Axisymmetric free vibration and instability phenomena of closed ellipsoidal nanoshells based on modified couple stress theory

This study presents an axisymmetric free vibration analysis of closed oblate, spherical, and prolate nanoshells based on the modified couple stress theory (MCST). The key contributions are: (i) the classification of natural frequencies and mode shapes into lower and upper branch modes, and (ii) the investigation of instability phenomena associated with the natural frequencies and mode shapes of specific vibration modes. Strain and rotation gradients are formulated using differential geometry, and the variational formulation—expressed in matrix form based on the principle of virtual work and incorporating membrane, bending, and MCST-related energy terms—is solved via the finite element method (FEM). Convergence and validation of the proposed approach are also demonstrated. The effect of the aspect ratio (b/a) for the first five vibration modes is examined over the range 0.25 ≤ b/a ≤ 1.5, and size-dependent effects are analyzed for each vibration branch. The study discusses the natural frequency parameters, their corresponding mode shapes, and the modal characteristics of normalized internal forces, bending moments, and couple stress moments. It finally assesses the influence of uniform external pressure for different shell geometries. The results reveal that the length scale ratio significantly affects the frequency parameter, but has little effect on the vibration mode shape. For preload-induced instability, MCST nanoshells with higher principal curvature yield larger critical pressures.