]. If such effect actually exists, then it should be reflected in the value of cross-linked epoxy polymers deformation-strength characteristics. Therefore, the authors of Ref [49] offered limiting properties (properties at fracture) prediction techniques, based on a methods of fractal analysis and cluster model of polymers amorphous state structure in reference to series of sulfur-containing epoxy polymers [50]. 

Hence, the stated above results have shown, that the correct description of fracture process of phenylone and particulate-filled nanocomposites on its basis can be obtained within the frameworks of fiiactal model only [54], In addition the fracture crack should be simulated by an isotropic fractal [55],... 

Let us note in conclusion, that the adduced above results have shown that the fractal Griffith crack model application can not only improve quantitative conformity of theoretical and experimental data, but also obtain qualitatively new picture of fracture processes. 

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