Scaling of fracture length and distributed damage

The linear theory of elasticity formulated in terms of dimensionless strain components does not allow the introduction of any space scaling except linear relations between fracture length and displacements and thus the determination theoretically of the strength of a body or structure directly. Self-similarity of a fracture process means the existence of a universal faulting mechanism. However, the general applicability of universal scaling to field observations and rock mechanics measurements remains the subject of some debate. Complete self-similarity of a fracture process is hardly ever found experimentally, except in some aluminium alloys. At early stages of the loading, material degrades due to increasing microcrack concentrations. Later, these microcracks where distributed in the process zone localize into a subcritically growing macrocrack, and finally the fracture process accelerates and rupture runs away, producing dynamic fracture. The macroscopic effects of distributed cracking and other types of damage require treatment by constitutive models that include non-linear stress-strain relations together with material degradation and recovery. The present model treats two physical aspects of the brittle rock behaviour: (1) a mechanical aspect, that is, the sensitivity of the macroscopic elastic moduli to distributed cracks and to the type of loading, and (2) a kinetic aspect, that is, damage evolution (degradation/recovery of elasticity) in response to ongoing deformation. To analyse the scaling of a fracture process and the onset of the dynamic events, we present here the results of numerical modelling of mode I crack growth. It is shown that the distributed damage and the process zone created eliminate the stress-strain crack-tip singularities, providing a finite rate of quasi-static crack growth. The growth rate of these cracks fits well the experimentally observed power law, with the subcritical crack index depending on the ratio between the driving force and the confining pressure. The geometry of the process zone around a quasi-static crack has a self-similar shape identical to that predicted by universal scaling of the linear fracture mechanics. At a certain stage, controlled by dynamic weakening and approximated by the reduction of the critical damage level proportional to the rate of a damage increase, the self-similarity breaks down and crack velocity significantly deviates from that predicted by the quasi-static regime. The subcritical crack growth index increases steeply, crack growth accelerates, the size of the process zone decreases, and the rate of crack growth ceases to be controlled by the rate of damage increase. Furthermore, the crack speed approaches that predicted by the elastodynamic equation. The model presented describes transition from quasi-static crack propagation to the dynamic regime and gives proper time and length scales for the onset of the catastrophic dynamic process.