Modified Nonlocal Strain Gradient Elasticity for Nano-Rods and Application to Carbon Nanotubes

Abstract

Nowadays, the modified nonlocal strain gradient theory provides a mathematically well-posed and technically reliable methodology to assess scale effects in inflected nano-structures. Such an approach is extended in this paper to investigate the extensional behavior of nano-rods. The considered integral elasticity model, involving axial force and strain fields, is conveniently shown to be equivalent to a nonlocal differential problem equipped with constitutive boundary conditions. Unlike treatments in the literature, no higher-order boundary conditions are required to close the nonlocal problem. Closed-form solutions of elastic nano-rods under selected loadings and kinematic boundary conditions are provided. As an innovative implication, Young’s moduli of Single-Walled Carbon Nanotubes (SWCNT) weare assessed and compared with predictions of Molecular Dynamics (MD). New benchmarks for numerical analyses were also detected.

1. Introduction

Components of Nano-Electro-Mechanical-Systems (NEMS), such as sensors and actuators, are usually modeled as nano-rods and nano-beams. It is well known that methods of local continuum mechanics cannot be adopted for such elements. Molecular Dynamics (MD) simulations are time-consuming and micro/nano-scaled experiments are usually difficult to implement. Hence, nonlocal continuum models have been developed for predicting the size-dependent mechanical behavior of nano-structures.

In the framework of nonlocal elasticity, Eringen’s strain-driven differential model [1] has been widely adopted in the literature (see, e.g., [2,3,4,5,6]). It is worth noting that recent papers on the strain-driven nonlocal model [7,8,9,10] prove that, if the bending field is expressed as convolution of elastic curvature with an averaging kernel assuming an exponential expression, a solution of this problem exists only if the bending field satisfies constitutive boundary conditions. Accordingly, it is shown in [11] that the nonlocal elastostatic problem is ill-posed in all cases of applicative interest, as acknowledged in the literature (see, e.g., [12,13,14,15,16,17,18]). A modified version of the Eringen integral model is proposed in [19] and has been recently applied to inflected nanobeams in [20].

The gradient elasticity theory [21] assumes that a material at the nano-scale is modeled via gradient terms. Many works investigate the small-scale effects on the static and dynamic behaviors of rods, beams and plates and the effect of stiffness enhancement has been often reported in these strain gradient models (see, e.g., [22,23,24,25,26]).

Recently, the Eringen’s integral law [1] has been combined with the strain gradient elasticity in [27] to formulate a higher-order nonlocal theory, thus collecting nonlocal theory and strain gradient theory into a single model.

Using such a model, many contributions have been provided to model the size-dependent behavior of nano-rods and beams (see, e.g., [28,29,30,31,32,33,34,35]) and plates [36,37].

The procedure consists in considering the integral nonlocal gradient method for structural problems defined on bounded domains equivalent to a differential law of higher-order than the one of the classical local problem. Therefore, additional non-classical suitable boundary conditions must be added to solve the nonlocal strain gradient elastostatic problem.

To solve the problem, different choices have been followed in the literature. Two usual choices consist in imposing higher-order boundary conditions pertaining to the strain gradient theory, of kinematic type [38] or static type [39]. It is worth noting that the structural behavior is greatly influenced by such choices.

The nonlocal strain gradient theory with higher-order boundary conditions has been recently adopted in [40] to study nano-rods in tension. The closed-form solutions for predicting the axial displacement and the variation of the Young’s modulus have been derived for four different nano-rods differing by the choice of the higher-order boundary conditions. In particular, for each nano-rod, the nonlocal parameters have been set to match the variation of the Young’s modulus obtained by the MD simulations.

The choice of the higher-order boundary conditions is disputed in the literature and is considered an open question (see, e.g., [41]). In the context of nano-beams subjected to flexure, a recent contribution [42] provides a definite solution to this issue. In fact, the non-classical boundary conditions to be imposed to solve the elastostatic problem of nonlocal strain gradient inflected nano-beams are given by constitutive boundary conditions (CBC) that naturally follow from the nonlocal strain gradient integral model. The consistent nonlocal strain gradient strategy has been successfully applied to free vibrations of nano-beams in [43].

In the present paper, the structural behavior of nano-rods in tension is formulated in the framework of the modified nonlocal strain gradient (NSG) theory. The expressions of the CBC for nano-rods are explicitly provided and it is shown that no unmotivated higher-order boundary conditions have to be prescribed to solve the nonlocal structural problem.

