Non-linear analysis of functionally graded microbeams using Eringen׳s non-local differential model

The primary objective of this paper is two-fold: (1) to formulate the governing equations of the Euler-Bernoulli and Timoshenko beams that account for (a) two-constituent material variation through beam thickness, (b) small strains but moderate displacements and rotations, and (c) material length scales based on Eringen׳s non-local differential model; and (2) develop the non-linear finite element models of beam theories with aforementioned features and obtain numerical results for static bending. The principle of virtual displacements is used to derive the non-linear equations governing functionally graded beams with Eringen׳s non-local constitutive models for both the Euler-Bernoulli and Timoshenko beam theories. A power-law model is used for the variation of the material properties of the two constituent materials. Finite element models of the resulting equations are developed and numerical results are presented for pinned-pinned and clamped-clamped boundary conditions, showing the effect of the non-local parameter and the power-law index on deflections and stresses.