Periodic structure composites have been widely used in engineering and intelligent manufacturing. The soft components in the composites, however, are subjected to large mechanical deformation upon the external stimuli. Hence, it is necessary to investigate the effect of deformation on the band structure. Based on the theoretical analysis and finite element model, we disclose the dynamic behaviors of the elastic wave propagating in such composites with various deformation levels and deformed configurations. It is demonstrated that the deformation level mainly affects the boundary of the bandgap, but the bandwidth remains constant. The elastic wave band structure is significantly changed by the tension and compression deformation while barely affected by the shear deformation. The conclusions can provide inspiration for the intelligent mechanical structure design controlling and tuning wave propagation.