A new nonlocal bending model for Euler–Bernoulli nanobeams

Mechanics Research Communications 62 (2014) 25–30

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A new nonlocal bending model for Euler–Bernoulli nanobeams Francesco Marotti de Sciarra ∗ , Raffaele Barretta Department of Structures for Engineering and Architecture, via Claudio 25, 80121 Naples, Italy

a r t i c l e

i n f o

Article history: Received 8 May 2014 Received in revised form 29 July 2014 Accepted 15 August 2014 Available online 23 August 2014 Keywords: Nonlocal elasticity Gradient elasticity Euler–Bernoulli nanobeam

a b s t r a c t This paper is concerned with the bending problem of nanobeams starting from a nonlocal thermodynamic approach. A new coupled nonlocal model, depending on two nonlocal parameters, is obtained by using a suitable definition of the free energy. Unlike previous approaches which directly substitute the expression of the nonlocal stress into the classical equilibrium equations, the proposed approach provides a methodology to recover nonlocal models starting from the free energy function. The coupled model can then be specialized to obtain a nanobeam formulation based on the Eringen nonlocal elasticity theory and on the gradient elastic model. The variational formulations are consistently provided and the differential equations with the related boundary conditions are thus derived. Nanocantilevers are solved in a closed-form and numerical results are presented to investigate the influence of the nonlocal parameters. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction The Euler–Bernoulli beam theory dates back to the 18th century. Such a model is based on the assumption that straight lines normal to the midplane before deformation remain straight and normal to the midplane after deformation. The resulting formulation for solving the deflection of local elastic beams is based on a fourthorder differential equation. A nonlocal continuum model has been introduced by Eringen (2002) to account for small-scale effects by specifying that the stress at a given point is dependent on the stress in neighbouring points of the body. An alternative methodology is based on an atomistic approach centred on the molecular dynamics and the molecular mechanics (see e.g. Cao and Chen, 2006; Chen and Cao, 2006). Starting from the study of Peddieson et al. (2003) which developed a nonlocal Euler–Bernoulli beam model, many contributions on this issue have been proposed following a similar approach, that is the nonlocal model is obtained by replacing the stress appearing in the classical equilibrium equations by its nonlocal counterpart (Reddy, 2007; Wang and Liew, 2007; Aydogdu, 2009; Arash and Wang, 2012).

∗ Corresponding author. Tel.: +39 081 768 3734. E-mail addresses: [email protected] (F. Marotti de Sciarra), [email protected] (R. Barretta). http://dx.doi.org/10.1016/j.mechrescom.2014.08.004 0093-6413/© 2014 Elsevier Ltd. All rights reserved.

An hybrid nonlocal beam model is developed in (Zhang et al., 2010) by postulating that the strain energy functional involves both local and nonlocal curvatures. Accordingly such a hybrid model shows nonlocal effects for an Euler–Bernoulli cantilever nanobeam under a transverse point load while the Eringen model is found to be free of small-scale effects for the same problem (Challamel and Wang, 2008). It is worth noting that, on the basis of a simple analogy (Barretta et al., 2014; Barretta and Marotti de Sciarra, 2014), the nonlocality effect on nanorods and nanobeams, formulated according to the Eringen model, can be simulated by prescribing suitable fields of axial and curvature distortions on corresponding local rods and beams. Hence a general procedure is available to establish if nonlocal nanorods and nanobeams are free of small-scale effects. In the present paper a new coupled nonlocal model is introduced by a suitable definition of the free energy which depends on two small length-scale parameters and on a participation factor which can make the nanobeam flexible or stiffer. Then nonlocal thermodynamics allows us to build up a consistent methodology to derive the related variational formulation and, as a consequence, the differential relations with the associated boundary conditions can be obtained in a straightforward manner. The proposed coupled model can be specialized to recover the Eringen model (1983, 1987) and the gradient model (Aifantis, 2003; Papargyri-Beskou et al., 2003; Giannakopoulos and Stamoulis, 2007; Akgöz and Civalek, 2012; Barretta and Marotti de Sciarra, 2013).