In addition, the variation of the Young’s modulus provided by MD simulations is recovered based on the NSG model for nano-rods developed in the present paper. In particular, carbon nanotubes are effectively described by NSG nano-rods with the usual boundary conditions, that is clamped at the one end and with a tensile force at the other end. As illustrated in Appendix A, the same result is obtained if a doubly-clamped nano-rod with an imposed axial displacement at one end is considered.

Finally, numerical analyses are presented as benchmark examples for applications and experimental tests on nonlocal nano-rods.

2. Modified Nonlocal Strain Gradient Law for Rods

Let us consider a functionally graded (FG) straight nano-rod of length L, the x-coordinate is taken along the length of the nano-rod with the y-coordinate along the thickness and the z-coordinate along the width of the nano-rod. The local Young’s modulus E of the FG nano-rod continuously changes in the thickness direction y, so that the Young’s elastic modulus at the point y isE(y)and the elastic area is_AE=_∫ΩEydA, beingΩthe nano-rod cross-section.

In the modified nonlocal strain gradient (NSG) model for FG nano-rods, the axial force N is defined in terms of elastic axial strain_εeland of its derivative_{∂x εel} is [27]

Nx,_λ0,_λ1,l=_ϕ0∗_AEεelx,_λ0−^l2 _∂xϕ1∗_{AE∂x εel}x,_λ1=_{∫0L ϕ0}x−ξ,_λ0AEεelξdξ−^l2 _{∂x ∫0L ϕ1}x−ξ,_{λ1AE∂ξ εel}ξdξ.

The smoothing kernels_ϕ0and_ϕ1depend on two non-dimensional nonlocal parameters_λ0>0and_λ1>0. The scale parameterl>0 , characteristic of the strain gradient elasticity [21], was subsequently introduced in [27] to make dimensionally homogeneous the convolutions in Equation (1) and to describe the importance of higher-order strain gradient fields.

Following [27,38], we consider that the nonlocal parameters are coincident, i.e.,λ:=_λ0=_λ1, and the kernels_ϕ0and_ϕ1are coincident with the bi-exponential averaging function given by

ϕ(x,_Lc)=12_Lcexp−x_Lc,

being_Lc=λL the characteristic length of Eringen nonlocal elasticity. The kernel (Equation (2)) fulfills positivity, symmetry, normalization and impulsivity [42].

Introducing the following fields

_N0x,_Lc=_∫0Lϕx−ξ,_LcAEεelξdξ_N1x,_Lc,l=_{l2 ∫0L}ϕx−ξ,_{LcAE∂ξ εel}ξdξ,

the modified nonlocal strain gradient elastic law (Equation (1)) can be rewritten as

Nx,_Lc,l=_N0x,_Lc−_{∂x N1}x,_Lc,l.

As proven in the next proposition, the modified nonlocal strain gradient integral relation (Equation (4)) for FG nano-rods is equivalent to a suitable differential law with constitutive boundary conditions.

Proposition 1 (Constitutive equivalence for FG nano-rods).

The modified nonlocal strain gradient constitutive law (Equation (4)) equipped with the bi-exponential kernel (Equation (2))

Nx,_Lc,l=_N0x,_Lc−_{∂x N1}x,_Lc,l,

withx∈0,L, is equivalent to the differential relation

_AE·_εelx−^l2 _∂x2AE·_εelx=Nx,_Lc,l−_Lc2∂x2Nx,_Lc,l

subject to the following two constitutive boundary conditions (CBC)

_{∂xNx,Lc,lx=0}=1_LcN0,_Lc,l+^l2 _{Lc2∂xAEεelxx=0∂xNx,Lc,lx=L}=−1_LcNL,_Lc,l+^l2 _{Lc2∂xAEεelxx=L}.

Proof.

Since the bi-exponential averaging function is given by

ϕ(x,_Lc)=12_Lcexp−x_Lc,

a direct evaluation provides the first derivative of the convolutions (Equation (3))

_{∂x N0}x,_Lc=1_Lc∫xLϕx−ξ,_LcAEεelξdξ+−_∫0xϕx−ξ,_LcAEεelξdξ_{∂x N1}x,_Lc,l=^l2 _Lc∫xLϕx−ξ,_{Lc∂ξAEεel}ξdξ+−_∫0xϕx−ξ,_{Lc∂ξAEεel}ξdξ.

The second derivative of the convolutions (Equation (3)) follows from Equation (9) using Equation (3) to get

_{∂x2 N0}x,_Lc=1_{Lc2l0 ∫2L}ϕx−ξ,_LcAEεelξdξ+−_AEεelx=1_Lc2N0x,_Lc−_AEεelx_{∂x2 N1}x,_Lc,l=^l2 _Lc2∫0Lϕx−ξ,_{Lc∂ξAEεel}ξdξ+−_∂xAEεelx==1_Lc2N1x,_Lc,l−^l2 _∂xAEεelx.

