A reinvestigation of the spring-mass model for metamaterial bandgap prediction

Metaconcrete and meta-truss bar are a new type of material and structure with extraordinary characteristics that cannot be found in nature. Metamaterials/metastructures possess the ability to manipulate wave propagation in certain frequency ranges, termed as bandgaps. Application of metamaterials/metastructures for structural protection is different from the traditional strategies which resist the external loads by using their strength or energy absorption through plastic deformation, metamaterials and/or metastructures stop incident stress waves from propagating through them if their frequency contents fall into the bandgaps, thus safeguarding the protected structures. Spring-mass models are commonly utilized to predict the wave propagation characteristics of local resonant metamaterials and metastructures. It is well understood that the formation of bandgaps is because of the generation of negative effective mass and negative effective stiffness owing to the out-of-phase local vibrations. However, in current literature, some studies derived the bandgaps associated with only the negative effective mass while others derived those from both the negative effective mass and negative effective stiffness. There has not been a systematic study and explanations on these differences, and there is also lack of understanding of the mechanics of bandgap formation, in particular the low-frequency bandgap. This paper presents a theoretical study to reinvestigate the formations of bandgaps in metaconcrete and meta-truss structure associated with the effective negative mass and stiffness, provides explanations of the discrepancies in the literature, and identifies the fundamental mechanism for the bandgap formation in metaconcrete and meta-truss structure. A comprehensive analysis is also provided for predicting bandgaps of metamaterials and metastructures, followed by a design procedure for engineering applications.