surface effect investigation for static bending of

Mechanics & Industry 13, 163–174 (2012)c© AFM, EDP Sciences 2012DOI: 10.1051/meca/2012010www.mechanics-industry.org

Mechanics&Industry

Surface effect investigation for static bending of nanowiresresting on elastic substrate using Timoshenko beam theoryin tandem with the Laplace-Young equation

Amin Khajeansaria and Gholam Hosein Baradaran

Department of Mechanical Engineering, Faculty of Engineering, Shahid Bahonnar, University of Kerman, Kerman 76188, Iran

Received 25 Septembre 2011, Accepted 16 April 2012

Abstract – In the present study, an enriched continuum mechanics framework is employed to study thesurface effects on bending behavior of silver nanowires (NWs) resting on elastic substrate. The Timoshenkobeam theory and the Laplace-Young equation are employed to investigate static behavior of silver NWslying on Winkler-Pasternak elastic substrate. Three types of boundary conditions are considered as doublysimply supported (S-S), doubly clamped (C-C) and cantilevered (C-F). Analytical solutions are obtainedfor NWs with surface crystallographic orientation of [001] subjected to a concentrated external force.By defining different normalized contact stiffness, extensive numerical results are carried out to studythe influence of effective parameters such as substrate, surface, aspect ratio (L/D) and diameter on thestiffness of NWs. According to the obtained results, the effect of surface and its rate of variation onstiffness of NWs lying on Winkler and Winkler-Pasternak elastic foundation models are more significant in(C-F) type of boundary condition compared to the NWs without foundation. By increasing the modulusof elastic substrate, the effect of shear deformation increases which it is more considerable in (C-C) and(S-S) NWs resting on the Winkler-Pasternak and Winkler substrate models, respectively.

Key words: Nanowire / elastic substrate / surface effect / size dependency / timoshenko beam

1 Introduction

Because of the novel physical properties of NWs, manyresearchers have tried to investigate the superior me-chanical properties of these one dimensional nanostruc-tures [1–7]. The extraordinary physical properties of NWsare mainly due to their large ratio of surface per volumeand quantum confinement effects. Quantum confinementtightly affects on the optical and electrical properties ofthe materials. Typically in nanoscale, this phenomenoncan also change the mechanical properties of materialsby increasing the difference of energy level and band gapin nanomaterial comparing to its bulk state, while in thebulk sample the band gap remains constant at its orig-inal energy due to a continuous energy state [8, 9]. Theexcess energy in a nanoscale material respect to corre-sponding bulk material is related to the particles nearthe surface of confined dimension. Therefore, the quan-tum confinement dominates the other processes leadingto change the potential energy, force field and mechanical

a Corresponding author:[email protected]

properties of nanomaterial respect to the bulk phase. Asa result, the balance of the surface particles is destroyedand the surface tension changes tremendously and thespecific tension and potential energy for the surface aredefined [10]. Because of their exceptional physical prop-erties, NWs have found several applications in nanoelec-tromechanical systems (NEMS). Interesting properties ofthe NEMS devices typically arise from the behavior of theactive parts, which in most cases, are in the forms ofcantilevers or doubly clamped beams with dimensions atnanometer scale [11–13].

In recent years, many methods and approaches havebeen implemented for studying the physical properties ofnanostructures such as nanowires, nanotubes, nanoplates,nanocomposites, etc. Atomistic simulations play an im-portant role to detect unexpected properties of NWs atsmall thickness [14, 15]. The thickness denotes the di-ameter of a circular cross section NW. In the case ofsquare cross section, the thickness denotes the lengthof cross section. For the NWs including a large clus-ter of atoms with the length scales above several ten ofnanometers, the atomistic simulations are too expensive.For investigation the mechanical properties of NWs by

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164 A. Khajeansari and G.H. Baradaran: Mechanics & Industry 13, 163–174 (2012)

experimental methods, the conventional tensile and creeptesting method have been prohibited, because these test-ing methods require the size of the sample be sufficientlylarge to be clamped rigidly by the sample holder withoutsliding [16]. Therefore, practically the bending deforma-tion experiments have been usually utilized to examinethe mechanical properties of NWs. However, the bend-ing experiments have also their individual complexities.Jing et al. [17] and Chen et al. [18] measured the impor-tance of the thickness and boundary conditions on theelastic properties of silver NWs respectively. To explainthe size related elastic properties of ZnO NWs in their ex-periments, Chen et al. [19] proposed a core-shell model,where the nanowire is treated as a composite with a shelland a core structure.

Enriched continuum mechanics models have been es-tablished to simplify the analysis of nanosystems. Asimplified model for prediction of the effective elasticproperties of nanosized structural elements based on thestandard continuum mechanics and atomistic simulationresults is demonstrated in [20]. A theoretical model is pre-sented in [21] to investigate the size dependent bendingbehavior of NWs with taking into account the influence ofboth the atomic configuration at the relaxed surface andthe surface tension. Li et al. [22] proposed a continuum ap-proach based on the strain gradient theory for determina-tion of the size effects in NWs under bending deformation.Among related studies, a valuable enriched continuummechanics model proposed by Gurtin and Murdoch [23].They introduced a non-classical phenomenon known asthe surface/interface effect which is taken into accountfor prediction of the size dependent elastic properties ofnanomaterials. This model is prepared by embedding thesurface energy into the conventional continuum mechan-ics model. In this continuum model, the main approachis that the nanoscale body is considered as a bulk volumesurrounded by a surface layer. Dingreville et al. [24] pro-posed a methodology to incorporate the surface energyinto the continuum theory to model the size-dependenteffective moduli of isolated nanosized elements (parti-cles, wires and films). The closed form relations havebeen proposed by Duan et al. [25–27] to characterize theeffective properties of nanocomposites containing inter-face stress. Sharma et al. [28, 29] have studied the size-dependent elastic state of non-homogenous nanomaterialscontaining inside inclusions. The numerical approachessuch as FEM and XFEM from the continuous models ofnanowires and nonocomposites were proposed by Yvonnetet al. [30, 31] to simulate the surface/interface effects. Intheir approaches, the surface elastic parameters were com-puted from atomistic simulations.

