A size-dependent element-free Galerkin method for analyzing the three-dimensional free vibration characteristics of functionally graded cylindrical microshells based on the consistent strain gradient theory

Within the framework of the consistent strain gradient theory (CSGT), we develop a size-dependent element-free Galerkin (EFG) method to analyze the three-dimensional (3D) free vibration characteristics of a functionally graded (FG) cylindrical microshell under simply supported boundary conditions. The formulation of the EFG method accounts for the effects of force-stress, couple-stress, and dilatational and deviatoric strain gradient tensors. Utilizing the CSGT-based EFG method, we subsequently determine the lowest natural frequency and associated vibration mode of the microshell. The accuracy and convergence rate of the CSGT-based EFG method are validated by comparing its results with those produced using size-dependent advanced and refined shear deformation theories documented in the literature. Finally, we conduct a parametric study to explore the effects of key factors on the lowest natural frequency of the microshell, including the material length-scale parameters, inhomogeneity index, length-to-mid-surface radius ratio, and mid-surface radius-to-thickness ratio, which are shown to be significant. The results also show that the material length-scale parameters consistently enhance the microshell’s overall stiffness, increasing its lowest natural frequency. The importance of different tensor effects on the lowest natural frequency of the microshell’s flexural mode is ordered as the couple-stress tensor effect > the deviatoric strain gradient tensor effect > the dilatational strain gradient tensor effect. Additionally, the applicable ranges of the structural scale for the CCST and CSGT are recommended to be approximately within the intervals, (Formula presented.) and (Formula presented.) respectively.