Subtracting the third derivative of Equation (9)₂ from Equation (9)₁, it turns out to be

_{∂x2 N0}x,_Lc−_{∂x3 N1}x,_Lc,l=1_Lc2N0x,_Lc−_AEεelx+−1_{Lc2∂x N1}x,_Lc,l−^l2 _∂x2AEεelx

so that, recalling Equation (5) and rearranging the terms, we have

_∂x2Nx,_Lc,l=_{∂x2 N0}x,_Lc−_{∂x3 N1}x,_Lc,l==1_Lc2N0x,_Lc−_{∂x N1}x,_Lc,l−1_Lc2AEεelx+^l2 _{Lc2∂x2AEεel}x

and Equation (6) is recovered.

The CBC in Equation (7) can be recovered as follows.

Using Equation (9)₂ , Equation (5) can be rewritten in the form

Nx,_Lc,l=_N0x,_Lc−^l2 _Lc∫xLϕx−ξ,_{Lc∂ξAEεel}ξdξ+−_∫0xϕx−ξ,_{Lc∂ξAEεel}ξdξ

and using Equations (9)₁ and (7)₂ , the first derivative of Equation (5) becomes

_∂xNx,_Lc,l=1_Lc∫xLϕx−ξ,_LcAEεelξdξ+−_∫0xϕx−ξ,_LcAEεelξdξ+−1_Lc2N1x,_Lc,l+^l2 _∂xAEεelx.

The CBC (Equation (7)) of modified nonlocal strain gradient nano-rods follows by evaluating Equations (13) and (14) at nano-rod boundary pointsx=0andx=L. In fact, we have atx=0

N0,_Lc,l=_N00,_Lc−^l2 _Lc∫0Lϕ−ξ,_{Lc∂ξAEεel}ξdξ_{∂xNx,Lc,lx=0}=1_Lc∫0Lϕ−ξ,_LcAE·_εelξdξ+−1_Lc2N10,_Lc,l+^l2 _{∂xAEεelxx=0}

so that Equation (15) provides the relations

N0,_Lc,l=_N00,_Lc−1_LcN10,_Lc,l_∂xN0,_Lc,l=1_LcN00,_Lc−1_Lc2N10,_Lc,l+^l2 _{Lc2∂xAEεelxx=0}

and the CBC in Equation (7)₁is recovered. Analogously, settingx=L in Equation (14), we get

NL,_Lc,l=_N0L,_Lc+^l2 _Lc∫0LϕL−ξ,_{Lc∂ξAEεel}ξdξ_{∂xNx,Lc,lx=L}=−1_Lc∫0LϕL−ξ,_LcAEεelξdξ+−1_Lc2N1L,_Lc,l+^l2 _{∂xAEεelxx=L}

so that Equation (17) provides the relations

NL,_Lc,l=_N0L,_Lc+1_LcN1L,_Lc,l_{∂xNx,Lc,lx=L}=−1_LcN0L,_Lc−1_Lc2N1L,_Lc,l+^l2 _{Lc2∂xAEεelxx=L}

and the CBC in Equation (7)₂ is recovered. Conversely, sufficient condition can be inferred from the uniqueness of the solution of Equation (6) consequent to the fact that the associated homogeneous equations

_AEεelx−^l2 _∂x2AEεelx=0Nx,_Lc,l−_Lc2∂x2Nx,_Lc,l=0

admit only the trivial solution under the homogeneous boundary conditions

_{∂xAEεelxx=0}=0_{∂xAEεelxx=L}=0

_{∂xNx,Lc,lx=0}=1_LcN0,_Lc,l_{∂xNx,Lc,lx=L}=−1_LcNL,_Lc,l

for (19)₁ and for (19)₂respectively. □

3. Elastic Equilibrium Problem

Let us consider a FG nano-rod subject to a distributed axial loadq(x)per unit length in the interval[0,L]and to concentrated axial forcesFat the end cross-sectionsx=0andx=L.

Differential condition of equilibrium can be written as

_∂xNx,_Lc,l=−q(x)

with the boundary conditionsNx,_Lc,l=∓Fatx=0andx=L.

The axial displacement at the abscissa x along the nano-rod axis is denoted byu(x)and the kinematically compatible axial strain has the form

ε(x)=_∂xu(x).