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F. Marotti de Sciarra, R. Barretta / Mechanics Research Communications 62 (2014) 25–30

An example of a nanocantilever subjected to a uniform load is illustrated and closed-form solutions are provided in order to investigate the influence of the nonlocal parameters. A comparison among the coupled, Eringen and gradient model is thus performed.

Table 1 Boundary conditions pertaining to the considered nanobeam model.

2. Kinematics An Euler–Bernoulli straight nanobeam occupying a domain V is considered. The cross-section of the nanobeam is denoted by , the centroid axis is indicated by x and the bending plane is defined by the Cartesian axes (x, y) originating at the cross-section centroid. The axis orthogonal to the bending plane  is denoted by z and the associated second moment of area is I =  y2 dA. The displacement field s of the nanobeam and the kinematically compatible deformation field D are then given by

⎡

−v(1) (x)y

s(x, y, z) = ⎣

v(x)

⎤

⎡

0

0

0

0

0⎦

0

0

0

0



ˇ˙ dV

(2)

V

where ˇ is the Helmholtz free energy of the nanobeam and  is the nonlocal axial stress. The superscript dot denotes differentiation with respect to the time. A new nonlocal model for nanobeams is proposed in the present paper by considering the following expression of the free energy: (1)

ˇ(ε, ε

1 1 1 ) = Eε2 + c12 Eε(1)2 + ˛c22 q(ε) 2 2 A

(3)

with E Young modulus, c1 , c2 nonlocal scalar parameters, ˛ participation factor, A cross-section area and q distributed transverse load intensity. The nondimensional scalar factor ˛ can assume any real value, as shown in the sequel, thus providing the weight of the third nonlocal term in Eq. (3). From an engineering point of view, the nanobeam becomes stiffer or not depending on the assumed value of the parameter ˛. Remark 3.1. The expression of the free energy (3) leads to a new variational formulation for nonlocal Euler–Bernoulli nanobeams, described by Eq. (5). The corresponding nonlocal model is conceived as a combination between the nonlocal Eringen and gradient models as pointed out in Remark 3.2. The term c22 q in Eq. (3) is peculiar of the model proposed by Eringen (1983). Indeed, as proved by Barretta and Marotti de Sciarra (2014), the elastostatic problem governing a Euler–Bernoulli nonlocal nanobeam is equivalent to the one of a corresponding local nanobeam subjected to a prescribed bending curvature given by c22 q/EI. The time derivative of the free energy is 1 ˙ ε(1) ) = Eεε˙ + c12 Eε(1) ε˙ (1) + ˛c22 q∂ε (ε)ε˙ ˇ(ε, A where ∂ε is the derivative with respect to the axial strain ε.

(2)

Substituting the time derivative of the free energy given by Eq. (4) into Eq. (2), using the kinematically compatible deformation field (1)2 and noting that ∂ε (ε)ε˙ = v˙ (2) , we get the following variational formulation



L



0



L

M v˙ (2) dx =

M0 v˙ (2) dx + ˛c22

0

L



qv˙ (2) dx + c12

0



(M, M0 , M1 ) = −

In nonlocal elasticity, the first principle of thermodynamic for an isotropic body can be written in a global form and the second one can be expressed in its usual local form (Eringen and Edelen, 1972; Polizzotto, 2003; Marotti de Sciarra, 2009a). Accordingly the vanishing of the body energy dissipation can be expressed as follows (Marotti de Sciarra, 2009b; Romano et al., 2010; Marotti de Sciarra and Barretta, 2014)

V

(1)

L

M1 v˙ (3) dx

0

(5)

where the stress resultant moments are given by

3. New nonlocal elastic model

 ε˙ dV =

−M (1) + ˛c22 q(1) = −M0 + c12 M1 (1) M − ˛c22 q = M0 − c12 M1 0 = c12 M1

(1)

where v is the transverse displacement along the y-axis and :=v(2) = −ε/y is the nanobeam bending curvature, with ε axial strain. The apex denotes the derivative along the nanobeam axis x.



