Based on Reissner's mixed variational theorem (RMVT), we developed finite cylindrical layer methods (FCLMs) to investigate the quasi-three-dimensional (3D) dynamic responses of simply-supported, two-layered functionally graded piezoelectric material (FGPM) film-substrate circular hollow cylinders with open- and closed-circuit surface conditions. The FGPM film-substrate cylinder considered in this work consists of a thick and soft FGPM substrate with a surface-bonded thin and stiff homogeneous piezoelectric material (HPM) film. The material properties of the FGPM layer are assumed to obey an exponent-law exponentially varying with the thickness coordinate, and the piezoelectric ceramic material PZT-4 is taken to be the reference material. The accuracy and convergence rate of FCLMs with different orders are assessed by comparing their solutions with the exact 3D ones available in the literature. The convergent solutions of FCLMs for the lowest frequency parameters and their corresponding modal electric and elastic variables of the FGPM film-substrate cylinder are presented. The influences of various factors with regard to some of the key dynamic responses of the cylinder are examined, such as the coupled electro-mechanical characteristics, material-property gradient index, surface boundary conditions, aspect ratio, and film-substrate thickness ratio.
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