A size-dependent 3D solution of functionally graded shallow nanoshells

Joao Carlos Monge, Jose Luis Mantari, Melchor Nicolas Llosa, Miguel Angel Hinostroza

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


An unavailable semi-analytical non-local 3D solution for functionally graded nanoshells with constant radii of curvature is presented. The small length scale effect is included in Eringen's nonlocal elasticity theory. The constitutive and equilibrium equations are written in terms of curvilinear orthogonal coordinates systems which are only valid for spherical and cylindrical shells, and rectangular plates. The stresses and displacements are assumed in terms of the Navier method which is applicable for simply supported structures. The derivatives in terms of thickness are approximated by the differential quadrature method (DQM). The thickness domain is discretized by the Chebyshev-Gauss- Lobatto grid distribution. Lagrange interpolation polynomials are considered as the basis function for DQM. The correct free surface boundary condition for out-of-plane stresses is considered. Several problems of isotropic and functionally graded shells subjected to different types of loads are analyzed. The results are compared with other three-dimensional solutions and higher-order theories. It is important to emphasize that the radii of curvature are crucial at nanoscale, so it should be considered in the design of nanodevices.

Original languageEnglish
Article number20220215
JournalCurved and Layered Structures
Issue number1
StatePublished - 1 Jan 2023


  • Eringen's nonlocal elasticity theory
  • equilibrium equations
  • functionally graded material
  • nanoshell


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