Bending analysis of functionally graded nanobeams based on the fractional nonlocal continuum theory by the variational Legendre spectral collocation method

In this article, the size-dependent bending behavior of nanobeams made of functionally graded materials is studied through a numerical variational approach. The nonlocal effects are captured in the context of fractional calculus. The nanobeams are modelled based on the Euler–Bernoulli beam theory whose governing fractional equation is derived utilizing the minimum total potential energy principle. In the solution procedure, which is directly applied to the variational form of governing equation, the truncated Legendre series in conjunction with the Legendre operational matrix of fractional derivatives are employed for numerical integration of fractional differential equation. The strong form of the governing equation is also derived and solved to examine the accuracy and efficiency of the proposed solution approach as well as for the comparison purpose. The influences of fractional order, nonlocality and material gradient index on the bending characteristics of nanobeams subject to different end conditions are investigated.