In many cases, the critical state of systems that reached the threshold ischaracterised by self-similar pattern formation. We produce an example ofpattern formation of this kind – formation of self-similar distribution ofinteracting fractures. Their formation starts with the crack growth due tothe action of stress fluctuations. It is shown that even when thefluctuations have zero average the cracks generated by them could grow farbeyond the scale of stress fluctuations. Further development of the fracturesystem is controlled by crack interaction leading to the emergence ofself-similar crack distributions. As a result, the medium with fracturesbecomes discontinuous at any scale. We develop a continuum fractal mechanicsto model its physical behaviour. We introduce a continuous sequence ofcontinua of increasing scales covering this range of scales. The continuumof each scale is specified by the representative averaging volume elementsof the corresponding size. These elements determine the resolution of thecontinuum. Each continuum hides the cracks of scales smaller than the volumeelement size while larger fractures are modelled explicitly. Using thedeveloped formalism we investigate the stability of self-similar crackdistributions with respect to crack growth and show that while theself-similar distribution of isotropically oriented cracks is stable, thedistribution of parallel cracks is not. For the isotropically orientedcracks scaling of permeability is determined. For permeable materials(rocks) with self-similar crack distributions permeability scales as cube ofcrack radius. This property could be used for detecting this specificmechanism of formation of self-similar crack distributions.
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