One of the main problems in homogenization of nonlinearly elastic composites is that long wavelength buckling prevents the possibility of homogenization by averaging over a single period cell. A careful convexification argument (that invokes a perturbation of the stored energy function by a Null Lagrangian) combined with regularity theory for monotone systems shows that long wavelength buckling cannot occur for small loads. Based on this observation we obtain a quantitative two-scale expansion for energy minimizers and we can discuss the linearization of periodic composites at strained equilibrium states. The talk is based on a joint work with M. Schffner.