Finite element solution to size-dependent nonlinear axisymmetric bending of FG-GPLRC circular/annular microplates using surface fundamental form: a comparison of MSGT and MCST

Abstract

In this paper, a finite element (FE) solution based on the modified strain gradient theory (MSGT) for size-dependent nonlinear bending analysis of ideal functionally graded graphene platelet-reinforced nanocomposite (FG-GPLRC) circular/annular microplates with smooth and continuous distributions of GPLs is compared with the modified couple stress theory (MCST). The effective Young’s modulus and Poisson’s ratio of the FG-GPLRC microplates are computed by the Halpin–Tsai (H–T) model and rule of mixtures (ROM), respectively. The strain–displacement relation and stress resultants are determined according to the surface fundamental form. The energy functional of the FG-GPLRC microplate is formulated based on the principle of virtual work and then solved numerically using the nonlinear FE method (FEM). Numerical results are examined to evaluate the influences of length scale parameters on the nonlinear bending response of FG-GPLRC microplates with different GPL patterns, weight fractions, and geometries. For l/h = 1.0, the nonlinear displacement results calculated using MSGT are consistently lower than those calculated using MCST, by approximately 2.7–3.0 times for clamped circular microplates and 3.8–4.1 times for clamped-pinned annular microplates. The results from these two theories diverge significantly under conditions with a small thickness-to-length scale ratio.

Keywords: Nonlinear bending analysis, Halpin–Tsai model, MSGT, MCST, FG-GPLRC microplates