Phononic bandgap engineering in cellular periodic structures and mechanical metamaterials

Abstract

A methodology has been developed for the computation of phononic band structure of cellular periodic structures, structures with spatially repetitive patterns, using Frequency domain Spectral Finite Element Method along with Bloch Theorem. Band structures depict the nature of wave propagation in periodic structures and are characterized by the presence of banded frequency zones where wave may or may not propagate. These zones are known as 'Pass Bands' and 'Stop Bands' respectively. Bloch theorem is conventionally used for such band structure computation in order to reduce computation of an infinite structure to that of a single unit cell- a unit of the structure containing the repeating pattern, and to enforce interrelationship between equivalent points in the unit cell. Frequency domain Spectral Finite Element Method uses exact generic solutions as shape functions, and can be very efficient for high frequency wave propagation problems. Wittrick-Williams method has been used to solve the resulting eigenvalue problem. The impact of periodic defects in the wave propagation has been studied through the use of supercell, a unit cell comprising multiples of the primitive unit cells, the smallest unit cells. Toward that end, the band structure of a supercell has been compared with that of the primitive cell and the differences are analyzed. A methodology has also been developed for the band structure computation of 3-D cellular structures with tapered elements using a Frequency domain Ritz Method. Wave propagation in various types of pentamodes has been taken as examples. A pentamode is a type of mechanical metamaterials possessing semi-fluidic properties. The auxetic cellular materials, structures having negative Poisson's ratio, is an important class of cellular structures. A novel design has been proposed that seamlessly combines conventional and auxetic honeycomb cores to develop cellular structures with a significantly larger bandgap than the pristine counterparts. Subsequently, a methodology has been developed to efficiently account for the effect of joints in the cellular structures using Spectral Superelement Method, the Finite Element Method that combines Frequency domain Spectral Finite Element with conventional Finite Element Method. The impact of various types of inclusions and features have also been investigated using this framework. Lastly, the methodologies developed so far have been used to present an atomistic-continuum modeling of graphene, the 2-D allotrope of carbon, by comparing the phononic band structure of the proposed model against the standard Density Functional Theory based band structure, as found in literature. The atomistic-continuum model parameters, thus arrived at, may be used for an approximate computation of very large systems where sophisticated methods could be prohibitive.