Static boundary conditions

v v(1) v(2)

⎤

−v(2) (x)y

⎦ , D(x, y, z) ⎣

Kinematic boundary conditions

(4)



(, Eε, Eε(1) )ydA

(, 0 , 1 )ydA = − 

(6)



The expression of the free energy (3) leads thus to the new variational formulation (5) for nonlocal Euler–Bernoulli nanobeams and the corresponding nonlocal model can be considered as a combination between the nonlocal Eringen and gradient models as pointed out in Remark 3.2. This new coupled model can thus be cast in the framework of the so-called hybrid nonlocal theory proposed in Challamel and Wang (2008), Zhang et al. (2010) where a different coupling between the Eringen and gradient models is provided. It is worth noting that the importance of providing a variational formulation associated with nonlocal models relies also in the fact that it is the starting point to formulate a nonlocal finite element (see e.g., Marotti de Sciarra, 2013, 2014). Differential and boundary conditions of equilibrium are recovered by integrating by parts the l.h.s. of Eq. (5) and imposing the equality with the external virtual power. In formulae  we get M (2) = q in [0, L] and T = −M (1) = F and M = M at x = 0, L , with T shear force and (F, M) transverse force and couple. Integrating by parts Eq. (5), the following nonlocal differential relation is provided (2)

(3)

q − ˛c22 q(2) = M0 − c12 M1

(7)

where the corresponding boundary conditions consistently follow from the related variational principle and are reported in Table 1. The nonlocal elastic equilibrium equation for nanobeams associated with the considered model can then be provided by expressing the differential equations (7) in terms of the transverse displacement v using Eqs. (1) and (6). In fact, noting the equalities



(Eε, Eε(1) )y dA = (EI v(2) , EI v(3) )

(M0 , M1 ) = −



(8)



being I =  y2 dA the second moment of area about the z-axis, the governing differential equation for the bending of the nonlocal Euler–Bernoulli nanobeam under distributed transverse loads is c12 EI v(6) − EI v(4) = −q + ˛c22 q(2)

(9)

where the related boundary conditions can be obtained from Table 1 and are reported in Table 2, with T = −M (1) . Hence the analysis performed above shows that the free energy (3) yields the sixth-order differential equations (9) governing the bending of the Euler–Bernoulli nanobeam. Accordingly six boundary conditions (three for each end of the nanobeam) are required (see Table 2) and the length-scale parameters appear in the differential equation as well as in the boundary conditions.

F. Marotti de Sciarra, R. Barretta / Mechanics Research Communications 62 (2014) 25–30 Table 2 Boundary conditions in terms of transverse displacement. Kinematic boundary conditions

Static boundary conditions

v v(1) v(2)

−EI v(3) + c12 EI v(5) = T + ˛c22 q(1) EI v(2) − c12 EI v(4) = M − ˛c22 q c12 EI v(3) = 0

The expression of the bending moment for the proposed model can then be recovered from the related variational formulation (5). In fact, integrating by parts the last integrals into Eq. (5) and recalling that the boundary conditions in Table 1 ensure M1 = 0, we get (1)

M = M0 − c12 M1 + ˛c22 q = EI v(2) − c12 EI v(4) + ˛c22 q

(10)

The considered nonlocal model tends to the classical (local) one for vanishing nonlocal parameters c1 and c2 . Remark 3.2. Uniqueness of the displacement solution, to within an additional rigid body motion, of the nonlocal elastic equilibrium problem governed by the differential equation (9) under the boundary conditions collected in Table 2 is ensured by the assumption that the nanobeam has a positive definite elasticity E > 0, so that the elastic stiffness fulfils the inequalities





2

Eε2 dV > 0 and

Eε(1) dV > 0

V

(11)

V

for any non-vanishing strain fields ε and ε(1) . Indeed, as shown below, the bilinear form of the elastic energy is positive. To this end, the variational formulation (5) can be integrated by parts using Eq. (8) and can be rewritten in terms of the displacement field in the following form:



L

0



L

EI v(2) v˙ (2) dx + c12





L

0

qv˙ dx − ˛c22

= 0

L



q(2) v˙ dx + M v˙ (1)

0





− ˛c22 qv˙ (1)

− M (1) v˙

x={0,L}

x={0,L}



x={0,L}

+ ˛c22 q(1) v˙

x={0,L}

(12)

