Static analysis of a hemispherical nanoshell under uniform pressure based on MCST: a comparison of FEM and GDQ solutions

Abstract

This study focuses on the static analysis of hemispherical nanoshells subjected to uniform pressure based on the modified couple stress theory (MCST). The strains, change of curvatures, and rotation gradients can be expressed by the surface fundamental form in orthogonal curvilinear coordinates. Here, the numerical results are calculated using two different approaches. Firstly, the finite element method (FEM) is used to solve the variational formulation, which is derived based on the principle of virtual work. Secondly, the generalized differential quadrature method (GDQ) is recommended in this study to solve the governing differential equations with essential and nonessential boundary conditions. Therefore, the novelty of this research is a comparison between the FEM and GDQ methods under different conditions, as no published studies to date have compared these approaches for this application based on MCST. The static behavior of hemispherical nanoshells made of fullerene C₄₈₆₀, silver, and gold with the effect of small-scale parameters are highlighted in this work, and the advantages and limitations of FEM versus GDQ are demonstrated and summarized. Overall, the results based on MCST from the two different methods match closely in terms of nanoshell displacement; however, the GDQ model without the nanoscale effect is validated with the published research and is in close agreement for all membrane forces and bending moments.