In the sequel, elastic_εeland kinematically compatibleεstrains are assumed to be coincident.

Exact solutions according to the proposed modified nonlocal strain gradient (NSG) model for FG nano-rods can be performed by the following steps.

1. Step 1: Solve the equilibrium Equation (22) to get the expression of the axial force

Nx,_Lc,l=−_∫0xqsds+_a1.

2. Step 2: Solve the second-order differential Equation (6) in the form

_AEεelx−^l2 _∂x2AEεelx==−_∫0xqsds+_a1+_Lc2∂xq(x)

obtaining the expression of the elastic axial strain_εelof the nano-rod in terms of three integration constants (_a1,_a2, and_a3) to be determined.

3. Step 3: Solve the first-order differential in Equation (23) in terms of the axial displacement u of the nano-rod to get the expression of u in terms of four integration constants (_a1,_a2,_a3, and_a4) to be determined.

4. Step 4: Determine the four integration constants (_a1,_a2,_a3, and_a4 ) by imposing the two CBC given by Equation (7) in terms of the axial displacement u

−q(0)=1_Lca1+^l2 _{Lc2AE∂x2uxx=0}−q(L)=1_Lc∫0Lqsds−_a1+^l2 _{Lc2AE∂x2uxx=L}.

and the two classical boundary conditions at the nano-rod end pointsx=0andx=Lby specifying

uorN.

It is worth noting that, in statically determinate rods, the axial force N can be obtained by Equation (24) by imposing the classical static boundary conditions.

4. Closed-Form Solutions for FG Nano-Rods

Closed-form elastic solutions for FG nano-rods with a clamped end atx=0and a free end atx=Land with both clamped ends are presented hereafter. The applied loads are a uniform load p, a concentrated forceFatx=L(or an imposed axial displacementδatx=L for doubly-clamped rods). Kinematic and static boundary conditions are enforced to the FG nano-rod ends according to classical rod theory and, in addition, the constitutive boundary conditions Equation (26) are imposed according to the proposed NSG model. Hence, the axial displacement u can be recovered following Steps 1–4 in Section 3.

In the sequel, the abbreviationsCFandCCstand for clamped-free andclamped-clamped, respectively. Moreover, let us assume that the elastic area_AEis constant along the nano-rod axis x.

For completeness sake, the FG nano-rod constraints, the considered applied load and the related boundary conditions are reported in Table 1.

To provide a non-dimensional analysis of FG nano-rods, the following non-dimensional variableξand the non-dimensional characteristic parametersλandμare adopted in the examples

ξ=xL,λ=_LcL,μ=lL.

The non-dimensional axial displacementu¯ depends on the kind of load applied to the nano-rod according to Table 2.

4.1. Case I: CF FG Nano-Rod with a Concentrated Load at the Free End

Let us consider a FG nano-rod of length L with a clamped end atx=0and a free end atx=Lsubject to a concentrated loadFat the free end.

Following the steps in Section 3, the axial force N can be evaluated by means of the equilibrium equation so that Equation (24), with the boundary conditionN(L)=F, yieldsN(x)=F. For simplicity, we drop the dependence on the nonlocal characteristic parameters_Lc,l. Then, the analytical solution of the NSG model for the FG nano-rod is obtained from the following nonlocal differential equation

_{AE ∂x}u(x)−^l2 _{AE ∂x3}u(x)=F

under the CBC (Equation (26))

F_Lc+^l2 _{Lc2AE∂x2uxx=0}=0−F_Lc+^l2 _{Lc2AE∂x2uxx=L}=0

and the kinematic boundary condition in Equation (27)

u(0)=0.

Hence, the solution of the differential equation (Equation (29)) with the boundary conditions in Equations (30) and (31) provides the axial displacement

u(x)=_ue(x)+F_LcAE^eLl−1^e−xlexl−1^eLl+^exl

where_ueis the rod axial displacement of the local model

_ue(x)=Fx_AE.

The maximum displacement takes place atx=L and can be obtained from Equation (32) settingx=L

u(L)=_ue(L)+2F_LcAE=FL+2_LcAE.

The classical (local) displacement_ueof the FG nano-rod is provided by_Lc→0. Moreover, the limit nano-rod displacement_u∞forl→+∞is given by

_u∞(x)=_ue(x)+2F_LcAELx=FL+2_LcAELx.

It is important to note that the NSG model for FG nano-rods exerts a softening effect, with respect to the local behavior, in terms of the nonlocal parameter_Lc. It is of interest that the maximum displacementu(L)of the NSG rod model does not depend on the gradient parameter l and the maximum axial displacement tends to the one of the classical (local) rod if_Lc→0.

