In the paper we study the homogenization method and its potential for research of some phenomenons connected with periodic elastic materials. This method will be applied on partial differential equations that describe the deformation of a periodic composite material. The next part of the paper will deal with applications of the homogenization method. The importance of the method will be discussed more detailed for the exploration of the so called bandgaps. Bandgap is a phenomenon which may appear during vibrations of some periodically heterogeneous materials. This phenomenon is not only observable during vibrations for the aforementioned materials, but we may also observe similar effects by propagation of electromagnetic waves of heterogeneous dielectric medias.