Analogies for simply supported nonlocal Kirchhoff plates of polygonal planform

Abstract

In the present paper, we consider nonlocal linear elastic Kirchhoff plates, where we restrict to thin, isotropic, and homogeneous plates under the action of static transverse forces, utilizing the differential equation form of the Eringen nonlocal continuum theory. For the case of simply supported plates of polygonal planform, we derive analogies between the solutions of the nonlocal Kirchhoff theory and its local counterpart. First, we extend the Marcus decomposition method for local Kirchhoff plates, where we show that, analogous to the local case considered by Marcus, the moment sum and the nonlocal deflection are both governed by Poisson boundary value problems, which correspond to auxiliary (local) membrane problems. In the present context, it eventually follows that there is a nonvanishing term responsible for a correction of the deflection due to the nonlocal effect, while the moments and shear forces of the local and the nonlocal plate coincide. The nonlocal correction of the deflection turns out to be governed by a membrane-type boundary value problem again. From this fact, it follows immediately that the correction for the deflection due to the nonlocal effect can be derived alternatively from both the local deflection and the local moment sum, which represents a substantial simplification of the nonlocal computations. For the sake of demonstration, we consider examples with closed form solutions. We first consider (as limiting case) infinite plates under the action of single forces, where the singular behavior of Green's function of the nonlocal Kirchhoff theory can be clarified. Then we discuss the bending of nonlocal plate strips for comparison sake, as well as equilateral triangular plates. To our best knowledge, no results for bending of nonlocal triangular plates have been presented in the literature so far. The present paper has been strongly influenced by a former contribution on analogies between simply supported polygonal Kirchhoff plates rigid in shear and shear-deformable plate solutions. This paper was co-authored together with our teacher, the late Professor Franz Ziegler, to whom we dedicate the present paper. In light of this latter contribution, and extending an idea promoted by Professor Franz Ziegler and his co-workers in the framework of various other fields, we finally show in the Appendix that the deflections of simply supported nonlocal Kirchhoff plates can be considered as deflections of a local "background" Kirchhoff plate under the action of additional fictitious eigenstrain loadings, such as thermal moments.

Exact and approximate analytical solutions for nonlocal nanoplates of arbitrary shapes in bending using the line element-less method

“…It is worth mentioning that, the above reported BCs considerably simplify for straight edges [22], as for instance in the case of polygonal plates. In this regard, consider a simply supported rectangular plate, of length a and width b.…”

“…Further, by skipping the common operator 1 À kr 2 ð Þ as in [22,41], Eq. (10) leads to the following more compact form of the equilibrium equation in terms of Marcus' moment…”

“…Recently, a solution to the issue pertaining to how appropriately handle simply-supported BCs in polygonal plates has been developed in [22], using an approach based on the Marcus' moment sum. In this regard, it has been shown that the BCs can equivalently be written as…”

“…However, it should be emphasized that the transformation of the original integral formulation to the differential one represents an approximation, whose validity can be proved for points far from the boundaries [11,22]. Thus, in the last decade several issues and paradoxical results have been also discussed in the literature related to the proper definition of the boundary conditions in nonlocal structures.…”

“…Few other studies have been devoted to the analysis of nonlocal plates of different shapes, with most contributions mainly related to circular shapes [36,37], where also some exact analytical solutions have been determined [38,39], while the bending analysis of nonlocal plates with general shapes has been recently discussed in [40] using a boundary element method. Finally, the case of nonlocal polygonal Kirchhoff plates has been analyzed in [22], where a solution of the bending problem has been found considering only simply-supported boundary conditions, and resorting to analogies with the solution of the classical (local) Kirchhoff plate. These results have been then exploited in [41] to compare the response of different plates models at the nano scale.…”