Semi-Analytical Differential Reproducing Kernel Element Method for the Size-Dependent Free Vibration Characteristics Analysis of Functionally Graded Doubly Curved Microscale Shells

Within the consistent couple stress theory (CCST) framework, the authors develop a semi-analytical differential reproducing kernel element method (DRKEM) for analyzing the three-dimensional (3D) free vibration characteristics of a functionally graded (FG) doubly curved (DC) microshell under fully simple supports. Employing Hamilton’s principle, the authors derive a Galerkin weak formulation to address this issue. The microshell’s material properties are considered nonhomogeneous and vary in the microshell’s thickness direction following an exponential function. Using the differential reproducing kernel (RK) interpolants as the unknown variables’ shape functions and then incorporating them into the derived Galerkin weak formulation, the authors develop a DRKEM for investigating the microshell’s free vibration characteristics. In numerical examples, the results revealed that the natural frequency solutions obtained using the DRKEM are in excellent agreement with the available 3D solutions of laminated composite DC macroshells and FG microplates by setting the material length-scale parameter and the inverse of the curvature radii at zero, respectively. The DRKEM’s convergence rate is also shown to be rapid.