Self-similar pattern formation and continuous mechanics of self-similar systems

In many cases, the critical state of systems that reached the threshold is characterised by self-similar pattern formation. We produce an example of pattern formation of this kind - formation of self-similar distribution of interacting fractures. Their formation starts with the crack growth due to the action of stress fluctuations. It is shown that even when the fluctuations have zero average the cracks generated by them could grow far beyond the scale of stress fluctuations. Further development of the fracture system is controlled by crack interaction leading to the emergence of self-similar crack distributions. As a result, the medium with fractures becomes discontinuous at any scale. We develop a continuum fractal mechanics to model its physical behaviour. We introduce a continuous sequence of continua of increasing scales covering this range of scales. The continuum of each scale is specified by the representative averaging volume elements of the corresponding size. These elements determine the resolution of the continuum. Each continuum hides the cracks of scales smaller than the volume element size while larger fractures are modelled explicitly. Using the developed formalism we investigate the stability of self-similar crack distributions with respect to crack growth and show that while the self-similar distribution of isotropically oriented cracks is stable, the distribution of parallel cracks is not. For the isotropically oriented cracks scaling of permeability is determined. For permeable materials (rocks) with self-similar crack distributions permeability scales as cube of crack radius. This property could be used for detecting this specific mechanism of formation of self-similar crack distributions.