The symmetric bilinear form of the elastic energy takes thus the form





L

L

EI v(2) v(2) dx + c12

a(v, v):=

EI v(3) v(3) dx

0

(13)

0

By considering compatible strains ε = −yv(2) and ε(1) = −yv(3) , see Eq. (1)2 , we have





L

0

EI v(3) v(3) dx 0



 Ey2 v(2) v(2) dV + c12

= V



 Eε2 dV + c12

= V

4. A nanocantilever subjected to a distributed load As an example, the elastic equilibrium of a nanocantilever with length L subjected to a uniform load q is formulated by considering the Eringen, the gradient-type constitutive models (see Remark 3.2) and the coupled model proposed in the present paper. These models will be labelled respectively by (i), (ii) and (iii) hereafter. The nonlocal solution pertaining to the Eringen model is obtained by solving the fourth-order differential equation −EI v(4) = −q + c 2 q(2) with the following boundary conditions:

⎧ v(0) = 0 ⎪ ⎪ ⎪ ⎨ v(1) (0) = 0 v(3) (L) = 0 ⎪ ⎪ ⎪ ⎩ (2)

(15)

(L) = −c 2 q

Note that the nonlocal parameter c enters only in the last boundary condition involving the bending moment. The nonlocal solution of the gradient model (ii) is obtained by solving the sixth-order differential equation c 2 EI v(6) − EI v(4) = −q with the following six boundary conditions:

⎧ v(0) = 0 ⎪ ⎪ ⎪ ⎪ v(1) (0) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ v(3) (0) = 0 v(3) (L) − c 2 v(5) (L) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v(2) (L) − c 2 v(4) (L) = 0 ⎪ ⎪ ⎪ ⎩ (3)

(16)

v (L) = 0

L

EI v(2) v(2) dx + c12

a(v, v) =

e0 represents a material constant. The new model defined by the free energy functional (3) collapses into the Eringen and gradient elasticity models (Eringen, 1983, 1987; Aifantis, 2003, 2014; Papargyri-Beskou et al., 2003; Giannakopoulos and Stamoulis, 2007; Akgöz and Civalek, 2012; Barretta and Marotti de Sciarra, 2013; Dell’Isola and Forest, 2014) by setting c1 :=c, ˛:=1, c2 :=c and c1 :=c, c2 :=0 respectively. These models are usually recovered following a different path of reasoning. In fact the starting point is the definition of the nonlocal stress which is commonly expressed in the Eringen form. The stress in the classical equilibrium equations is replaced by the corresponding nonlocal quantity and the nonlocal model is thus derived. On the contrary the proposed approach starts from the definition of the free energy and allows us to provide the related variational formulation so that the nonlocal model can be consistently developed.

EI v

EI v(3) v˙ (3) dx

27

Ey2 v(3) v(3) dV V 2

Eε(1) dV ≥0

(14)

V

for any displacement field v. The inequality above is vanishing if and only if ε = 0 and ε(1) = 0. The r.h.s. of Eq. (12) provides the power of the effective loading. Equilibrium of the loading system on the nanobeam implies equilibrium of the effective loading and ensures existence of the solution of nonlocal elastostatic problem, see e.g. Romano (2003). Remark 3.3. Let c = e0 l be the length-scale parameter introduced by Eringen (1983, 2002), where l is the material length scale and

The nonlocal solution of the new coupled model (iii) is achieved by solving the sixth-order differential equation (9) for the following values of the nonlocal parameterss c1 :=c and c2 :=c, where the participation scalar factor ˛ ∈  changes the nonlocal term describing the Eringen free energy in Eq. (3). The relevant six boundary conditions to be prescribed are given by

⎧ v(0) = 0 ⎪ ⎪ ⎪ ⎪ v(1) (0) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ v(3) (0) = 0 v(3) (L) − c 2 v(5) (L) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ EI v(2) (L) − c 2 EI v(4) (L) = −˛c 2 q ⎪ ⎪ ⎪ ⎩ (3) v (L) = 0

(17)

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F. Marotti de Sciarra, R. Barretta / Mechanics Research Communications 62 (2014) 25–30

