Two-scale asymptotics for non-local effects in composites with highly anisotropic fibres

Abstract

The problem of periodic homogenisation for a conducting two-phase composite exhibiting non-locality in the overall behaviour is considered. The matrix is isotropic and its conductivity is constant, while the fibre-shaped inclusions are assumed to be anisotropic, with high contrast between conductivities along the fibres and in the transverse directions. The contrast is set to correspond to the so-called double-porosity scaling, which is of special interest. A full two-scale asymptotic expansion for the solution is constructed and rigorously justified via appropriate error estimates. An example of the non-local homogenised operator in the case of fibres with circular cross-sections is given.