Fractal Cracks

The studies carried out earlier have shown that polymer film samples strength to a considerable extent is defined by growth parameters of stable crack in local deformation zone (ZD) at a notch tip [1-3], As it has been shown in Refs. [4, 5], the fiactal concept can be used successfully for the similar processes analysis. This concept is used particularly successfully for the relationships between fracture processes on different levels and subjecting fracture material microstructure derivation [5]. This problem is of the interest in one more respect. As it has been shown earlier, both amorphous polymers structure [7] and Griffith crack [4] are fractals. Therefore, the possibility to establish these objects fractal characteristics intercommunication appears. The authors of Refs. [8, 9] consider stable cracks in polyarylatesul-fone (PASF) film samples treatment as fractals and obtain intercommunication of this polymer structure characteristics with samples with sharp notch fracture parameters. 

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]. 

FIGURE 8.1 The dependences of stable crack opening 5 on its length in double logarithmic coordinates for PAST samples, obtained from solutions in dichloroethane (1) chloroform (2) and N, N-dimethylformamide (3) [8], 

FIGURE 8.2 The electron microphotograph of stable crack boundary for PASF sample (solvent - chloroform). Enlargement 15,000 [10], 

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- 

---

Fracture strength of brittle solids with small disorder and rough (fractal) cracks... 

It is clear that both for the flat surface case (with Z >> as in (3.10c), when the crack length is much higher than the correlation length of the fractal crack), and shallow surface case (with Zmax << 0 tnd dz/dr = 1 as in (3.106)), equating Es with E one gets... 

In Ref [4], it has been shown that between stress intensity factors (resistance to crack propagation) for fractal crack and smooth one-dimen-... 

And at last, for fractal cracks the following empirical relationship is valid... 

Fractal Model of Stability to the Cracking of Modified Polyethylene... 

Fractals are of great interest because of their exquisite combination of beauty, complexity, and endless structure. They are reminiscent of natural objects like mountains, clouds, coastlines, blood vessel networks, and even broccoli, in a way that classical shapes like cones and squares can t match. They have also turned out to be useful in scientific applications ranging from computer graphics and image compression to the structural mechanics of cracks and the fluid mechanics of viscous fingering. 

Situation C Edges, cracks, etc. (b) could place at disposal energy richer reaction centres than the surfaces with the normal number of free valencies (a). Therefore, the active centres can be distributed more in case of fractal surface (b) than the non-fractal surface... 

Preston, S., B.S. Griffiths, and I.M. Young. 1997. An investigation into sources of soil crack heterogeneity using fractal geometry. Eur. J. Soil Sci. 48 31-44. 

Natural fractals such as clouds, polymers, aerogels, porous media, dendrites, colloidal aggregates, cracks, fractured surfaces of solids, etc., possess only statistical self-similarity, which, furthermore, takes place only in a restricted range of sizes in space [1,4,16]. It has heen shown experimentally for solid polymers [22] that this range is from several angstroms to several tens of angstroms. 

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]. 

As an example of fractality displayed in polymers at a macroscopic level (a secondary structural element), we shall consider the growth of a crack in a film of amorphous vitreous polyarylate-sulfone [100] (see Figure 11.6). 

---