Vibration of a Functionally Graded Doubly Curved Shallow Nanoshell: An Improved FSDT Model and its Nonlocal Finite Element Implement

PurposeThe present work aims to thoroughly analyze the free vibration behavior of size-dependent functionally graded (FG) doubly-curved shallow nanoshells with spatial variations in microstructure. The most advantage of the current work is that the material properties including nonlocal parameter are continuously graded through the thickness direction. Furthermore, the investigation focuses on understanding the impact of various parameters, such as the thickness-to-side ratio, radius ratio, nonlocal parameters, different material compositions, and power-law index, on the free vibration behavior of FG doubly-curved nanoshells.MethodsTo achieve this goal, an eight-node quadrilateral finite element model with five degrees of freedom is then developed based on the enhanced FSDT to examine, for the first time, the free vibration response of nanoplates and nanoshells considering constant and variable non-local parameters. Four types of hyperbolic nanoshells are considered: flat plates, spherical shells, hyperbolic parabolic shells, and cylindrical shells. The effective material properties of FG curved nanoshells are continuously graded in the thickness direction according to a power-law function. The shear correction factor is a function rather than a constant factor as in the conventional FSDT, with the goal of increasing the accuracy of nanoshell modelling. The suggested shear correction function eliminates the need for the traditional correction factor commonly used in FSDT in addition to satisfying the free conditions on both the upper and lower surfaces. Moreover, to develop physically consistent reduced-order continuum models of natural discrete nanostructures with accuracy comparable to molecular dynamics models, we employ a modified nonlocal theory to capture microstructure-related effects. The governing equations are derived via Hamilton's principle and subsequently solved via the finite element method.ResultsFirstly, the accuracy, fast rate of convergence, and robustness of the developed FE model are validated through comparisons with established benchmarks from the literature, confirming the soundness of the proposed formulation. An extensive parametric analysis is conducted to examine the effects of thickness-to-side ratio, radius ratio, nonlocal parameters, different material compositions, and power-law index on the free vibration of FG doubly-curved nanoshells.ConclusionsThis investigation revealed that size-dependent effects significantly influence the vibrational characteristics of FG doubly curved nanoshells. The Nonlocal parameters and their spatial variations induce a stiffness-softening effect on the natural frequencies of FG doubly curved nanoshells. In addition, the findings demonstrate that the parameters such as the side-to-thickness ratio and radius ratio were found to exert significant influences on nanoshell vibration behavior, emphasizing the importance of considering geometric factors alongside material properties in optimizing structural performance. The findings from this study can be utilized to provide a powerful method to develop efficient continuum models with nanostructural details to capture the potential nonlocal interactions between atoms.