Scattering of harmonic waves from a nonlinear elastic inclusion

This article considers the scattering of harmonics stemming from the interaction of a primary wave with a heterogeneous and elastically nonlinear inclusion present in an otherwise linearly elastic host medium. The elastodynamic equations of motion are derived for general elastic anisotropy up to a third-order in displacement nonlinearity (cubic nonlinearity). The method of successive approximations is applied in order to decouple the equations of motion into a linear system of equations. The linear equations permit the use of Green's functions to obtain the scattering amplitudes from an arbitrarily shaped inclusion. General forms of the scattering amplitudes are given as functions of scattering-based quadratic and cubic acoustic nonlinearity parameters. Shape factors are offered for some simple geometries in order to arrive at closed-form solutions. An explicit example is given in the case of a spherically shaped inclusion with isotropic elastic moduli. The influence of the second-, third-, and fourth-order elastic stiffnesses, primary and scattered wave mode types, and scattering angles are highlighted. Potential experimental techniques, based on the present scattering model, offer an alternative method of probing the nonlinear elastic properties of materials.