Nonlinear analysis of functionally graded nanoscale beams incorporating the surface energy and microstructure effects

Abstract

A nonclassical size–dependent beam model is developed to study the geometrically nonlinear response of functionally graded (FG) nanoscale beams including the combined effect of surface elasticity, surface residual stress, microstructure and size-dependency. To this, the Gurtin–Murdoch surface elasticity theory in combination with the modified couple stress theory is employed to capture the simultaneous effects of surface energy and microstructure size phenomenon. Euler–Bernoulli and Timoshenko beam models considering the von Kármán geometric nonlinearity are adopted. The material properties of the bulk and surface of the FG beam are assumed to be changed continuously through the thickness according to a power law scheme and Mori–Tanaka homogenization technique. The geometrically nonlinear size–dependent governing equations and corresponding nonclassical boundary conditions are exactly obtained through Hamilton's principle. The generalized differential quadrature method (GDQM) is employed to discretize the nonlinear nonclassical governing differential equations and the nonclassical boundary conditions for various end supports. A good agreement between the results of the present model and those available in the literature is obtained. Selected numerical results show explicitly that the material length scale parameter, surface elastic modulus, surface residual stress, gradient index and material distribution have a significant influence on the geometrically nonlinear bending response of FG nanobeams.