3D elastodynamic fields of non-uniformly coated obstacles: Notion of eigenstress and eigenbody-force fields

Based on wave-function expansion, the time harmonic wave scattered by a circular and spherical inhomogeneity has been studied by numerous investigators. This method has also been employed to axisymmetrically coated circular and spherical inhomogeneities by some authors. When the geometry of the obstacle is not axisymmetric, the wave-function expansion is no longer applicable. In this paper, it is proposed to employ the dynamic equivalent inclusion method (DEIM) which is more general than the methods presented in the literature. It will be seen that DEIM may be used to treat a wide range of situations in a unified manner and is not bound to certain symmetries. The DEIM was first proposed by Fu and Mura Fu, L.S., Mura, T., 1983. The determination of elastodynamic fields of an ellipsoidal inhomogeneity. ASME J. Appl. Mech. 50, 390–396], and no further developments have been done on it ever since. Its original formulation has some shortcomings with regard to the concept of homogenizing eigenstrains, and for usage of polynomial eigenstrains. Moreover, it is limited to single ellipsoidal inhomogeneity without coating. The new viewpoints of homogenizing eigenstress and eigenbody-force fields which are compatible with the physics of the problem are given. Expressing the eigenstress, eigenbody-force fields and the Green’s function associated with the governing Helmholtz equation in terms of the spherical wave-functions is the natural choice and is very effective. Another important task is the development of the three dimensional DEIM for inhomogeneities having homogeneous or functionally graded (FG) coating with variable thickness, which eliminates any possible symmetries.