Damage mechanics with long‐range interactions: correlation between the seismic b‐value and the fractal two‐point correlation dimension

Abstract

A modified Griffith criterion for a two‐dimensional array of aligned elliptical cracks with a long‐range interaction potential is presented. In accordance with observation, the pattern of cracks is assumed to be fractal, with a two‐point correlation dimension Dc indicating a power‐law distribution of crack spacings r, and a power‐law exponent D of the crack length distribution. From a simple dislocation theory of the seismic source D is proportional to the seismic b‐value if an individual earthquake or acoustic emission is produced by displacement on a specific fault or crack in the population. As a result, the theory is applicable to incremental damage rather than the long‐term evolution of crack systems with large displacements. The long‐range interaction between cracks is taken to be elastic, implying a positive interaction potential proportional to r‐1. Two models are presented for the spatio‐temporal evolution of the resulting seismicity due to: (A) progressive alignment of epicentres along an incipient fault plane; and (B) clustering of epicentres around potential nucleation points on an existing fault trace. The modified Griffith criterion predicts either an increase or a decrease in the potential energy release rate G′, depending on the sign of ∂Dc/∂D and the nature of the concentration of deformation. For model (A), if ∂Dc/∂D > 0 (corresponding to an implied positive correlation between the b‐value and Dc), then G′ increases in the presence of an interaction potential. In contrast G′ increases if ∂Dc/∂D < 0 for model (B). Both results lead to a mechanical weakening effect associated with more concentrated deformation. Such an association of mechanical weakening with concentration of deformation is fundamental to the development of fault systems. On the other hand if ∂Dc/∂D <0 (corresponding to an implied negative correlation between the b‐value and Dc), G′ decreases for model (A) in the presence of an interaction potential, implying a hardening of the material due to the interaction. For model (B) G′ decreases when ∂Dc/∂D >0. The mechanical hardening (lowering G′) is associated with geometrically distributed damage in either case. Equivalently this can be seen as a shielding effect, with the zone of damage reducing the local stresses on a particular crack. If there is no correlation the interaction potential has a slight mechanical hardening effect with no strong geometric effect. These predictions are also consistent with the usual tenets of damage mechanics, in which early crack growth is stable, distributed and is associated with mechanical hardening, and material failure occurs later in the cycle due to localized, unstable crack coalescence, associated with mechanical weakening. The main difference between the theory presented here and standard damage mechanics is that crack coalescence is organized, and hence instability can develop at lower crack densities. Copyright © 1992, Wiley Blackwell. All rights reserved