Tunable elastic wave propagation in planar functionally graded metamaterials

Structures made of functionally graded materials (FGM) are successful attempts to enhance the mechanical properties of homogeneous materials. On the other hand, periodically architected structures provide phenomenal opportunities to design structures much lighter than their bulk counterparts showing exceptional mechanical characteristics. In the present study, to utilize the advantages of both technologies, wave propagation properties of functionally graded metamaterials (FGMM), i.e., periodically architected structures made of FGM, are investigated for three different planar topologies, and their vibration filtering performances are analyzed. The mathematical formulations to obtain the equation of motion for the FGMM are developed using the finite element method, and Floquet-Bloch's theorem is employed to find their dispersion curves. Periodically architected structures with hexagonal, rectangular, and triangular unit cells are considered, and the effects of the FGM on their stop-band percentages are investigated. A comparison between band structures of pure steel (St), pure alumina (Al2O3) and St-Al2O3 reveals that using FGM in the periodically architected structures can greatly enhance wave propagation properties by opening new stop-band regions leading to structures with much more versatility and tunability. The material distribution is assumed to vary according to both power-law and exponential-law rules along the beam axis and thickness, and the effects of Young's modulus ratio, density ratio, relative density, and non-negative power-law exponent are scrutinized on the bandgap properties. The results indicate that periodically architected structures made of FGM exhibit much higher percentages of stop-bands, and playing with corresponding FGM parameters can tune this value for desired engineering needs. In addition, a mathematical approach is presented to investigate the polarization of the studied FGMM in the longitudinal, transverse, and rotational directions, and to measure the effects of material distribution on the polarization of the first three branches of the dispersion curves. It is revealed that the polarization factors of the three first dispersion branches are mainly geometry-dependent and change slightly with the material distribution.