Dynamic analysis of elastic waves by an arbitrary cavity in an inhomogeneous medium with density variation

Based on complex function theory, an analytical solution for the dynamic stress concentration due to an arbitrary cylindrical cavity in an infinite inhomogeneous medium is investigated. Two different conformal mappings are introduced to solve scattering by an arbitrary cavity for the Helmholtz equation with variable coefficient through the transformed standard Helmholtz equation with a circular cavity. By assuming that the medium density continuously varies in the horizontal direction with an exponential law and the elastic modulus is constant, the complex-value displacements and stresses of the inhomogeneous medium are explicitly obtained. The distribution of the dynamic stress for the case of an elliptical cavity are obtained and discussed in detail via a numerical example. The results show that the wavenumber, inhomogeneous parameters, and different values of aspect ratio have a significant influence on the dynamic stress concentration factors around the elliptical cavity.