  An energy for first-order structured deformations in the context of periodic
homogenization is obtained. This energy, defined in principle by relaxation of
an initial energy of integral type featuring contributions of bulk and
interfacial terms, is proved to possess an integral representation in terms of
relaxed bulk and interfacial energy densities. These energy densities, in turn,
are obtained via asymptotic cell formulae defined by suitably averaging, over
larger and larger cubes, the bulk and surface contributions of the initial
energy. The integral representation theorem, the main result of this paper, is
obtained by mixing blow-up techniques, typical in the context of structured
deformations, with the averaging process proper of the theory of
homogenization.
