Crack Fractal Dimension

Let us consider in conclusion the physical significance of stable crack fractal dimension in PASF samples. As it has been shown in Ref [12], the stress concentration coefficient Kg of triangular crack is given by the Eq. (5.10). In Fig. 8.6 the relation between parameters and for PASF sample (solvent - chloroform) is adduced, fi om which linear reduction follows at growth. Thus, the dimension for stable crack has the simple physical significance - it is the value, reciprocal to stress concentration level at crack tip. 

TABLE13.1 The Experimental and Theoretical Values of Crack Fractal Dimension for PASF Films, Prepared For Polymer Solutions in Different Solvents [1]... 

Secondly, polymers are known to possess multilevel structures (molecular, topological, supermolecular, and floccular or block levels), the elements of which are interconnected [43, 44]. In addition, an external action on a polymer can induce the formation of new (secondary) structural elements — cracks, fractured surfaces, plastic flow regions, etc. These primary and secondary structural elements as well as the processes forming them are characterised by miscellaneous parameters therefore, only empirical correlations have been obtained, at best, between these parameters. If each of the above-mentioned elements (processes) is described by a standard parameter, for example, fractal dimension, one can derive analytical equations relating them to one another and containing no adjustable parameters. This is very significant for the computer synthesis of structure and for the prediction of properties and behaviour of polymeric materials during performance. Note that fractal analysis has been used successfully to describe the phenomena of rubber elasticity [16, 45, 46] and fluidity [25, 47-49]. 

It has been found that the basic elements of structure of initial and deformed polymers are homogeneous fractals that can be characterised by their fractal dimension. Examples of such elements are macromolecular coil, supermolecular organisation as seen in the cluster structure and a stable crack in film samples of polymers. These examples are given as examples of the possible variants of the term "multifractality with reference to polymers. 

The surface of a crack depends on the properties of the material and on loading characteristics. The surface may be more or less rough and developed. It appears that the surface of the crack in cement-based materials has fractal nature, which indicates that the effective determination of its area is related to the scale of magnification. General remarks about fractals and fractal dimension may be found in a book by Mandelbrot (1983). The methods on how to characterize the cracks and the fracture surfaces using the notion of fractal dimension is briefly described in Section 10.5. 

The question as to whether there is a general and reliable relationship between the fractal dimension of a surface and the fracture toughness of the material is considered by a few authors and it has been shown that a crack surface profile can be effectively described in terms of fractal geometry (e.g. Lange et al, 1993). 

The tests of concrete specimens subjected to Mode II fracture were aimed at a further investigation of relations between fractal dimension and roughness of the fracture surface after Mode II crack propagation and are presented below based on paper by Brandt and Prokopski (1993). 

Hiroaki Hara and Seiji Okayama, Fractal dimension and scaling behavior of cracks in a random medium Frequency-rank distribution described by generalized random walks, Phys. Rev. B 37, 9504-9511 (1988). 

Thus, the direct usage of fractal dimension of failure surface, formed by instable crack, for pol miers limiting properties characterization is deprived of sense. The multifractal treatment application [46] in the given case does not change the situation. Nevertheless, if fractal dimension reflects local plastic deformation level, then similar correlations has, at any rate, applied significance [47]. 

Long, Q. Y, Suqin, L., Lung,C. W. (1991). Studies ofthe Fractal Dimension of a Fracture Surface Formed by Slow Stable Crack Propagation. J. Phys. D Appl. Phys., 24(4), 602-607. 

In Fig. 8.3, the relation between fractal dimensions of structure and stable crack for PASF samples is adduced. As it was to be expected from the most general considerations, the intercommunication existed between di-... 

FIGURE 8.3 The relation between fractal dimensions of stable crack and polymer structure for PAST samples. The straight lines drawing mode explanations are given in the... 

The dependence of stable crack length / on its fractal dimension D for PASF... 

Hence, the stated above results have shown, that the stable crack in PASF samples, obtained from different solvents, can be treated as stochastic fi actal, whose dimension has clear physical significance - its value is reciprocal to stress concentration degree at a crack tip. The value is determined unequivocally by polymer structure fractal dimension and in its turn, influences on PASF samples fracture process parameters [8]. 

FIGURE 8.6 The dependence of stress concentration coefficient of stable crack on its fractal dimension D for PAST [8],... 

The fiacture surface fractal dimension at quasibrittle fracture is determined according to the Eq. (4.50) and at quasiductile fracture - according to the Eq. (4.51). And at lastthe value for fractal Griffith crack d can be estimated according to the Eq. (5.17) at the condition tn = 2. 

Figure 11.6 shows the plots for the variation of versus r j for two amorphous polymers the plots correspond to relationship (11.28). In other words, a stable crack in polymer film samples is a stochastic fractal with the dimension 1.48. The linearity of plots shown in... 

In Figure 14.4 double logarithmic dependences 2 In 8 = /(In r) are presented for two amorphous polymers which have appeared to be linear and by virtue of it, correspond to Equation (14.7). Otherwise, the stable crack in film polymeric samples is a stochastic fractal with dimension 1.48. Linearity of the diagrams presented on Figure 14.4 reflects the self-similarity of a crack at different stages of its growth. Thus, at the macroscopic level polymers the fractal properties are also displayed. 

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. ... 

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