Hence the transverse displacement field of the Eringen model is obtained in the following form (i) v1 (x) = v0 (x) − c 2

qx2 , 2EI

with v0 (x) =

qLx3 qL2 x2 qx4 − + 24EI 6EI 4EI (18)

where v0 is the classical (local) transverse displacement. The transverse displacement field of the gradient model is (ii) v2 (x) = v0 (x) + +

c 3 (1 + e2L/c )qL (−1 + e2L/c )EI

c 3 e(2L/c)−(x/c) qL (1 − e2L/c )EI

−

+

c 3 ex/c qL (1 − e2L/c )EI

2qc 2 x2 c 2 qLx + EI 4EI

(19)

and the transverse displacement field of the coupled model is (iii)

v3 (x) = v0 (x) + +

c 3 (1 + e2L/c )qL (−1 + e2L/c )EI

c 3 e(2L/c)−(x/c) qL (1 − e2L/c )EI

−

+

c 3 ex/c qL (1 − e2L/c )EI

c 2 qLx 2qx2 c 2 (−1 + ˛) − EI 4EI

(20)

The transverse displacement fields v1 , v2 , and v3 reduce to the classical (local) one v0 for c = 0. The upper bound v∞ of the displacement field in terms of the scale parameter c for the gradient model can be obtained by evaluating the limit of v2 for c → +∞. Hence the displacement field of the gradient model must belong to the strip bounded by the functions:

v0 (x) =

qx4 qLx3 qL2 x2 − + 24EI 6EI 4EI

and

v∞ (x) =

qL2 x2 12EI

(21)

The bending moment of the three models is obtained by substituting Eqs. (18)–(20) into Eq. (10) with the appropriate c1 and c2 . For all the considered models, the expressions of the bending moment coincide to the classical (local) function (1/2)q(L − x)2 . 5. Examples and discussion For convenience sake, we introduce the following dimensionless quantities: =

x , L

=

y , L

=

c , L

B=

b , L

H=

h , L

v∗i () = vi (x)

EI , qL4 (22)

with i = {1, 2, 3}, so that only the material parameters and ˛ are required in computations. The dimensionless transverse displacements of the nanocantilever under a uniform load for the three models and for different values of and ˛ are reported in Fig. 1(a) and (b). The maximum deflection of the nanocantilever occurs at the free end-section  = 1 but such a deflection, evaluated using the Eringen model (i), decreases for increasing and vanishes for = 0.5, see Fig. 1(a) and (b). Such a physically counterintuitive behaviour exhibited by the Eringen model (i) for = 0.5 is completely absent in the new coupled model (iii), see the solid lines depicted in Fig. 1(a) and (b). The dimensionless transverse displacement pertaining to the coupled model (iii) is plotted for the considered values of with ˛ = −2.0 in Fig. 1(a) and with ˛ = 0.5 in Fig. 1(b). It is apparent that the nanocantilever softens with increasing for a negative participation factor ˛. On the contrary, the nanocantilever stiffens with increasing if the parameter ˛ is positive. Fig. 2 shows the dimensionless transverse displacements v∗3 of the nanocantilever for = {0.1, 0.2, 0.3} and ˛ = {−3.0, −2.0, 0.1, 0.5, 1.0}. It can be readily seen that the deflection of the nanocantilever increases with increasing if the

Fig. 1. (a) Dimensionless transverse displacements of the nanocantilever under a uniform load for ␶ = {0.1, 0.2, 0.3, 0.5}. Comparison among the Eringen model v∗1 , gradient model v∗2 and coupled model v∗3 with ˛ = −2. (b) Dimensionless transverse displacements of the nanocantilever under a uniform load for ␶ = {0.1, 0.2, 0.3, 0.5}. Comparison among the Eringen model v∗1 , gradient model v∗2 and coupled model v∗3 with ˛ = 0.5.

parameter ˛ is negative and decreases with increasing if ˛ is positive. The relation between the ratio of the dimensionless transverse deflection at the free end-section v∗i (1), with i = {1, 2, 3}, and the classical (local) dimensionless deflection v∗0 (1) versus the dimensionless length-scale parameter for increasing values of the participation factor ˛ is plotted in Fig. 3. The magnitude of ˛ is varying from −3.0 to 1.0 and it should be noted that the coupled model (iii) corresponds to the gradient model (ii) if ˛ = 0. Evidently all the nonlocal models yield the classical (local) deflection of the free end-section for = 0. The plots of v∗1 (1)/v∗0 (1) and v∗2 (1)/v∗0 (1) show that both the nonlocal Eringen and gradient models stiffer the nanocantilever with respect to the local behaviour for any value of . Moreover the gradient

Fig. 2. Dimensionless transverse displacements v∗3 of the nanocantilever under a uniform load for increasing values of the participation factor ˛.

