Weak-form differential quadrature finite elements for functionally graded micro-beams with strain gradient effects

This paper proposes weak-form differential quadrature finite elements for strain gradient functionally graded (FG) Euler–Bernoulli and Timoshenko micro-beams. The elements developed both have six degrees of freedom per node and do not require shape functions. The effective material properties are assumed to change continuously along the thickness direction. To guarantee the inter-element continuity conditions, we construct sixth- and fourth-order differential quadrature-based geometric mapping schemes. The two mapping schemes are combined with the minimum potential energy principle to derive their respective element formulations. Several illustrative examples are presented to demonstrate the convergence and adaptability of our elements. Finally, we utilize the latter element to explore the size-dependent vibration characteristics of multiple-stepped FG micro-beams. Numerical results reveal that our elements have distinct convergence and adaptability advantages over the related standard finite elements. The step location, thickness ratio, power-law index, and material length scale parameter have notable impacts on the structural vibration frequencies and mode shapes.