On Bending of Bernoulli-Euler Nanobeams for Nonlocal Composite Materials

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.

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.

Nonlocal Elasticity
In local linear elasticity for isotropic materials, stress and strain at a point x of a Cauchy continuum are functionally related by the following classical law: with  and  Lamé 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) t  is the nonlocal stress, (2)   is the macroscopic stress given by ( 5), (3)  is the influence function, (4)  =  0 / a dimensionless nonlocal parameter defined in terms of the material constant  0 and of the internal and external characteristic lengths  and , respectively.
In agreement with the Eringen proposal in choosing the following influence function 1− 2 ∇ 2 , the nonlocal elastic law (2) rewrites as where ∇ 2 denotes the Laplace operator.The differential form adopted for bending of nanobeams, analogous to (3), is provided by where  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 by with  Young modulus.

Bending of Nonlocal Nanobeams
Let us consider a bent nanobeam of length , with Young modulus  functionally graded in the cross section Ω 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 G 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 [0, ], F  , concentrated forces at the end cross sections {0, }, M, concentrated couples at the end cross sections {0, }.
The bending stiffness is defined by Differential and boundary conditions of equilibrium are expressed by where   is the bending moment.
The bending curvature, corresponding to the transverse displacement V, is given by The 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 Ω: with the axial dilation provided by the known formula () = −().
Enforcing ( 8) and ( 7) 1 and imposing the static equivalence condition we obtain the relation This equation can be interpreted as decomposition formula of the bending curvature   =  2 V/ 2 into elastic  EL and inelastic  IN parts with Accordingly, 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  IN (13) 2 .

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 V * () = (100  )/(   4 )V(), for selected values of the nonlocal parameter  fl  0 /.Details of the calculations and some comments are reported below.The l.h.s. of ( 11) is hence known, so that the differential condition of nonlocal elastic equilibrium to be integrated writes explicitly as

Cantilever Nanobeam. The bending moment is given by
The general integral of ( 15) takes the form where is a particular solution of (15).The evaluation of the integration constants  and  is carried out by prescribing the boundary conditions The transversal deflection follows by a direct computation The maximum displacement is given by Nanocantilever becomes stiffer with increasing the nonlocal parameter .Indeed, according to the analysis proposed in Section 3, the sign of the prescribed distortion  IN , describing the nonlocality effect, is opposite to the one of the elastic curvature  EL .In particular, the displacement of the free end vanishes if  = 0.5 (see Figure 1) according to (20).
It is worth noting that, with the structure being statically determinate, the bending moment ( 14) is not affected by the scale phenomenon.
In agreement with the equivalence method exposed in Section 3, ( 24) and ( 25) provide the deflection and A plot of the dimensionless bending moment  * = −(100/   2 ) versus the dimensionless parameter  = / is given in Figure 3 for increasing values of the nonlocal parameter .

Conclusion
The Eringen nonlocal law has been used in order to assess size effects in nanobeams formulated according to the Bernoulli-Euler kinematics.The treatment extends to functionally graded materials the analysis carried out in [24] under the special assumption of elastically homogeneous nanobeams.Transverse deflections of cantilever and clamped-simply supported nanobeams have been established for different values of the nonlocal parameter.Such analytical solutions could be conveniently adopted by other scholars as simple reference examples for numerical evaluations in nonlocal composite mechanics.