Scattering of elastic waves by an orthotropic sphere

Other conference contribution, 2023

Scattering of elastic waves in materials with a single spherical inclusion is an archetype of many scattering problems in physics and geophysics with applications in material characterization, non-destructive testing, medical ultrasound, etc. The study of spherical inclusions, besides their simplicity, provides enlightening details of the scattering phenomenon and a good approximation for more realistic objects. The analytical solution of the scattering by a single isotropic sphere is addressed comprehensively in the literature. However, anisotropic behavior is observed in plenty of natural and synthetic materials and only a few studies have been performed on the investigation of mechanical wave scattering by anisotropic obstacles. In this regard, Jafarzadeh et al. (Wave Motion, 112 (2022) 102963) studied the scattering of elastic waves by a transversely anisotropic sphere . The aim of the present work is to study the more general case with an orthotropic spherical inclusion. Wave scattering by a single spherical obstacle with orthotropic anisotropy inside an infinite, three-dimensional, homogeneous, and isotropic elastic medium is considered. In the isotropic surrounding, the displacement field is constructed as a superposition of the incident and scattered waves. Using the classical approach, these waves are expressed as expansions in the regular and outgoing spherical vector wave functions, respectively. The objective is to find the transition (T) matrix elements that relate the expansion coefficients of the scattered wave to those of the incident wave. The spherical inclusion, on the other hand, has orthotropic symmetry in Cartesian coordinates. Transforming the anisotropic stress-strain relations and the elastodynamic equations into spherical coordinates shows that the governing equations are inhomogeneous due to the appearance of trigonometric functions in the polar and azimuthal coordinates. To deal with the inhomogeneous governing equations, the same methodology as in the prior studies of the authors for a sphere with hexagonal and cubic symmetry is followed. The displacement field is expanded into a series of vector spherical harmonics in the angular coordinates and each coefficient in turn is expanded into a power series in the radial coordinate. It follows from the stress-strain relation and the equation of motion that there is coupling among partial waves inside the sphere which is more complex than for the transversely isotropic and cubic cases. The equation of motion inside the sphere leads to recursion relations among the unknown coefficients in the power series. Then the rest of the unknowns are determined by the continuity of the displacement and traction on the surface of the sphere, and the T matrix elements are calculated. Specifically, in the low frequency limit, where the sphere is much smaller than the wavelength of the incident wave, these elements can be expressed explicitly. Furthermore, the T matrix of a single sphere is used in combination with Foldy theory, to study attenuation and phase velocity of polycrystalline materials. Comparisons are performed with other theories and with numerical FEM computations from the literature