Flexibility-based stress-driven nonlocal frame element: formulation and applications

Abstract

In the present work, a novel flexibility-based nonlocal frame element for size-dependent analyses of nano-sized frame-like structures is proposed. The material small-scale effect is consistently represented by the stress-driven nonlocal integral model and the element equation is constructed within the framework of flexibility-based finite element formulation. The merits of both stress-driven nonlocal integral model and flexibility-based finite element formulation render the proposed nonlocal frame element “consistent” and “exact”. Therefore, the “one-element-per-member” modeling approach is applicable. The modified Tonti’s diagram is utilized to show the overview of the element formulation and to summarize the formulation step. The element state determination process as well as the displacement recovery procedure are also discussed. Three numerical examples are employed to show accuracy, characteristics, and applications of the proposed nonlocal frame element. The first example shows the model capability to eliminate the “constant-force” paradoxical responses inherent to the Eringen’s nonlocal frame model and the simplified strain-gradient frame model; the second presents and characterizes the essence of the material small-scale effect on global and local responses of a propped-cantilever nanobeam; the third investigates the material small-scale effect on the tensile response of an auxetic metamaterial. All analysis results demonstrate that the material nonlocality associated with the stress-driven nonlocal integral model consistently yields a stiffer system response.