F. Marotti de Sciarra, R. Barretta / Mechanics Research Communications 62 (2014) 25–30

Fig. 3. Ratios of the dimensionless transverse deflection at the free endsection v∗1 (1)/v∗0 (1), v∗2 (1)/v∗0 (1) and v∗3 (1)/v∗0 (1) versus the dimensionless length-scale parameter for increasing values of the participation factor ˛ = {−3.0, −2.0, 0.1, 0.5, 1.0}.

Fig. 4. Ratios of the dimensionless transverse deflection at the free endsection v∗1 (1)/v∗0 (1), v∗2 (1)/v∗0 (1) and v∗3 (1)/v∗0 (1) versus the participation factor ˛ for increasing values of the dimensionless length-scale parameter = {0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.4, 0.5}.

model is stiffer than the Eringen one for any dimensionless lengthscale parameter and, as previously stated, it has an asymptotic behaviour for → +∞. For a given dimensionless length-scale , the ratio v∗3 (1)/v∗0 (1) of the deflection of the nanocantilever is equal to one if the parameter ˛ is obtained in terms of according to the function ˛( ) = [−1 − 2 + e1/ (−1 + 2 )]/(1 + e1/ ) plotted in Fig. 3. Hence for a given dimensionless length-scale parameter , the corresponding participation factor ˛( ) is such that the coupled model (iii) has the same deflection at the nanocantilever free endsection of the classical (local) behaviour. For a given , the ratio v∗3 (1)/v∗0 (1) is greater than one if the parameter ˛ is less than the value ˛( ) so that the coupled model softens the nanocantilever leading to a larger tip deflection compared to the local case. On the contrary if the parameter ˛ is greater than the value ˛( ) the ratio v∗3 (1)/v∗0 (1) is less than one and the nonlocal effect in the coupled model stiffens the nanocantilever. If the regularizing parameter ˛ increases the behaviour of the nanobeam tends to that of the Eringen model (ii), see the green plot obtained with ˛ = 0.1. The free end-section has a null deflection for = 0.5, according to the Eringen model, if ˛ = (−1 + e2 )/(1 + e2 ) ∼ = 0.76. The relationships between the ratios v∗i (1)/v∗0 (1) of the transverse deflection at the free end-section, with i = {1, 2, 3}, versus the participation factor ˛ is plotted in Fig. 4 for different values of the dimensionless length-scale parameter . The plot of v∗3 (1)/v∗0 (1) for the coupled model (iii) has a linear behaviour in terms of ˛ and

