Weakly nonlinear two dimensional wave scattering in incompressible pre-stressed materials: Far field approximation

Abstract

We develop a series expansion approach to the calculation of the scattering coefficients of in-plane transverse elastic waves. The waves are scattered by a cylindrical obstacle as they propagate through an incompressible nonlinear material which is inhomogeneously pre-stressed. We deal with the calculation using a distorted wave Born approximation in the far field. A neo-Hookean strain energy function is considered for the incompressible elastic material, and a rigid cylinder is considered as an obstacle. From the static equilibrium configuration, a small-on-large approach is constructed via a series expansion of the wave equation which allows us to compute both the scattered wave and the induced pressure field, and their dependence on the applied pre-stressed. In the far field limit, the pre-stress effect on the scattering coefficients is calculated via partial waves. The formalism can be expanded directly to three-dimensional media with different obstacle symmetries.