Generalized Eshelby Problem in the Gradient Theory of Elasticity

A generalized Eshelby problem of arbitrary order in the gradient elasticity for a multilayer inclusions of spherical shape with a polynomial strain field at infinity is considered. For this problem we propose a constructive method of solution in a closed finite form, using generalized Papkovich-Neuber representation and the system of canonical potentials based on harmonic polynomials. We use also the Gauss theorem on the decomposition of an arbitrary homogeneous polynomials. The solutions of the generalized Eshelby problem are applied in the method of asymptotic homogenization of the gradient elasticity to accurately calculation of the effective characteristics of composite materials with scale effects.