Attenuation and speed of antiplane waves in a cracked solid using the Kramers-Kronig relations

The propagation of antiplane waves in an unbounded linearly elastic solid that contains a distribution of flat cracks is considered. The cracks are parallel to each other, the distribution is dilute, and the incident wave is normal to the planes of the cracks. Using the Kramers-Kronig relations, the speed c of the coherent wave in the cracked solid is derived as a function of the dimensionless frequency wa/c_T(where w is the angular frequency, a is the half crack width, and c_Tis the speed of antiplane waves in an uncracked solid) and of crack density. Numerical results for the speed c are obtained, based on the assumption that the limit of c for high frequencies is equal to c_T. These results show that the speed of the coherent wave in a cracked solid is always less than the limiting speed at high frequencies. The results of this work are valid for the entire range of frequencies.