Elastic wave scattering by an elliptical crack

In crack scattering problems, analytical solutions have been derived only for circular or slit cracks. But real defects in solids often take the form of elliptical cracks, for which analytical elastodynamic solutions remain unavailable. For this reason, this study addresses this long-standing problem. We obtain a series-expansion solution for elastic wave scattering by a three-dimensional elliptical crack under incident plane waves with arbitrary propagation directions, polarizations and wavelengths and express crack opening displacements, stress intensity factors, scattering far fields and scattering cross sections in terms of expansion coefficients. The principal advantage of the proposed method over purely numerical approaches is that it allows the derivation of explicit, rigorous asymptotic solutions at both low and high frequencies. We have validated the solution by comparing the results with a variety of existing results in the corresponding cases. In the low-frequency limit the solution naturally reduces to static solutions, while in the high-frequency limit the solution coincides with Kirchhoff approximation for the problem of wave reflection by a traction-free half-plane. At intermediate frequencies, the solution converges as truncation number increases and shows good agreement with the results of boundary element method. Moreover, the closed-form expressions for low- and high-frequency asymptotic solutions can reveal prominent frequency-dependent characteristics of the cracks. Potential applications of the solutions include acoustic nondestructive testing of defects in solids, interpretation of seismic dispersion and attenuation in fractured reservoirs, and evaluation of dynamic effective properties of cracked materials. The solutions may also serve as a benchmark to check numerical methods for the same problem.