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

A. V. Dyskin (1)

1 School of Civil and Resource Engineering

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. The driving force of the fracture system formation is the crack growth due to the action of stress fluctuations. The importance of this mechanism is that even when the fluctuations have zero average the cracks generated by them could growth far beyond the scale of stress fluctuations. Further development of the fracture system is controlled by crack interaction, which in the case of isotropically oriented cracks leads to the emergence of self-similar distributions.

The presence of self-similar distributions of fractures in a material poses a challenge in continuum modelling, since this material becomes discontinuous at any scale. We develop a continuum fractal mechanics to model mechanical behaviour of such materials. 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 volume elements of the corresponding size over which averaging is performed in the process of defining the field variables in the continuum. Subsequently, at each scale the material is modelled by a continuum that hides the cracks of smaller scales while explicitly introducing larger structural elements. The properties assigned to the continuum are effective characteristics accounting for the macroscopic effect of the hidden cracks.

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 the crack distribution produced by the action of stress fluctuations permeability increases as cube of crack radius. This property could be used for detecting this specific mechanism of formation of self-similar crack distributions.

Domaines: Interfaces continentales, environnement; Océan, Atmosphère; Sciences de la Terre

A. V. Dyskin. Self-similar pattern formation and continuous mechanics of self-similar systems. Hydrology and Earth System Sciences Discussions, 2006, 3 (4), pp.1879-1913. ⟨hal-00298741⟩