A modified Griffith criterion for the evolution of damage with a fractal distribution of crack lengths: application to seismic event rates and b ‐ values

SUMMARY The Griffith criterion for dynamic crack growth results from a calculation of a minimum in the Gibbs free energy of an infinite medium containing a single elliptical crack. However rock failure in the laboratory or during large earthquakes is usually preceded by the evolution of an aureole of damage in the form of subsidiary microcracks or faults. A characteristic of such precursory damage is that it is fractal, having a power‐law crack length distribution, and also a power‐law spatial and temporal correlation. Such fractal damage is consistent with the concept of faulting or cracking as a self‐organized critical phenomenon, and has been widely confirmed by empirical observation of faults, laboratory fractures and indirect seismic monitoring in the laboratory and field. Here we consider the free energy change Δ F associated with an ensemble of N T weakly interacting, aligned, elliptical cracks of different lengths, with semilength expectation value 〈 c 〉, under a constant tensile stress s̀ applied at the boundary of each element. The crack ensemble represents a state of damage which may evolve in a quasi‐static way due to subcritical crack growth. By considering ∂(Δ F )/∂〈 c 〉= o, a modified strain energy release rate G ′=∂ U /∂ A d = f (〈 c 〉, 〈 c 2 〉) is defined, where U is the potential strain energy, and A d is the total surface area of the array of cracks. For a given N T , G ′ is proportional to the rate of change of the total volume of damage with respect to the total area of damage A d . This reflects the fact that mechanical energy is stored in a volume, and released on a surface. As the number of cracks tends to 1, G ′ tends naturally to the strain energy release rate G for a single crack. For a fractal distribution of crack lengths N c ( c ) = N T ( c / c o ) ‐D , limited to a range ( c o , c 1 ), it can be shown that G ′ is negatively correlated to D for a given constant value of N T . Also, the curves for higher N T are associated with higher G ′. A similar result can be obtained more simply by using the expectation value 〈 G 〉 directly as an appropriate parameter, with the advantage that both N T and D can be allowed to vary independently. These predictions are respectively consistent with the positive (negative) correlation established between acoustic emission event rates (seismic b ‐values) and G in the laboratory for quasi‐static tensile failure by mode I subcritical crack growth due to stress corrosion reactions in double torsion loading. The theory also correctly predicts the order of magnitude of the stress corrosion index for these experiments, and the observation that more heterogeneous materials have higher stress corrosion indices. However the correlations established between event rate, D and G ' (or 〈 G 〉) are completely general, and can apply in principle to other forms of fault development or crack growth with weak long‐range interactions. Most studies of the statistics of damage evolution are by their very nature indirect or posthumous, unless the material of interest is optically transparent. Seismic monitoring of damage in the form of small earthquakes or acoustic emissions can be used to measure the parameters a and b of the earthquake frequency‐magnitude relation log N, = a ‐ bm, where a = log N, b = CD /3. C is the slope of the scaling relation between magnitude and the common algorithm of seismic moment, and the event rate N is assumed proportional to N T . Thus changes in a and b can in principle be used to infer G′ (or 〈G〉) during the evolution of damage in the intermediate term prior to dynamic failure of laboratory rock samples or the Earth.