Scattering of elastic waves by a sphere with cubic anisotropy

Other conference contribution, 2021

Scattering of elastic waves in materials with inhomogeneities is a classical problem in physics and geophysics, and have applications in non-destructive testing, material characterization, medical ultrasound, etc. The classical analytical solution of the scattering by a single isotropic spherical obstacle provides a good approximation and a basis for more complicated problems and gives a deep understanding of the scattering phenomenon [1].

However, plenty of natural and synthetic materials, specifically the grains in a metal, are known to be anisotropic. Recently, the scattering of elastic waves by a circle with cubic anisotropy is studied in 2D by Bostrom [2, 3], and Jafarzadeh et al. [4] use the same method to study the 3D scattering problem for a transversely isotropic sphere. The present work is a continuation of these studies and investigates the 3D scattering by a sphere with
cubic anisotropy. Consider the scattering of a single spherical obstacle with cubic anisotropy contained in a three-dimensional, homogeneous, isotropic and infinite elastic medium. In the isotropic surrounding the classical approach is used with the displacement field constructed as a superposition of incident and scattered waves, which are expanded in spherical vector wave functions. Inside the sphere the stress-strain relations are given in Cartesian coordinates for the cubic material and these are first transformed to spherical coordinates. These relations then become inhomogeneous in that they contain factors with trigonometric functions in both angular coordinates and the same becomes true also for the equations of motion. To proceed it is useful to expand the displacement into a series of vector spherical harmonics where each coefficient in turn is expanded into a power series in the radial
coordinate. It follows from the equation of motion that the coefficients in the power series obey certain recursion relations, thus reducing the number of unknowns inside the sphere. Expressing also the stresses as a series in the vector spherical harmonics, the rest of the unknowns are determined by the continuity of the displacement and traction on the sphere boundary. As a result, the transition (T) matrix elements, relating the expansion coefficients of the scattered wave to those of the incident wave, are calculated. It is, in particular, possible to obtain explicit expressions for the leading order T matrix elements for low frequencies.

References

[1] V. Varadan, A. Lakhtakia, and V. Varadan, Field representations and introduction to scattering. North-Holland, Amsterdam, 1991.

[2] A. Bostrom, "Scattering by an anisotropic circle," Wave Motion, vol. 57, pp. 239-244, 2015.

[3] A. Bostrom, "Scattering of in-plane elastic waves by an anisotropic circle," The Quarterly Journal of Mechanics and Applied Mathematics, vol. 71, pp. 139-155, 2018.

[4] A. Jafarzadeh, P. D. Folkow, and A. Bostrom, "Scattering of elastic sh waves by transversely isotropic sphere," in Proceedings of the International Conference on Structural Dynamic, EURODYN, vol. 2, pp. 2782-2797, 2020.