Elastostatics of Nonuniform Miniaturized Beams: Explicit Solutions Through a Nonlocal Transfer Matrix Formulation

Abstract

A mathematically well-posed nonlocal model is formulated based on the variational approach and the transfer matrix method to investigate the elastostatics of nonuniform miniaturized beams. These beams are composed of an arbitrary number of sub-beams with diverse material and geometrical properties, as well as small-scale size dependency. The model adopts a stress-driven nonlocal approach, a wellestablished framework in the Engineering Science community. The curvature of a sub-beam is defined through an integral convolution, considering the bending moments across all cross-sections of the subbeam and a kernel function. The governing equations are solved and the deflections are derived in terms of some constants. The formulation employs local and interfacial transfer matrices, incorporating continuity conditions at cross-sections where sub-beams are joined, to define relations between constants in the solution of a generic sub-beam and those of the first sub-beam at the left end. The boundary conditions are then imposed to derive an explicit, closed-form solution for the deflection. This solution significantly facilitates the study of nonuniform beams with many sub-beams. The study explores nonuniform beams made of two to five different sub-beams. The results are presented with an emphasis on the effects of the material properties, nonlocalities, and lengths of the sub-beams on the deflection. The results reveal the intricate behavior of nonlocal nonuniform beams, showcasing complexity in contrast to the predictable responses observed in the bending of the local nonuniform beams.