Modelling and Simulation in Engineering

Volume 2016 (2016), Article ID 6369029, 5 pages

http://dx.doi.org/10.1155/2016/6369029

## On Bending of Bernoulli-Euler Nanobeams for Nonlocal Composite Materials

Department of Civil Engineering, University of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, Italy

Received 4 December 2015; Accepted 24 April 2016

Academic Editor: Theodoros C. Rousakis

Copyright © 2016 Luciano Feo and Rosa Penna. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Evaluation of size effects in functionally graded elastic nanobeams is carried out by making recourse to the nonlocal continuum mechanics. The Bernoulli-Euler kinematic assumption and the Eringen nonlocal constitutive law are assumed in the formulation of the elastic equilibrium problem. An innovative methodology, characterized by a lowering in the order of governing differential equation, is adopted in the present manuscript in order to solve the boundary value problem of a nanobeam under flexure. Unlike standard treatments, a second-order differential equation of nonlocal equilibrium elastic is integrated in terms of transverse displacements and equilibrated bending moments. Benchmark examples are developed, thus providing the nonlocality effect in nanocantilever and clampled-simply supported nanobeams for selected values of the Eringen scale parameter.

#### 1. Introduction

Analysis of nanodevices is a subject of special interest in the current literature. Particular attention is given to the static behavior of beam-like components of nanoelectromechanical systems (NEMS). Nonlocal constitutive behaviors are adequate in order to evaluate the size phenomenon in nanostructures; see, for example, [1–9]. Investigations on random elastic structures have been carried out in [10–13]. Many research efforts have been devoted to theoretical and computational advances about specific structural models [14–19]. Recent variational formulations of nonlocal continua have been developed in [20–22]. Noteworthy theoretical results on functionally graded nanobeams have been contributed in [23–25]. Nevertheless, exact solutions are not always available so that finite element strategies are needful; see, for example, [26]. Micromechanical approaches are broadly used in order to analyze the effective behavior of composite structures [27].

Innovative applications of engineering interest are proposed in [28–30]. Numerical and experimental methodologies for composite structures are developed in [31, 32]. Effective applications of tensionless models concerning crack propagation are reported in [33–37]. A skillful analysis of equilibrium configurations of hyperelastic cylindrical bodies and compressible cubes is carried out in [38].

The present paper deals with one-dimensional nanostructure by making recourse to the tools of nonlocal continuum mechanics. Small-scale effects exhibited by functionally graded nanobeams under flexure are analyzed in Section 2.

#### 2. Nonlocal Elasticity

In local linear elasticity for isotropic materials, stress and strain at a point of a Cauchy continuum are functionally related by the following classical law:with and L constants.

Such a constitutive behavior is not adequate to evaluate size effects in nanostructures. An effective law able to capture scale phenomena was developed by Eringen in [39] who defined the following nonlocal integral operator:where (1) is the nonlocal stress,(2) is the macroscopic stress given by (5),(3) is the influence function,(4) a dimensionless nonlocal parameter defined in terms of the material constant and of the internal and external characteristic lengths and , respectively.In agreement with the Eringen proposal in choosing the following influence function , the nonlocal elastic law (2) rewrites aswhere denotes the Laplace operator. The differential form adopted for bending of nanobeams, analogous to (3), is provided bywhere is the nonlocal normal stress and is the macroscopic normal stress on cross sections. Note that the stress is expressed in terms of elastic axial strains bywith Young modulus.

#### 3. Bending of Nonlocal Nanobeams

Let us consider a bent nanobeam of length , with Young modulus functionally graded in the cross section and uniform along the beam axis . The cross-sectional elastic centre and the principal axes of elastic inertia, associated with the scalar field , are, respectively, denoted by and by the pair .

The nanobeam is assumed to be subjected in the plane to the following loading conditions: **, distributed load per unit length in the interval , **, concentrated forces at the end cross sections , **, concentrated couples at the end cross sections .The bending stiffness is defined byDifferential and boundary conditions of equilibrium are expressed bywhere is the bending moment.

The bending curvature, corresponding to the transverse displacement , is given byThe differential equation of nonlocal elastic equilibrium of a nanobeam under flexure is formulated as follows. Let us preliminarily multiply (4) by the coordinate along the bending axis and integrate on the cross section :with the axial dilation provided by the known formula .

Enforcing (8) and (7) and imposing the static equivalence conditionwe obtain the relationThis equation can be interpreted as decomposition formula of the bending curvature into elastic and inelastic partswithAccordingly, the scale effect exhibited by bending moments and displacements of a FG nonlocal nanobeam can be evaluated by solving a corresponding linearly elastic beam subjected to the bending curvature distortion (13)_{2}.

#### 4. Examples

The solution methodology of the nonlocal elastic equilibrium problem of a nanobeam enlightened in the previous section is here adopted in order to assess small-scale effects in nanocantilever and clamped-simply supported nanobeams under a uniformly distributed load . The nonlocality effect on the transverse displacement is thus due to the uniform bending curvature distortion formulated in (13)_{2}. Graphical evidences of the elastic displacements are provided in Figures 1 and 2, in terms of the following dimensionless parameters and , for selected values of the nonlocal parameter . Details of the calculations and some comments are reported below.