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

Authors: Luciano Feo; Rosa Penna
Journal: Modelling and Simulation in Engineering
 (2016)

---

## 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.


---

## Body

## 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 Section2.


## 2. Nonlocal Elasticity

In local linear elasticity for isotropic materials, stress and strain at a pointx of a Cauchy continuum are functionally related by the following classical law:(1)σijx=2μeijx+λerrxδij,with μ and λ 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:(2)tijx=∫VKx′-x,τσijx′dV,where (1)
tij is the nonlocal stress,(2)
σij is the macroscopic stress given by (5),(3)
K is the influence function,(4)
τ=e0a/l a dimensionless nonlocal parameter defined in terms of the material constant e0 and of the internal and external characteristic lengths a and l, respectively.In agreement with the Eringen proposal in choosing the following influence function 1-c2∇2, the nonlocal elastic law (2) rewrites as(3)1-e0a2∇2tij=σij,where ∇2 denotes the Laplace operator. The differential form adopted for bending of nanobeams, analogous to (3), is provided by(4)σ¯-c2d2σ¯dx2=σ,where σ¯ is the nonlocal normal stress and σ is the macroscopic normal stress on cross sections. Note that the stress σ is expressed in terms of elastic axial strains by(5)σ=Eε,with E Young modulus.


## 3. Bending of Nonlocal Nanobeams

Let us consider a bent nanobeam of lengthL, with Young modulus E functionally graded in the cross section Ω and uniform along the beam axis x. The cross-sectional elastic centre and the principal axes of elastic inertia, associated with the scalar field E, are, respectively, denoted by G and by the pair (y,z).The nanobeam is assumed to be subjected in the plane(x,y) to the following loading conditions:⁡qt, distributed load per unit length in the interval [0,L],⁡Ft, concentrated forces at the end cross sections {0,L},⁡M, concentrated couples at the end cross sections {0,L}.The bending stiffness is defined by(6)kb≔∫ΩEy2dA.Differential and boundary conditions of equilibrium are expressed by(7)d2Mbdx2=qt,in0,L,dMbdx=±Ft,at0,L,Mb=∓M,at0,L,where Mb is the bending moment.The bending curvature, corresponding to the transverse displacementv, is given by(8)χx=d2vdx2x.The differential equation of nonlocal elastic equilibrium of a nanobeam under flexure is formulated as follows. Let us preliminarily multiply (4) by the coordinate y along the bending axis and integrate on the cross section Ω:(9)∫Ωσ¯ydA-e0a2∫Ωd2σ¯dx2ydA=∫ΩEεydA,with the axial dilation provided by the known formula ε(x)=-χ(x)y.Enforcing (8) and (7)1 and imposing the static equivalence condition(10)Mb=-∫ΩσydA,we obtain the relation(11)Mb-e0a2qt=kbd2vdx2.This equation can be interpreted as decomposition formula of the bending curvature χb=d2v/dx2 into elastic χEL and inelastic χIN parts(12)χb=χEL+χIN,with(13)χEL=Mbkb,χIN=-e0a2kbqt.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 χIN (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 loadqt. 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 ξ=x/L and v∗(ξ)=(100kb)/(qtL4)v(ξ), for selected values of the nonlocal parameter τ≔e0a/L. Details of the calculations and some comments are reported below.Figure 1
Dimensionless transverse displacementsv∗ for the nanocantilever versus ξ for τ∈{0,0.05,0.1,0.15,0.2,0.3,0.5}.Figure 2
Dimensionless transverse displacementv∗ for the clamped-simply supported nanobeam versus ξ for τ∈{0,0.05,0.1,0.15,0.2,0.3,0.5}.


### 4.1. Cantilever Nanobeam

The bending moment is given by(14)Mbx=qt2x2-qtLx+qtL22.The l.h.s. of (11) is hence known, so that the differential condition of nonlocal elastic equilibrium to be integrated writes explicitly as(15)kbd2vdx2x=Mbx-τ2L2qt.The general integral of (15) takes the form(16)vx=ax+b+v¯x,where(17)v¯x=-qt24kbx4+qtL6kbx3-qtL24x2-τ2L2qt2kbx2is a particular solution of (15). The evaluation of the integration constants a and b is carried out by prescribing the boundary conditions(18)v0=0,dvdx0=0.The transversal deflection follows by a direct computation(19)vx=qt24kbx4-qtL6kbx3+qtL24kbx2-τ2qtL22kbx2.The maximum displacement is given by(20)vm=vL=18-τ22qtL4kb.Nanocantilever becomes stiffer with increasing the nonlocal parameter τ. Indeed, according to the analysis proposed in Section 3, the sign of the prescribed distortion χIN, describing the nonlocality effect, is opposite to the one of the elastic curvature χEL. In particular, the displacement of the free end vanishes if τ=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.