The effects of the non-dimensional characteristic parametersλandμ on the elastic response of nano-rods are examined in Figure 1 and Figure 2.

Figure 1a,b shows the non-dimensional axial displacementu¯in terms of the gradient non-dimensional parameterμforλ=0.4andλ=0.8, respectively. The local non-dimensional axial displacement_u¯eis recovered byλ→⁰⁺and is reported with the dot dashed line. The limit non-dimensional axial displacement_u¯∞ follows from Equation (35) and is reported with the dotted line in terms of the non-dimensional nonlocal parameterλ.

The comparison between nonlocal FG nano-rods and classical (local) FG rods in Figure 1 highlights the increment of the axial displacementu¯due to the behavior of the nonlocal model. The parameterλhas the effect of increasing the axial displacement, i.e. a largerλinvolves greater axial displacementsu¯for a given value of the non-dimensional gradient parameterμ.

Figure 2a shows the non-dimensional axial displacementu¯of the FG nano-rod in terms of the nonlocal non-dimensional parameterλforμ=0.15. The local non-dimensional axial displacement_u¯eis recovered byλ→⁰⁺and is reported with the dot dashed line. The limit nano-rod non-dimensional axial displacement_u¯∞in terms ofλis plotted with the dotted line.

Figure 2b shows the non-dimensional axial displacementu¯of the FG nano-rod in terms of the nonlocal non-dimensional parameterλforμ=0.15(thick lines) andμ=0.30(dotted lines). The non-dimensional maximum axial displacementu¯(1)increases for increasing values of the nonlocal parameterλand is independent of the non-dimensional gradient parameterμ. The limit nano-rod non-dimensional axial displacement_u¯∞is plotted with the black dotted line.

The 3D plot of the non-dimensional maximum axial displacementu¯(1)for the proposed NSG method versus the non-dimensional characteristic parametersλandμ is reported in Figure 3. The horizontal plane is the non-dimensional local maximum axial displacement_u¯e(1)=1.

The innovative nonlocal model exhibits a hardening behavior in terms of the non-dimensional characteristic parameterλand the maximum axial displacement does not depend on the non-dimensional gradient parameterμ.

The nonlocal model coincides with the classical (local) model of rods for non-dimensional characteristic parametersλtending to vanishing, i.e., andλ→⁰⁺.

4.2. Reduced Young’s Modulus

In applications, the evaluation of the Young’s modulus of micro- and FG nano-rods is of great interest. Hence, we evaluate the reduced rigidity_Kr from Equation (34) to get

_Kr=LL+2_LcAE.

Note that the classical rigidity_Kr=_AEcan be recovered if_Lc→0.

Using the reduced rigidity in Equation (36), the axial displacement of the modified nonlocal strain gradient model can be calculated by adopting the classical (local) analysis for rods. Let us consider a constant Young’s elastic modulusE(y)=Eso that the elastic area is_AE=_∫ΩEydA=EA. Hence, the reduced Young’s modulus_Er can be defined from Equation (36) as

_Er=LL+2_LcE

depending on the nonlocal parameter_Lc. The upper bound of the nano-rod reduced Young’s modulus_ErforL→∞is provided by the local Young’s modulus E. Analogously, for_Lc→0, we recover the local Young’s modulus E and for_Lc→+∞, the nano-rod reduced Young’s modulus tends to vanish.

To make a comparison, we considered the data presented in [40] for a SWCNT of armchair (10, 10). The diameter d of the SWCNT (n, m) can be calculated by

d=aπ3ⁿ²+n·m+^m2=1.356nm

where the carbon–carbon bond length isa=0.142nm. The effective thickness of the considered SWCNT ist=0.34 nm and the classical (local) Young’s modulus assumed in [40,44] isE=909.5GPa.

The Young’s modulus predicted by the MD simulations is reported in [40,44]. Note that the strain gradient elastic theory with high-order boundary conditions was adopted in [40] and the necessity to impose higher-order (non-classical) boundary conditions has the effect that the classical boundary conditions of the classical theory (such as free and clamped boundary conditions) may no longer be meaningful for the modified nonlocal strain gradient rod. As a result, following [28,35], one needs to take into account further boundary conditions involving higher-order stress and strain distributions. Accordingly, four nonlocal nano-rods are considered in [40], depending on the considered higher-order boundary conditions, and for each of them the small-scale parameters are set to match the results of the MD simulations.

