Green's function for a two–dimensional exponentially graded elastic medium

Abstract

The free–space Green function for a two–dimensional exponentially graded elastic medium is derived. The shear modulus Âμ is assumed to be an exponential function of the Cartesian coordinates (x,y), i.e. μ ≡ μ(x,y) = μ0e2(β1x+β2y), where μ0, β1, and β2 are material constants, and the Poisson ratio is assumed constant. The Green function is shown to consist of a singular part, involving modified Bessel functions, and a non–singular term. The non–singular component is expressed in terms of one–dimensional Fourier–type integrals that can be computed by the fast Fourier transform.