Scattering of waves in elastic media containing passive and active cracks

Scattering of wavefields in a 3-D medium that includes passive and/or active structures, is numerically solved by using the boundary integral equation method (BIEM). The passive structures are velocity anomalies that generate scattered waves upon incidence, and the active structures contain endogenous fracture sources, which are dynamically triggered by the dynamic load due to the incident waves. Simple models are adopted to represent these structures: passive cracks act as scatterers and active cracks as fracture sources. We form cracks using circular boundaries, which consist of many boundary elements. Scattering of elastic waves by the boundaries of passive cracks is treated as an exterior problem in BIEM. In the case of active cracks, both the exterior and interior problems need to be solved, because elastic waves are generated by fracturing with stress drop, and the growing crack boundaries scatter the incident waves from the outside of the cracks. The passive cracks and/or active cracks are randomly distributed in an infinite homogeneous elastic medium. Calculations of the complete waveform considering a single scatter show that the active crack has weak influence on the attenuation of first arrivals but strong influence on the amplitudes of coda waves, as compared with those due to the passive crack. In the active structures, multiple scattering between cracks and the waves triggered by fracturing strongly affect the amplitudes of first arrivals and coda waves. Compared to the case of the passive structures, the attenuation of initial phase is weak and the coda amplitudes decrease slowly.     

Wave speeds and attenuation of elastic waves in material containing cracks

Summary. Expressions now exist from which may be calculated the propagation constants of elastic waves travelling through material containing a distribution of cracks. The cracks are randomly distributed in position and may be randomly orientated. The wavelengths involved are assumed to be large compared with the size of the cracks and with their separation distances so that the formulae, based on the mean taken over a statistical ensemble, may reasonably be used to predict the properties of a single sample. The results are valid only for small concentrations of cracks.
Explicit expressions, correct to lowest order in the ratio of the crack size to a wavelength, are derived here for the overall elastic parameters and the overall wave speeds and attenuation of elastic waves in cracked materials where the mean crack is circular, and the cracks are either aligned or randomly orientated. The cracks may be empty or filled with solid or fluid material. These results are achieved on the basis of simply the static solution for an ellipsoidal inclusion under stress.
The extension to different distributions of orientation or to mixtures of different types of crack is quite straightforward.