Free vibration analysis of four-unknown shear deformable functionally graded cylindrical microshells based on the strain gradient elasticity theory

In this paper, we present a four-unknown shear deformation theory and then develop a shear deformable functionally graded cylindrical microshell model using the strain gradient elasticity theory. Unlike the existing shear deformable cylindrical shell models, the present one contains four independent displacement functions only and introduces three material length scale parameters. In addition, the trapezoidal shape factor (1 + z/R) of a shell element is taken into account in the expressions of classical and non-classical stresses to obtain the accurate stress-resultants over the thickness. The material properties of the microshell are estimated through the Mori–Tanaka homogenization technique. By using Hamilton’s principle, the equations of motion and boundary conditions are obtained. Closed-form solutions are derived for the free vibration problem of cylindrical microshells with simply supported ends and fixed ends respectively. Comparison studies are performed to establish the validity of the derived formulation. Finally, some illustrative examples are presented to investigate the influences of the material length scale parameter, gradient index, thickness-to-radius ratio, thickness-to-length ratio and boundary conditions on the vibration characteristics of cylindrical microshells. Numerical results indicate that both the frequency and higher-order mode shapes exhibit significant size-dependence when the thickness of the microshell approaches to the material length scale parameter.