Semi-Analytical Finite Element Modeling of Nonlinear Nanoshells

The nonlinear response of shell structures is highly sensitive to geometry, loading conditions, and imperfections. At the micro- and nanoscale, the problem is further complicated by size-dependent effects, which can significantly influence the mechanical response. Capturing the combined effects of geometric nonlinearity and material nonlocality therefore remains a major modelling challenge, requiring specialized and sophisticated computational tools. To address this need, the present work introduces a new nonlinear finite element formulation for nanoshells analysis. The framework is implemented within the Sanders' thin shell theory where geometric nonlinearities are introduced through the von K\'arm\'an hypothesis. Size dependency is incorporated via the second-order strain-gradient constitutive model, leading to a final set of nonlinear partial differential equations of order six. The solution of these equations is achieved through a combination of finite elements and a perturbation approach. In this solution strategy, the generalized displacements of the shell are interpolated using a novel hierarchical Hermite polynomial space which satisfies the high continuity requirements of the nonlocal shell model. Subsequently, the finite element coefficients are expanded into an asymptotic series, leading to a semi-analytical solution of the nonlinear governing equations. This combination enables efficient and accurate prediction of nonlinear responses of nanoshells. Numerical results demonstrate the capability of the proposed framework to accurately capture the coupled effects of nonlinearity and nonlocality. Our results show significant deviations from classical models and highlight the importance of incorporating size-dependent effects in nonlinear nanoscale shell analysis.