29

its slope depends by the dimensionless length-scale parameter in agreement with the expression − 2 /2. The intersections of the coupled model plots with the horizontal axis passing at 1 are given by the lozenge points in Fig. 4 and provide the participation factor ˛ 0 which ensures that the coupled model has the same behaviour of the local one for a given dimensionless length-scale parameter , i.e. v∗3 (1)/v∗0 (1) = 1. Note that the values of ˛ 0 are negative. It is apparent that for increasing the participation factor ˛ increases to recover the local behaviour. At the same value, the deflection of the nanocantilever tip is greater than that of its local counterpart for ˛ < ˛ 0 whereas the deflection is smaller than its local counterpart for ˛ > ˛ 0 . Hence the dimensionless length-scale parameter of the coupled model makes the nanocantilever flexible or stiff depending upon whether the regularization parameter ˛ is less than ˛ 0 or not. The intersections of the coupled model plots with the vertical axis ˛ = 0 provide the ratios v∗2 (1)/v∗0 (1) for different values of the dimensionless length-scale parameter . For a given , the ratio of the deflection v∗2 (1)/v∗0 (1) is always less than unit so that the gradient model yields a nanocantilever which is stiffer than its local counterpart and becomes stiffer with increasing the dimensionless length-scale parameter . At the same value, the deflection of the nanocantilever free end-section acquired by the coupled model (iii) is greater than that obtained by the gradient model for ˛ < ˛ 2 = 0 whereas the deflection is smaller than that obtained by the gradient model for ˛ > ˛ 2 = 0. In other words, when compared with the gradient model, the dimensionless length-scale parameter softens the nanocantilever for ˛ < 0 leading to a larger deflection and stiffens the nanobeam for ˛ > 0 giving a smaller deflection. In Fig. 4 the dimensionless deflection v∗1 (1)/v∗0 (1) of the nanocantilever tip in the Eringen model is given by the dot-dashed horizontal lines tending to the horizontal reference axis for increasing values of . The intersections of the plots obtained by the coupled model v∗3 (1)/v∗0 (1) and by the Eringen model v∗1 (1)/v∗0 (1) for increasing values of are provided by the bullet points in Fig. 4. These intersections provide the participation factor ˛ 1 ensuring that the tip deflection in the coupled and Eringen models has the same value for a given dimensionless length-scale parameter . It is apparent that the participation factor ˛ 1 is positive and increases for increasing . At the same value, the deflection of the nanocantilever free end-section is greater than that obtained by the Eringen model for ˛ < ˛ 1 whereas such a deflection is smaller than that obtained by the Eringen model for ˛ > ˛ 1 . Hence the dimensionless length-scale parameter in the coupled model makes the nanocantilever flexible or stiff depending upon whether the regularization parameter ˛ is less than ˛ 1 or not. Thus for a given , the small length-scale effect introduced by the proposed coupled nonlocal method makes the nanocantilever flexible or stiff, with respect to the local model, Eringen model and gradient model, depending on whether the participation factor ˛ is less than its corresponding limit value (˛ 0 , ˛ 1 , ˛ 2 = 0) or not.

6. Conclusion A new coupled nonlocal Euler–Bernoulli nanobeam model is presented based on a consistent thermodynamic approach. The governing equations and the related high-order boundary conditions are derived by using a variational formulation associated with the nonlocal model. The proposed coupled model encompasses the gradient elastic beam model and the Eringen model as special cases and reduces to the classical (local) Euler–Bernoulli beam model as the small-scale parameters vanish.

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Closed-form solutions for the bending problem of a nanocantilever under a distributed load are presented and numerical results are compared with the one obtained using the gradient elastic and Eringen models. The participation factor introduced in the coupled model can make the nanobeam flexible or stiffer when compared with the gradient elastic model, the Eringen model and the classical (local) beam model. Acknowledgement The support of “Polo delle Scienze e delle Tecnologie” — University of Naples Federico II — through the research project FARO 2012 is gratefully acknowledged. References Aifantis, E.C., 2003. Update on a class of gradient theories. Mech. Mater. 35, 259–280. Aifantis, E.C., 2014. Gradient material mechanics: perspectives and prospects. Acta Mech., http://dx.doi.org/10.1007/s00707-013-1076-y. Akgöz, B., Civalek, Ö., 2012. Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory. Arch. Appl. Mech. 82, 423–443. Arash, B., Wang, Q., 2012. A review on the application of nonlocal elastic models in modelling of carbon nanotubes and graphenes. Comput. Mater. Sci. 51, 303–313. Aydogdu, M., 2009. A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E 41, 1651–1655. Cao, G., Chen, X., 2006. The effect of displacement increment on the axial compressive buckling behavior of single-walled carbon nanotubes. Nanotechnology 17, 3844–3855. Challamel, N., Wang, C.M., 2008. The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology 19, 1–7. Chen, X., Cao, G., 2006. A structural mechanics approach of single-walled carbon nanotubes generalized from atomistic simulation. Nanotechnology 17, 1004–1015. Barretta, R., Marotti de Sciarra, F., 2014. Analogies between nonlocal and local Bernoulli–Euler nanobeams. Arch. Mech. Appl., http://dx.doi.org/10.1007/ s00419-014-0901-7.

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