On the contrary, the proposed NSG model has no higher-order boundary condition to add so that the classical definitions of external constraints must not be modified. Hence, a unique model of rod can be considered and a unique value of the nonlocal parameter has to be set to match the results of the MD simulations.

In Figure 4 the results provided by Equation (37) are plotted together with the MD data versus the SWCNT length. Upper and lower bounds of the nonlocal parameter_Lc=0.04272nm and_Lc=0.06942 nm are reported in Figure 4a to include the values provided by the MD simulations. A good agreement between the Young’s modulus obtained by the NSG model and the MD results could be obtained by setting the nonlocal parameter_Lc=0.0534 nm, as shown in Figure 4b.

The NSG model provides values of the reduced Young’s modulus_Ertending to the classical (local) value E for increasing values of the SWCNT length L. On the contrary, the values of the MD simulations appear to be constant for values of the SWCNT’s length greater than 27 nm.

The small-scale effect on displacements can be clearly observed in Figure 5 where the non-dimensional maximum displacementu(L)/L, pertaining to the NSG model, is plotted versus the SWCNT’s length L for an applied forceF=1nN and the nonlocal parameter_Lcranging in the set0.04272,0.0534,0.06942nm. As can be seen, the small-scale effect on the displacement can be observed when the length of SWCNT is small and the small-scale effect increases for increasing values of the nonlocal parameter_Lc. IfL→+∞, the non-dimensional maximum displacementu(L)/Ltends to the corresponding local one1/_AE=7.59117×¹⁰⁻⁴.

In [45], a continuum mechanics model has been proposed to predict the effective wall thickness of a SWCNT and to calculate its Young’s modulus. The deformation of a central long SWCNT in a bundle of SWCNTs, subjected to an external pressure, has been considered in plane-strain and has been modeled as a thin ring with a mean radiusR, thicknesstin the radial direction and a unit width in the axial direction. Hence, it has been obtained that the radius of the nanotube isR=0.7066nm and the predicted thickness ist=0.0617nm so that the related Young’s modulus isE=4880 GPa. Considering the nonlocal parameter (dependent on the longitudinal atom spacing in armchair CNTs [46,47])_Lc=0.0534 nm as previously calibrated by means of the MD data, the variation law of Young’s modulus obtained by the proposed NSG model for the SWCNT investigated in [45] is reported in Figure 6, with radiusR=0.7066nm, thicknesst=0.0617nm and Young’s modulusE=4880GPa. The NSG model provides values of the reduced Young’s modulus_Ertending to the valueE=4880GPa for increasing values of the SWCNT lengthL.

It is apparent that, for a nanotube with L = 4 nm, the reduced Young’s modulus is_Er=4753GPa, achieving97.4%of the value ofE . Hence, we can conclude that, for the SWCNT considered in [45], the variation of the Young’s modulus_Eris really small if the length of the SWCNT is greater that4 nm. The small-scale effect on displacements is observed in Figure 7, in which the non-dimensional maximum displacementu(L)/Lof the NSG model is plotted versus the SWCNT’s length L for an applied forceF=1nN and the nonlocal parameter_Lcranging in the set(0.04272,0.0534,0.06942)nm. The small-scale effect on the displacement is apparent and the small-scale effect increases for increasing values of the nonlocal parameter_Lc. The non-dimensional maximum displacementu(L)/Ltends to the corresponding local displacement1/AE=7.48069×¹⁰⁻⁴ifLtends to+∞.

4.3. Case II: CF FG Nano-Rod Subject to a Uniformly Distributed Axial Load

Let us now consider a FG nano-rod with a clamped end atx=0and a free end atx=Lsubject to a uniformly distributed axial loadq(x)=p.

Following the steps reported in Section 3, the axial force N can be evaluated by means of the equilibrium equation so that Equation (24), with the boundary conditionN(L)=0, yieldsN(x)=p(L−x). Hence, the closed-form solution of the NSG model for the considered FG nano-rod is obtained by the following nonlocal differential equation

_{AE ∂x}u(x)−^l2 _{AE ∂x3}u(x)=p(L−x)

under the CBC in Equation (26)

pL_Lc+^l2 _{Lc2AE∂x2uxx=0}=−p^l2 _{Lc2AE∂x2uxx=L}=−p

and the classical boundary condition in Equation (27)

u(0)=0.

Hence, the axial displacement is

u(x)=_ue(x)+p_AE^e2Ll−1^e−xlexl−1·^eLll2−_Lc2+^eL+xll2−_Lc2++^e2Ll−^l2+L_Lc+_Lc2+^exl−^l2+L_Lc+_Lc2

where_ueis the axial displacement of the local model

_ue(x)=px2_AE2L−x.

