Harmonic waves in an elastic solid containing a doubly periodic array of cracks

The propagation of time-harmonic longitudinal waves in a solid containing a periodic distribution of cracks is investigated in a two-dimensional configuration. The cracks are parallel to the x-axis, and their centers are located at positions x=2ml, y=2nd (m, n=0, ± 1, ±2,...). For normal incidence, a dispersion equation is obtained. The derivation is based on the assumptions that the frequency is below a certain cut-off frequency and that the cracked planes, y = 2nh, are sufficiently distant from each other. These assumptions allow one to view the cracked planes as planes of homogeneous reflection and transmission. The reflection and transmission coefficients corresponding to a planar array of cracks are used to obtain numerical results. Dispersion curves are plotted for three values of the spacing between the cracked planes. An important feature of the results is the presence of passing and stopping bands. A simple method to calculate the slopes of the dispersion curves at the origin is also presented.