Three-dimensional free vibration analyses of functionally graded laminated shells under thermal environment by a hierarchical quadrature element method

Abstract This paper carries out three-dimensional (3D) free vibration analyses of laminated shells made of functionally graded materials (FGMs) in both thermal and non-thermal environments by a hierarchical quadrature element method (HQEM). Two configurations of FGM laminated shells were considered, the first with FGM face sheets and homogeneous core, while the second with homogeneous face sheets and FGM core. Effective material properties of the FGMs were estimated in terms of two micromechanical models called Voigt’s rule of mixture (ROM) and Mori-Tanaka (MT) scheme. For shells in thermal environment, a nonlinear temperature distribution along thickness direction was considered while the elasticity properties were assumed to be temperature-dependent. Natural frequencies obtained by the presented formulation for FGM laminated shells under non-thermal conditions but varied geometric and boundary conditions were compared with those obtained by two-dimensional (2D) finite element method (FEM) and semi-analytical approaches in literatures first to assess the correctness and accuracy. Then the influence of volume fraction index, core thickness, and temperature gradient on natural frequencies of FGM laminated shells under thermal environment were investigated. The three-dimensional formulations of this work were shown to need similar computational cost as 2D ones but with the accuracy of 3D theory.