Superposing scheme for the three-dimensional Green’s functions of an anisotropic half-space

• This paper presents the 3D Green’s functions of a generally anisotropic half space.
• This method provides an extension of infinite Green’s function to half space case.
• The derivation is based on superposing an extra term to infinite Green’s function.
• The infinite Green’s functions are also used to obtain the complementary terms.
• Application is made to obtain explicit result for transversely isotropic materials.

This paper presents a method of superposition for the half-space Green’s functions of a generally anisotropic material subjected to an interior point loading. The mathematical concept is based on the addition of a complementary term to the Green’s function in an anisotropic infinite domain. With the two-dimensional Fourier transformation, the complementary term is derived by solving the generalized Stroh eigenrelation and satisfying the boundary conditions on the free surface with the use of Green’s functions in the full-space case. The inverse Fourier transform leads to the contour integrals, which can be evaluated with the application of Cauchy residue theorem. Application of the present results is made to obtain analytical expression for the orthotropic materials which were not reported previously. The closed-form solutions for the transversely isotropic and isotropic materials derived directly from the solutions as being a special case are also given in this paper.