Scattering of Longitudinal Elastic Waves From an Anisotropic Spherical Shell

An exact solution for scattering of ultrasound from a spherically orthotropic shell is presented. The shell is assumed to be embedded in an isotropic elastic medium, and the core surrounded by the shell is also assumed to be isotropic. The shell itself is assumed to be “spherically orthotropic,” with five independent elastic constants (the spherical analog of a transversely isotropic material in Cartesian coordinates). Field equations for this material are presented, and these equations are shown to be separable. Working with the displacement vector, we find that the radius dependent part of the solution satisfies coupled second-order ordinary differential equations. This system of equations is solved using the method of Frobenius, and results in four independent series determined by material properties to within a multiplicative constant. Use of boundary conditions expressed in terms of stresses and displacements at the inner and outer shell radii completes the solution. Numerical results for a range of shell elastic constants show that this solution matches known analytic results in the special case of isotropy and matches previously developed finite difference results for anisotropic elastic constants. The effect of shell anisotropy on far-field scattering amplitude is explored for an incident plane longitudinal wave.