Modelling and Simulation in Engineering

Volume 2016 (2016), Article ID 8574129, 9 pages

http://dx.doi.org/10.1155/2016/8574129

## Euler-Bernoulli Nanobeam Welded to a Compressible Semi-Infinite Substrate

^{1}Department of Sciences and Methods for Engineering (DISMI), University of Modena and Reggio Emilia, Via Amendola 2, 42122 Reggio Emilia, Italy^{2}Department of Engineering Enzo Ferrari (DIEF), University of Modena and Reggio Emilia, Via P. Vivarelli 10, Int 27, 41125 Modena, Italy

Received 23 December 2015; Accepted 5 October 2016

Academic Editor: Julius Kaplunov

Copyright © 2016 Pietro Di Maida and Federico Oyedeji Falope. 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

The contact problem of an Euler-Bernoulli nanobeam of finite length bonded to a homogeneous elastic half plane is studied in the present work. Both the beam and the half plane are assumed to display a linear elastic behaviour under infinitesimal strains. The analysis is performed under plane strain condition. Owing to the bending stiffness of the beam, shear and peeling stresses arise at the interface between the beam and the substrate within the contact region. The investigation allows evaluating the role played by the Poisson ratio of the half plane (and, in turn, its compressibility) on the beam-substrate mechanical interaction. Different symmetric and skew-symmetric loading conditions for the beam are considered, with particular emphasis to concentrated transversal and horizontal forces and couples acting at its edges. It is found that the Poisson ratio of the half plane affects the behaviour of the interfacial stress field, particularly at the beam edges, where the shear and peel stresses are singular.

#### 1. Introduction

The mechanical interaction between bars, strips, rings, thin films, and so forth and an elastic substrate has been widely investigated because of its great importance in many practical engineering tasks. As an example, in the framework of civil engineering, the load transfer between FRP sheets (or other reinforcing elements like discrete fibres [1, 2]) applied to existing concrete structures is often performed by modeling the reinforcing stiffeners as 2D membranes [3–6] welded to half plane (for the equilibrium configurations of cylindrical and cubic bodies [2]). By following such an approach, retrofitted concrete structures [7], elastic foundations [8–10], rigid road pavements [11], and retrofitted masonry elements [12, 13] have been studied (for the behaviour of high performance concrete based on fly ash, see [14]).

In microelectronics, various NEMS and MEMS involve thin films and coatings [5, 15–17] (examples about numerical studies performed on thin films can be found in [6, 18]), and the stress concentrations arising in the neighbourhood of geometric discontinuities and/or across bimaterial interfaces provide useful information about the risk of occurrence of fracture phenomena [19, 20] in such devices and, in turn, about their stability in time [21, 22].

In mechanical engineering, the film-substrate contact mechanics is relevant in order to study the mechanical response of thermal barriers and protective alloys typically involved in a variety of mechanical devices, like blades of centrifugal pump impellers, components of compressors, gas turbines, and so forth. In most of these examples, the problem can be studied by neglecting the bending stiffness of the covering; namely, the coating is modeled like a membrane element. However, in a wide class of mechanical systems, the flexural behaviour of the covers can not be neglected without introducing rough approximations. As an example, the flexural behaviour must be taken into account to properly study beams [23] and multilayer systems [24]. As an example, foundations of buildings are often characterized by high values of flexural rigidity and, typically, they are simulated as beams or plates resting on an elastic support [8, 9]. Similarly, the mechanical behaviour of MEMS or NEMS based on coatings characterized by small length-to-thickness ratio must be studied by taking into account the bending stiffness of the layers.

The contact problem of an Euler-Bernoulli beam perfectly bonded to an elastic half plane has been performed in [23]. These authors carried out their investigation by considering an incompressible half plane (i.e., by considering , where is the Poisson ratio of the substrate), thus finding a nonoscillatory behaviour of the interfacial stress field. This allowed the authors to straightforwardly express the unknown peel and shear stresses as infinite series of Chebyshev polynomials having square-root singularities at the edges of the beam.

Studying the singular nature of the stress and strain fields in such contact problems is an important topic [6, 25] in order to assess if such systems are resistant or prone to delamination, interfacial crack propagation, and other damage phenomena [5, 26–28].

In the present work, the effect induced by the Poisson ratio of the half plane on the beam-half plane mechanical interaction has been studied. The oscillatory index characterizing the interfacial stress field is found to depend on the Poisson ratio of the half plane only. The strain compatibility condition leads to a system of two integral equations, which is reduced to algebraic systems by expanding the interfacial stresses in series of orthogonal Jacobi polynomials having complex index. In the present investigation, both the beam and the half plane have been assumed homogeneous bodies displaying linear elastic behaviour. However, the analysis can be extended to nonhomogeneous bodies by following the approach performed in [29–31]. The time dependence can be taken into account also (for a general approach concerning the nonlocal behaviour of functionally graded materials see, e.g., [32, 33]).

The paper is organized as follows. The formulation of the problem is given in Section 2. The main results are presented and discussed in Section 3. Finally, conclusions are reported in Section 4.

#### 2. Formulation of the Problem

The problem of Euler-Bernoulli nanobeams bonded to an isotropic elastic half plane is formulated in the present section. The reference system is centred at the middle span of the beam, as shown in Figure 1. Perfect adhesion between the half plane and beam is assumed. Both the axial and flexural stiffness of the beam are taken into account. The beam and the underlying half plane have an isotropic constitutive law, characterized by the following elastic constants , , , and denoting the Young moduli and the Poisson ratio, respectively (the subscript , with stands for beam or semi-infinite substrate, resp.). Furthermore, , and denote the height and the total length of the beam, respectively, whereas and indicate the area and the moment of inertia of the beam cross section.