Static and Dynamic Behaviors of Microstructured Membranes within Nonlocal Mechanics

Abstract

In this paper, the static and dynamic behaviors of a finite microstructured rectangular membrane were studied. The microstructured membrane model comprised an equal number of elastic springs in both directions forming a rectangular lattice membrane. The authors considered the out-of-plane displacement of each node of this two-dimensional lattice. The rectangular lattice membrane was fixed at its boundary. A nonlocal continuous membrane model was developed to approximate the behavior of the finite lattice model. A continualization procedure was applied to the discrete equations in which the difference operators were approximated by differential operators. The resulting nonlocal continuum was governed by some length scales in each direction that depended on the size of the microstructure. This nonlocal continuum exactly coincided with the one elaborated by Rosenau in the 198s in the dynamics range. The authors showed that such a nonlocal medium also can be used for static deflection of microstructured membranes. A comparison of both discrete (the reference lattice model) and continualized nonlocal responses brought out the effectiveness of this micromechanics-based approach. The continualized nonlocal continuum also was compared with a phenomenological Eringen’s nonlocal model.