An effective medium model for elastic waves in microcrack damaged media

Direct numerical simulations of waves traveling through microcrack-damaged media are conducted and the results are compared to effective medium calculations to determine the applicability of the latter for studying wave propagation. Both tensile and compressive waves and various angular distributions of randomly-located cracks are considered. The relationships between the input wavelength and the output wave speed and output signal strength are studied. The numerical simulations show that the wave speed is nearly constant when 1/ka > 60 for tensile waves and 1/ka > 10 for compressive waves, where k is the wave number and a is the average half-crack length. The direct simulations also show that when the input wavelength is much longer than the crack length, 1/ka > 60, the wave can pass through the damaged medium relatively unattenuated. On the other hand, when the input wavelength is shorter than a “cut off” wave length, the output wave magnitude decreases linearly with the input wavelength. The effective medium wave speed and magnitude calculations are not dependent on the input wavelength and therefore the results correspond well with the numerical simulations for large 1/ka. This suggests a minimum wavelength for which the homogenized methods can be used for studying these problems.