Three-dimensional interfacial Green’s functions in anisotropic bimaterials

Abstract

In this paper, we derive, for the first time, the complete set of three-dimensional interfacial elastostatic Green’s functions in anisotropic bimaterials, including displacements, stresses, and their derivatives with respect to the source coordinates. We make use of the extended Stroh formalism and the Mindlin’s superposition method, and express these Green’s functions in terms of one-dimensional finite-part integrals with variable θ over [0, π]. Denoting by r the distance between the field and source points on the interfacial planes, we show that the interfacial displacements, stresses and derivatives of displacements, and derivatives of stresses are proportional, respectively, to 1/ r, 1/ r 2, and 1/ r 3, while their finite-part integrals are, respectively, in the orders of 1/cos θ, 1/cos 2 θ, and 1/cos 3 θ. Because of the special dependence upon the distance r, the interfacial Green’s functions on the whole interfacial plane are completely determined by their values on the unit circle on the interfacial plane. An efficient and accurate method is also proposed for the evaluation of the involved finite-part integrals, and some typical numerical examples are given to show the general features of the interfacial Green’s functions. In particular, it is remarked that some of them are discontinuous across the interface. These interfacial Green’s functions are essential to various integral-equation methods in solving inclusion and interfacial crack problems in anisotropic bimaterials. Furthermore, they are also required in the study of strained quantum dot semiconductor devices should the Green’s function method be applied.