Modeling of a functionally graded micro-ring segment for the analysis of coupled extensional–flexural waves

Established in this study is a microstructure-dependent continuum model of a micro-ring with a functionally graded material composition in the thickness direction. The developed model is employed to furnish the nature of elastic waves propagation within a segment of the micro-ring. The development of the model is grounded on the modified couple stress theory, the Hamilton’s principle, and the classical rule of mixture. For validation purposes, patterns of wave propagation within homogenous rings and homogeneous micro-rings are analyzed from the reduced forms of the model. Key effects incorporated in the new model include small-scale parameter, power-law index, Winkler–Bach and rotary inertia effects. Dispersion analyses revealed a profile of frequency spectra characterized by distinct frequency zones, each of which is engendered by a bifurcation point. Explicit closed-form expressions are provided for high and low frequency limits as well as the long and short wavelength limits of the coupled waves. Phase speed dispersion curves for the micro-ring are found to have different branches that signify the distinctly flexural and extensional waves. Higher values of the power-law index are found to lead to a marked decrease in the phase speed and a noticeable increase of the group speed of the extensional wave.