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 lin ear relations between fracture length and displacements and thus the determ ination theoretically of the strength of a body or structure directly. Self -similarity of a fracture process means the existence of a universal faulti ng mechanism. However, the general applicability of universal scaling to fi eld observations and rock mechanics measurements remains the subject of som e debate. Complete self-similarity of a fracture process is hardly ever fou nd experimentally, except in some aluminium alloys. At early stages of the loading, material degrades due to increasing microcrack concentrations. Lat er, these microcracks where distributed in the process zone localize into a subcritically growing macrocrack, and finally the fracture process acceler ates and rupture runs away, producing dynamic fracture. The macroscopic eff ects of distributed cracking and other types of damage require treatment by constitutive models that include non-linear stress-strain relations togeth er with material degradation and recovery. The present model treats two phy sical 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 evol ution (degradation/recovery of elasticity) in response to ongoing deformati on. To analyse the scaling of a fracture process and the onset of the dynam ic events, we present here the results of numerical modelling of mode I cra ck growth. It is shown that the distributed damage and the process zone cre ated eliminate the stress-strain crack-tip singularities, providing a finit e rate of quasi-static crack growth. The growth rate of these cracks fits w ell the experimentally observed power law, with the subcritical crack index depending on the ratio between the driving force and the confining pressur e. 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 linea r fracture mechanics. At a certain stage, controlled by dynamic weakening a nd 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 re gime. The subcritical crack growth index increases steeply, crack growth ac celerates, the size of the process zone decreases, and the rate of crack gr owth ceases to be controlled by the rate of damage increase. Furthermore, t he crack speed approaches that predicted by the elastodynamic equation. The model presented describes transition from quasi-static crack propagation t o the dynamic regime and gives proper time and length scales for the onset of the catastrophic dynamic process.