On propagation of time-harmonic elastic waves through a double-periodic array of penny-shaped cracks

Improved boundary integral equation method for the investigation of time-harmonic longitudinal elastic wave penetration through a plane of penny-shaped cracks with a periodic square or rectangular lattice in 3D infinite elastic solid is proposed. Under the assumption of normal incidence of wave, the corresponding symmetric wave scattering problem is reduced to a boundary integral equation for the displacement jump across the crack-surfaces in a unit-cell by means of a 3D double-periodic Green's function in terms of the exponentially convergent Fourier integrals. A regularization technique for this Green's function involving special lattice sums in closed forms is adopted, which allows its effective calculation in a wide range of wave numbers. A collocation method is used for the solution of boundary integral equation. The reflected and transmitted far-field displacements are shown to be a superposition of a finite number of propagating wave modes and expressed by the obtained solution. The crack-opening-displacements, wave reflection and transmission coefficients in dependence on the wave number, lattice and crack sizes are computed and analyzed.