The maximum axial displacement of theCFFG nano-rod is attained at the free endx=Land is given by

u(L)=_ue(L)+pL_LcAE=pLL+2_Lc2_AE.

It is apparent from Equation (42) that the axial displacement u depends on the nonlocal parameters l and_Lc. On the contrary, the maximum axial displacementu(L) (see Equation (44)) is independent of the nonlocal gradient parameter l.

The limit maximum axial FG nano-rod displacement for the nonlocal parameter_Lc→0is given by the classical (local) displacement_ue(L)=p^L22_AEof the FG nano-rod.

The non-dimensional axial displacementu¯of the FG nano-rod versus the FG nano-rod non-dimensional lengthξ is reported in Figure 8 in terms of the nonlocal non-dimensional parameterλforμ=0.15(thick lines) andμ=0.30(dotted lines). The non-dimensional maximum axial displacementu¯(1)increases for increasing values of the nonlocal parameterλand is independent of the non-dimensional gradient parameterμ. The limit FG nano-rod non-dimensional axial displacement for the non-dimensional nonlocal parameterλtending to vanish is given by

_u¯lim(ξ)=−1+^eξμeξμ−^e1μ1+^{e1μμ2 e−ξμ}−12ξ−2ξ

and is reported with magenta thick line forμ=0.15and dotted line forμ=0.30. Note that_u¯lim(1)=_u¯e(1)=0.5.

The limit non-dimensional axial displacement_u¯∞forμ→0 is obtained from Equation (42) and is given by

_u¯∞(ξ)=12+λξ.

The corresponding plot is reported with the black dotted line in Figure 8 for the considered values of the non-dimensional nonlocal parameterλ.

4.4. Case III: CC FG Nano-Rod Subject to a Uniformly Distributed Axial Load

Let us consider a fully clamped FG nano-rod subject to a uniformly distributed axial loadq(x)=p.

Following the steps reported in Section 3, the axial force isNx=−px+_a1so that the analytical solution of the FG nano-rod is provided as

_{AE ∂x}u(x)−^l2 _{AE ∂x3}u(x)=−px+_a1

under the CBC in Equation (26)

_{a1 Lc}+^l2 _{Lc2AE∂x2uxx=0}=−p1_Lcpx−_a1+^l2 _{Lc2AE∂x2uxx=L}=−p

and the classical boundary condition in Equation (27)

u(0)=0uL=0.

Hence, the axial displacement is

u(x)=_ue(x)+p2_AE^eLl+1^e−xlexl−1·^exl−^eLl2^l2−L_Lc−2_Lc2

where_ueis the axial displacement of the local model

_ue(x)=px2_AEL−x.

Note that we have_a1=pL/2so that the axial force of the NSG method isN(x)=p(L−2x)/2and coincides to the local one.

The non-dimensional axial displacementu¯of the FG nano-rod versus the FG nano-rod non-dimensional lengthξ is reported in Figure 9 in terms of the nonlocal non-dimensional parameterλforμ=0.15(thick lines) andμ=0.30(dotted lines). The non-dimensional midspan axial displacementu¯(1/2)increases for increasing values of the nonlocal parameterλfor a givenμ. The limit FG nano-rod non-dimensional axial displacement for the non-dimensional nonlocal parameterλ tending to vanish follows from Equation (50)

_u¯lim(ξ)=−1+^eξμeξμ−^e1μ1+^{e1μμ2 e−ξμ}−12ξ−1ξ

and is reported with magenta thick line forμ=0.15and dotted line forμ=0.30.

The limit non-dimensional axial displacement_u¯∞ is obtained from Equation (50) forμ→∞and is the vanishing one, i.e.,_u¯∞(ξ)=0 . The corresponding plot is reported with the black dotted line in Figure 9.

The 3D plot of the non-dimensional maximum axial displacementu¯(1/2)for the proposed NSG method versus the non-dimensional characteristic parametersλandμ is reported in Figure 10. It is apparent that the NSG method stiffness or soften the nano-rod depending on the values of the non-dimensional nonlocal and gradient parametersλ,μ.

The cuts of the 3D plot for given values of the non-dimensional parametersλandμ are provided in Figure 11. In particular, the plots of the non-dimensional maximum axial displacementu¯(1/2)for the NSG method versus the non-dimensional characteristic parameterλforμ=0.2andμ=0.6 are provided in Figure 11a. It is apparent that forμ=0.2andλ<0.070156orμ=0.6andλ<0.4the NSG model is stiffer than the local model and forμ=0.2andλ>0.070156orμ=0.6andλ>0.4the NSG model softens the nano-rod. The limit values of the non-dimensional maximum axial displacementu¯(1/2)forλ→⁰⁺are given by0.0915228forμ=0.2and by0.0281989forμ=0.6.

