Unified theory of 2n+1 order size-dependent beams: Mathematical difficulty for functionally graded size-effect parameters solved

New insights on theoretical modeling of size-dependent functionally graded (FG) nanobeams are provided by establishing a unified theory of 2n+1 order shear deformable model with the aids of nonlocal strain gradient elasticity. The unified model covers Euler-type (n = 0), Reddy-type (n = 1), 5th (n = 2), 7th (n = 3) order beam and etc., and the limiting situation n → ∞ predicts nonlocal strain gradient Timoshenko model. The mathematical difficulty for FG nonlocal parameter is particularly emphasized, and an attempt is made for the first time to overcome the difficulty. Theoretically, the governing equations and boundary conditions of 2n+1 order nonlocal strain gradient beams, especially with FG nonlocal parameter and FG strain gradient parameter, are systematically formulated. The difficulty for FG nonlocal parameter is satisfactorily solved with by adopting the present 2n+1 order beam theory. Analytically, solutions to bending and buckling analyses within the unified model are obtained, from which the analytical solutions for Euler- and Timoshenko-type beam can be recovered. Numerically, bending deflection and buckling critical load for Euler beam, Reddy beam, 5th-11th order beam and Timoshenko beam are depicted, of which the benchmark solutions for the 5th to 11th order beam are given for the first time. Meanwhile, potential extensions of the unified model into fractional order is discussed, where benchmark solutions for n = 1.1, 0.88, 0.77, 0.4and0.2 are listed. The influences of FG nonlocal parameter, dimensionless height and Poisson's ratio (or the ratio E/G) on the bending deflection and buckling critical load are systematically studied. The present work mainly contributes to theoretical developments and greatly facilitates the mechanical analysis of beam-type structures.