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

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.


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][4][5][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][9][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][16][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 ]  = 0.5, 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][27][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][30][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.

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, ℎ, and 2 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.
The shear and peeling stresses, () and (), respectively (Figure 2), which arise along the contact region, represent the unknown of the problem.In the following, the apex prime (  ) denotes the derivative with respect to the spatial coordinate .
By considering the symmetric as well as skew-symmetric loading cases reported in Figure 2, the balance condition of the beam along  ∈ [−, +] leads to where , , and  are the internal axial force, shear force, and bending moment of the beam, respectively, whereas  * ,  * , and  * are the corresponding external concentrated loads acting at the beam edges.By using the equilibrium conditions (1) and the isotropic elastic constitutive law, the axial strain    , and the slope   , take the expressions where φ is an unknown integration constant.Concerning the semi-infinite substrate, the longitudinal strain and slope at the contact region are known in closed form [34] as Figure 2: Mechanical interaction between the beam and the substrate for symmetric and skew-symmetric loading cases.
Thus, the unknowns of the problem are the stress distributions () and () at the interface within the contact region.
The interfacial shear and peeling stresses can be straightforwardly expanded in series of Jacobi orthogonal polynomials, where  (,)  (/) represents the Jacobi polynomial of order  (see Appendix) and  is the order of the singularity of the stress field at the beam edges, namely, at  = ±.Indeed, as shown by [23], with  = log(3 − 4]  ) or  = log(3 − ]  )/(1 + ]  ) for a plane strain or plane stress state, respectively.The problem is governed by the following compatibility equations between the beam and the substrate: By introducing ( 2)-( 4) into (6) and truncating series (4) to the first  elements, the imposition of the compatibility conditions at  + 1 collocation points   over the contact region allows reducing the system of singular integral equations (6) into a linear algebraic system for the unknown   and   .The collocation points   , with  = 1, 2, . . .,  + 1, have been set as the roots of the Chebyshev polynomial of first kind of order  + 1, ] .
Once the coefficients   and   are found, the strain and displacement fields can be determined and, in turn, the internal forces in the beam can be assessed.The displacement components along the substrate surface [34] take the form where  and V represent the longitudinal and transversal displacement.The displacement field of the beam reads All the integrals involved in the above expressions can be evaluated in closed form (see Appendix).
The strength of the interfacial stresses at the beam ends can be assessed through the peeling and shear stress intensity factors   and   :

Results and Discussion
The behaviour of the beam-substrate system under the symmetric and skew-symmetric loading conditions reported in Figures 2(a) and 2(b) has been investigated.The series expansions (4) have been truncated to 13 terms and the compatibility conditions (6) have been imposed at 14 collocation points within the contact region.Refereed to the dimensionless spatial coordinate  = /, Figures 3-5 show the dimensionless shear and peeling stress distributions due to different symmetric or skew-symmetric loading cases.
For the beam loaded with two axial forces acting at the film edges, the shear stress (Figure 3) shows a monotonic trend over the entire contact region.Conversely, the behaviour displayed by the peeling tractions is not monotonic as they change sign in the neighbourhood of the film edge.
For the case of shear forces acting at the beam edges, the shear contact stresses are reported in Figure 4.As shown, the results are almost invariant with respect to the Poisson ratio of the substrate, with particular reference to the peeling stress.The interfacial shear stress displays a nonmonotonic trend, in particular for the loading case of two skew-symmetric shear forces, except for ]  = 0.5, namely, for an incompressible half plane, as found by [23].The loading case of two symmetric couples acting at the beam edges induce an almost linear trend for the shear stress (Figure 5) in the region closer to the middle of the beam for any value assumed by the Poisson ratio of the half plane.Conversely, when two skew-symmetric couples act at the beam ends, the shear stress decreases as the Poisson ratio increases.Also for the loading cases of two end couples, the Poisson ratio affects the shear stress more than the peel tractions.
The trend exhibited by the peeling distribution on the contact surface allows satisfying the balance of momentum induced by shear stress along contact region and the external load acting on the beam.

Conclusions
The contact problem of an Euler-Bernoulli nanobeam perfectly welded to a homogeneous elastic half plane has been addressed in the present work.The problem has been solved by expressing the unknown shear and peel interfacial stresses as infinite series of Jacobi orthogonal polynomials.The compatibility condition between the strains of the beam and those of the underlying half plane leads to a system of two Fredholm integral equations with Cauchy kernel.The complex singular behaviour of such stresses has been handled by assuming properly the singularity of the stress field at the beam edges.This allows reducing the singular integral equations into a linear system of algebraic equations, which has been solved for the unknown coefficients of the interfacial stress field.The shear and peel stresses have been evaluated for different values of the Poisson ratio for the half plane, thus finding the effect played by this elastic parameter on the beam-substrate mechanical interaction.In particular, it has been shown that the Poisson ratio of the half plane affects mostly the shear stress distribution than the behaviour of the peel tractions.For both symmetric or skew-symmetric couples or shear forces acting at the beam edges Poisson ratio does not affect significantly the mechanical response of the system.
The analysis allows properly assessing the stress concentrations at the edges of the beam (note that high stress levels can increase the time-dependent effects, as shown in [35][36][37] for concrete frames), an important task in many engineering problems involving beams and plates bonded to an elastic support.For example, the obtained results can be used to properly design FRP stiffeners applied to concrete structures that are prone to various damaging phenomena (see, e.g., [38]).It should be noted that the case of a thermal variation Δ (thermal variations can sensibly affect the mechanical behaviour of structural elements, as shown in [39,40]) acting on the beam (for the effect produced by thermal loads on concrete structures, see, e.g., [38]) can be studied as the loading case of two symmetric axial end forces  by assuming where  denotes the coefficient of thermal expansion.Note also that the contact mechanics of a nonhomogeneous composite beam with random composition and microstructure should be investigated by following the approach reported in [41,42].Section 2 to evaluate in closed form the integrals involved in the compatibility equations [45]:

Figure 3 : 25 ]Figure 4 :
Figure 3: Dimensionless mechanical interactions of the beam subjected to axial forces acting at the edges.

Figure 5 :
Figure 5: Dimensionless mechanical interactions of the beam subjected to couples acting at the edges.