Size-dependent nonlinear mechanical behavior of third-order shear deformable functionally graded microbeams using the variational differential quadrature method

By developing a nonlinear microstructure-dependent third-order shear deformable beam model based on the most general form of Mindlin's strain gradient elasticity theory (SGT) and the von Karman hypothesis, the size-dependent nonlinear mechanical behavior of a microbeam made of functionally graded materials (FGMs) are described. The matrix representations of classical and non-classical kinematic and constitutive relations are obtained first. Then, the variation of energy functional is obtained in a matrix form. Afterwards, the variational differential quadrature (VDQ) rule is utilized to directly derive the discretized form of nonlinear governing equations of motion on the space domain. The periodic time differential operators and the pseudo arc-length continuation method are also used to solve the nonlinear problems of microbeams including the nonlinear free and forced vibration as well as nonlinear bending and postbuckling. The effects of length-scale parameter and boundary conditions on the microstructure-dependent nonlinear mechanical characteristics of FGM microbeams are investigated. The present model accommodates the simple forms of microstructure-dependent formulations based on the modified strain gradient theory (MSGT) and modified couple stress theory (MCST) for some specific values of the material length scale parameters. The results predicted by MSGT, MCST and classical theory are compared. (C) 2015 Elsevier Ltd. All rights reserved.