  This work is a follow-up to our previous work "A numerical approach related
to defect-type theories for some weakly random problems in homogenization"
(preprint available on this archive). It extends and complements, both
theoretically and experimentally, the results presented there. Under
consideration is the homogenization of a model of a weakly random heterogeneous
material. The material consists of a reference periodic material randomly
perturbed by another periodic material, so that its homogenized behavior is
close to that of the reference material. We consider laws for the random
perturbations more general than in our previous work cited above. We prove the
validity of an asymptotic expansion in a certain class of settings. We also
extend the formal approach introduced in our former work. Our perturbative
approach shares common features with a defect-type theory of solid state
physics. The computational efficiency of the approach is demonstrated.
