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Inductive shunt circuits have thus far been explored mainly for low-frequency bandgap formation in locally reso- nant piezoelectric metamaterials. Other than the well-studied bandgap phenomenon, the substantial sensitivity of the dispersion curves to variations in the target frequency (i.e. to variations in the inductance value) right below or above the bandgap offers a very rich potential for applications ranging from on-demand spatial tailoring of the refractive index profile to dynamic stiffness modification for leveraging in novel problems of wave propa- gation with spatial and temporal property modulation. As a specific instance, if the unit cells of a metamaterial are shunted to resonate at gradually varying frequencies above the design frequency, one can achieve a smooth variation of both group velocity and phase velocity in space for wavelengths much longer than the lattice size, as a low-frequency electromechanical gradient-index metamaterial. In this work, we explore flexural waves in a one-dimensional piezoelectric metamaterial with unit cells that are shunted to inductive circuits of varying inductance values. Unit cell dispersion characteristics for an infinite metamaterial are studied to demonstrate various phenomena, such as the modification of the phase velocity and the refractive index. Specifically, case studies are given for the formation of a hyperbolic secant refractive index profile to enable plane wave focusing. The effects of dissipation and frequency variation are also studied, revealing that the proposed concepts can enable significant refractive index variation even in the presence of damping (e.g. sufficient for lens design). The advantages of this approach span from low-frequency gradient-index metamaterial design to stiffness modulation beyond the limits of short- and open-circuit values even in the absence of a negative capacitance circuit.
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