### 4.2. Clamped-Simply Supported Nanobeam

Equilibrated bending moments are provided by the relation(21)Mbx=a1x-L+qtL22-qt2x2,with a1∈R being an arbitrary constant. The differential condition of nonlocal elastic equilibrium (11) to be integrated takes thus the form(22)-kbd2vdx2x=a1x-L+qtL22-qt2x2+τ2L2qt.Equation (22) is solved by imposing the following kinematic boundary conditions:(23)v0=0,dvdx0=0,vL=0.A direct computation gives the transversal displacement field(24)vx=qt24kbx4-5qtL48kbx3+qtL216kbx2+τ2qtL4kbx2L-xand the bending moment(25)Mbx=qt2x2-qtL254+3τ2x+qtL223τ2+14.The maximum displacement is given by(26)vm=vx¯,with x¯=3L(3/2)(5/24+τ2/2)-(9/16)τ4+11/768+(13/96)τ2 solution of the equation (the known local maximum point x¯≃0.578L is recovered by setting τ=0)   (dv/dx)(x)=0.In agreement with the equivalence method exposed in Section3, (24) and (25) provide the deflection and bending moment of a corresponding local nanobeam under the transversal load distribution qt and the distortion χIN equivalent to the nonlocality effect.A plot of the dimensionless bending momentM∗=-(100/qtL2)M versus the dimensionless parameter ξ=x/L is given in Figure 3 for increasing values of the nonlocal parameter τ.Figure 3
Dimensionless bending momentM∗ versus dimensionless abscissa ξ of a clamped-simply supported nanobeam.


## 4.1. Cantilever Nanobeam

The bending moment is given by(14)Mbx=qt2x2-qtLx+qtL22.The l.h.s. of (11) is hence known, so that the differential condition of nonlocal elastic equilibrium to be integrated writes explicitly as(15)kbd2vdx2x=Mbx-τ2L2qt.The general integral of (15) takes the form(16)vx=ax+b+v¯x,where(17)v¯x=-qt24kbx4+qtL6kbx3-qtL24x2-τ2L2qt2kbx2is a particular solution of (15). The evaluation of the integration constants a and b is carried out by prescribing the boundary conditions(18)v0=0,dvdx0=0.The transversal deflection follows by a direct computation(19)vx=qt24kbx4-qtL6kbx3+qtL24kbx2-τ2qtL22kbx2.The maximum displacement is given by(20)vm=vL=18-τ22qtL4kb.Nanocantilever becomes stiffer with increasing the nonlocal parameter τ. Indeed, according to the analysis proposed in Section 3, the sign of the prescribed distortion χIN, describing the nonlocality effect, is opposite to the one of the elastic curvature χEL. In particular, the displacement of the free end vanishes if τ=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.


## 4.2. Clamped-Simply Supported Nanobeam

Equilibrated bending moments are provided by the relation(21)Mbx=a1x-L+qtL22-qt2x2,with a1∈R being an arbitrary constant. The differential condition of nonlocal elastic equilibrium (11) to be integrated takes thus the form(22)-kbd2vdx2x=a1x-L+qtL22-qt2x2+τ2L2qt.Equation (22) is solved by imposing the following kinematic boundary conditions:(23)v0=0,dvdx0=0,vL=0.A direct computation gives the transversal displacement field(24)vx=qt24kbx4-5qtL48kbx3+qtL216kbx2+τ2qtL4kbx2L-xand the bending moment(25)Mbx=qt2x2-qtL254+3τ2x+qtL223τ2+14.The maximum displacement is given by(26)vm=vx¯,with x¯=3L(3/2)(5/24+τ2/2)-(9/16)τ4+11/768+(13/96)τ2 solution of the equation (the known local maximum point x¯≃0.578L is recovered by setting τ=0)   (dv/dx)(x)=0.In agreement with the equivalence method exposed in Section3, (24) and (25) provide the deflection and bending moment of a corresponding local nanobeam under the transversal load distribution qt and the distortion χIN equivalent to the nonlocality effect.A plot of the dimensionless bending momentM∗=-(100/qtL2)M versus the dimensionless parameter ξ=x/L is given in Figure 3 for increasing values of the nonlocal parameter τ.Figure 3
Dimensionless bending momentM∗ versus dimensionless abscissa ξ of a clamped-simply supported nanobeam.


## 5. 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.



---

*Source: 6369029-2016-12-25.xml*