Based on the local resonance effect of elastic waves, three single-phase acoustic metamaterials are proposed in this paper. Based on the finite element method and Bloch's theorem, the energy band structure diagrams and vibration modes are plotted, and the band gap properties and band gap opening mechanism of these structures are explored. New structures possessing lower frequency band gaps are obtained by topological optimization. The transmission curves verify the accuracy of the band gap and the vibration attenuation ability of the structure. Finally, the structural parameters were adjusted and the effect of each parameter change on the band gap characteristics was analyzed. The results show that the proposed structure has a maximum band gap coverage of 72.4 % and a strongest attenuation peak of less than −400 (dB) due to the occurrence of a local resonance. This paper provides a methodology for analyzing the vibration and noise reduction performance of single-phase material phononic crystals, as well as a three-dimensional phononic crystal with potential for practical applications. •Based on the local resonance effect, we obtained three models by adjusting the model structure.•Four forms of vibration that open the bandgap have been discovered.•Verification of bandgap accuracy based on transmission curves and stress clouds.•The analytical and validation methods in this paper provide ideas for studying the damping characteristics of 3D models.