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

The free-space Green function for a two-dimensional exponentially graded elastic medium is derived. The shear modulus It is assumed to be an exponential function of the Cartesian coordinates (x, y), i.e. mu equivalent to mu(x, y) = mu(0)e(2(beta1x+beta2y)), where mu(0), beta(1) and beta(2) 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.