Scattering of time-harmonic elastic waves by an elastic inclusion with quadratic nonlinearity

This paper considers the scattering of a plane, time-harmonic wave by an inclusion with heterogeneous nonlinear elastic properties embedded in an otherwise homogeneous linear elastic solid. When the inclusion and the surrounding matrix are both isotropic, the scattered second harmonic fields are obtained in terms of the Green's function of the surrounding medium. It is found that the second harmonic fields depend on two independent acoustic nonlinearity parameters related to the third order elastic constants. Solutions are also obtained when these two acoustic nonlinearity parameters are given as spatially random functions. An inverse procedure is developed to obtain the statistics of these two random functions from the measured forward and backscattered second harmonic fields.