3D numerical solution for the bending analysis of simply supported functionally graded shallow nanoshells based on nonlocal strain gradient theory

A three-dimensional nonlocal strain gradient theory and numerical solution for the bending analysis of functionally graded shallow nanoshells are presented in this manuscript. The small-scale length effects are addressed by introducing two parameters: one for nonlocal stress and another for capturing the size effect on strains. A curvilinear orthogonal coordinate system is utilized to formulate the equilibrium and constitutive equations. The Navier solution, applicable to simply supported spherical, cylindrical panels, and rectangular plates, is employed to derive stresses and displacements in terms of the midsurface domain. The Differential Quadrature Method (DQM) is applied to approximate the derivatives in the thickness domain, using Chebyshev-Gauss-Lobatto grid distribution and Lagrange interpolation polynomials as basis functions. Proper traction conditions for out-of-plane stresses at the top and bottom surfaces are considered. Various problems of functionally graded nanoshells subjected to bisinusoidal and uniformly distributed loads are analyzed and compared with existing literature. The results are unique due to their 3D nature, providing a benchmark solution to an existing problem that has traditionally been addressed using 2D or quasi-3D approaches with just one nonlocal parameter. Consequently, these solutions can be considered as a referential for evaluating distinctive 2D theories for plates and shallow shells.