The plots of the non-dimensional maximum axial displacementu¯(1/2)for the NSG method versus the non-dimensional characteristic parameterμforλ=0.2andλ=0.6 are provided in Figure 11b. It is immediate to note that forλ=0.2andμ<0.374166orλ=0.6andμ<0.812404the NSG model is stiffer than the local model and forλ=0.2andμ>0.374166orλ=0.6andμ>0.812404the NSG model softens the nano-rod. The limit values of the non-dimensional maximum axial displacementu¯(1/2)forμ→⁰⁺are given by0.265forλ=0.2and by0.785forλ=0.6.

Finally, a standard numerical analysis of Equation (50) shows that the non-dimensional maximum axial displacementu¯(1/2) for the proposed NSG method is attained at the midspan and is reported in Table 3 in terms of the non-dimensional nonlocal parameterλand gradient parameterμ. It worth noting that the maximum displacement is attained at the midspanξ=1/2of the FG nano-rod independent of the values ofλandμ. The non-dimensional maximum axial displacement of the classical (local) model is_u¯e(1/2)=1/8=0.125. Accordingly, the non-dimensional maximum axial displacementsu¯(1/2)in terms of the pairsλ,μ, which are less than the non-dimensional maximum classical axial displacements_u¯e(1/2)=0.125 of the FG nano-rod, are reported in italic in Table 3. Hence, the italic values of the non-dimensional maximum axial displacementu¯(1/2) of the NSG method are smaller than the one of the classical (local) model, thus Table 3 allows one to identify the corresponding pairsλ,μhaving the effect of stiffen or soften the FG nano-rod with respect the classical (local) behavior.

5. Conclusions

FG elastic nano-rods under tension have been investigated by the modified nonlocal strain gradient (NSG) theory. The new formulation contains a nonlocal parameter and a material length scale parameter to incorporate the scaling effects of nonlocal stress and microstructure-dependent strain gradient. In comparison to other strain-driven methodologies, the new proposal has been shown to be well-posed and does not require higher-order boundary conditions. In fact, in addition to the classical static and kinematic boundary conditions, closure of the NSG model has to be carried out by prescribing suitable constitutive boundary conditions. Closed-form nonlocal solutions of FG clamped-free and clamped-clamped nano-rods have been provided, exhibiting stiffening or softening effects depending on the values of nonlocal and gradient parameters. Single-Walled Carbon Nanotubes (SWCNT) of armchair (10, 10) were modeled as NSG nano-rods, showing that the new approach could capture the small-scale behavior of Young’s modulus as predicted by the MD simulations. The nonlocal parameter was thus tuned to characterize Young’s modulus vs. the SWCNT length.

Author Contributions

Conceptualization, Methodology, Software, Validation, Investigation, Writing—Review & Editing: R.B., M.Č. and F.M.d.S. All the authors contributed equally to this work.

Funding

This research received no external funding.

Acknowledgments

Financial supports from the Italian Ministry of Education, University and Research (MIUR) in the framework of the Project PRIN 2015 “COAN 5.50.16.01”—code 2015JW9NJT—and from the research program ReLUIS 2018 are gratefully acknowledged.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

The reduced stiffness can be alternatively obtained by considering aCC−δnano-rod, which is a fully clamped nano-rod with an imposed axial displacementδat the end pointx=L.

Following the steps reported in Section 3, the axial force is expressed in terms of an unknown parameter in the formN(x)=_a1so that the solution of the nano-rod is provided by the nonlocal differential equation

_{AE ∂x}u(x)−^l2 _{AE ∂x3}u(x)=_a1

under the CBC (Equation (26))

_{a1 Lc}+^l2 _{Lc2AE ∂x2uxx=0}=0−_{a1 Lc}+^l2 _{Lc2AE ∂x2uxx=L}=0

and the classical boundary conditions (Equation (27))

u(0)=0u(L)=δ.

Hence, the axial displacement is

u(x)=_ue(x)+δ_Lc^eLl−1LL+2_Lc^e−xl·−^eLlL+^e2xlL+^eL+xlL−2x+^exl2x−L

where_ueis the axial displacement of the local model

_ue(x)=δLx.

Moreover, the value of the parameter_a1provides the axial force

N(x)=_AEL+2_Lcδ

and we recover the same value of the reduced stiffness reported in Equation (37).

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