On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media

Abstract

Higher order (so-called strain gradient) homogenised equations are rigorously derived for an infinitely extended periodic elastic medium with the periodicity cell of a small size ε, in the presence of a fixed body force f , via a combination of variational and asymptotic techniques. The coefficients of these equations are explicitly related to solutions of higher order unit cell problems. The related higher order homogenised solutions are shown to be best possible in a certain variational sense, and it is shown that these solutions are close to the actual solutions up to higher orders in ε. We derive a rigorous full asymptotic expansion for the energy I ε, f and also show that its higher order terms are determined by the higher order homogenised solutions. The resulting variational construction generates higher order effective constitutive relations which are in agreement with those proposed by phenomenological strain gradient theories.