Recently an enriched continuum mechanics approachhas been presented by Wang and Feng [32–35] to in-vestigate the size dependent mechanical properties ofNWs by taking into account both the surface elasticityand the surface stress. They evaluated the effect of sur-face on contact, free vibration and axial buckling behav-ior of NWs under static loads using the Euler-Bernoulliand the Timoshenko beam theories and linear elasticity.

Comparing variation of the Young modulus versus di-ameter of NWs with the associated experimental resultsin He and Lilley [36] and Jiang and Yan [37], confirmsthe applicability of the above method to predict the sizedependency of mechanical properties of NWs. In noveltechnologies or materials the NWs implement as key com-ponents in flexible electronics, nanocircuitry, nanosensors,nanoprocessors, electronic logic gates, renewable energytechnologies and biological or gas sensing applications [1–3, 5–7, 11–13]. Thereby the NWs can be deposited on asubstrate. In NEMS, are mentioned nanodevices at aboveand/or other similar future application [38, 39]. There-fore surveying the mechanical behavior of NWs restingon the substrate medium can be considerable. In re-cent years, many researchers have studied the mechanicalbehavior of nanoscale materials especially carbon nan-otubes and graphene sheets embedded in various sur-roundings [40–42].

To the best of our knowledge, effect of surface on staticbending of NWs on elastic substrate by considering theshear deformation effect has not investigated yet. In thepresent paper, an enriched continuum mechanics modelis employed to investigate the effects of residual surfacestress and surface elastic constants on the bending defor-mation of NWs lying on elasticsubstrate. The linear sur-face elasticity theory combined with the Laplace-Youngequation to model the surface effects and Winkler andPasternak elastic foundations model is also applied tosimulate the substrate medium. The shear deformationeffect is considered by assuming Timoshenko beam the-ory and an explicit solution is obtained for transverse dis-placement and stiffness of NWs with three different loadsand boundary conditions: (a) cantilever (C-F) with a con-centrated force at the free end, (b) simply supported (S-S)and (c) clamped (C-C) under concentrated forces at themiddle point.

2 Formulation of the problem

The Winkler elastic foundation model for a beam isrecognized as groups of closely spaced vertical linear elas-tic springs lying under the beam surface. The foundationmodulus is determined by the stiffness of the springs andthe reaction force of foundation taken to be linearly pro-portional to the beam deflection at any point [43,44]. TheWinkler model does not take into account the continuityand cohesion of the elastic environment. A more realis-tic and accurate model of the elastic foundation can beobtained by a two parameters elastic foundation whichis known as the Winkler-Pasternak model. The Paster-nak elastic foundation model [45] simulate the transverseshear stress due to the shear deformation of the medium,while the Winkler foundation model account for the nor-mal pressure from the surrounding elastic medium. InFigure 1, a part of deformed NW with length under bend-ing deformation and external static transverse load q(x) isshown. The NW is modeled with a Timoshenko beam ele-ment lying on Winkler-Pasternak elastic substrate model.

A. Khajeansari and G.H. Baradaran: Mechanics & Industry 13, 163–174 (2012) 165

Fig. 1. (a) Shear deformation effect on the configuration of an element of NW on Winkler-Pasternak elastic type foundation,(b) the circular cross section of NW including different elastic modulus of bulk and surface layer with the assumption of t � D.

The model contains the outer surface layer with the resid-ual surface stress which is represented by the blue arrowsin longitudinal direction of NW and the bulk volume atthe core.

The surface stress τ s is related to the surface strainεs through the Shuttleworth’s equation [46],

τ s = τs0I +

∂γ(εs)∂εs

(1)

where γ is the surface strain energy density and τ s0 is the

strain-independent surface/interfacial stress at εs, whichis known as residual surface stress and I is the unit tensorfor the surface. The surface elasticity tensor S is definedby

τ s = τ s0I + S : εs (2)

where S is specified as

S =∂τ s(εs)∂εs

∣∣∣∣εs=0

(3)

In fact the positive definiteness of the bulk elastic modu-lus tensor which satisfies the stability of the solid cannotbe applied to the surface elastic tensor. This is arisingfrom the fact that the surface cannot exist independentof the bulk, and the total energy which is compound ofthe bulk energy and the surface energy, needs to satisfythe positive definiteness condition of the elastic modulustensor of the whole material [10].

In one dimensional bending problems presented in thisstudy, the relation between surface stress and infinitesimalsurface strain in longitudinal direction is reduced to

τs = τs0 + Esεs (4)

where Es (N.m−1) is the surface elastic modulus which iscalculated from the surface elastic constants [36]:

Es =(S1111 + 2S1122)(S1111 − S1122)

(S1111 + S1122)(5)

According to Figure 1b the circular cross section of theNW is a composite material compound of a bulk volumewith the elastic modulus of Eb, the thickness of D and asurface layer around the cross section by the elastic modu-lus and thickness of Es

1 and ts respectively. The thicknessof the surface layer ts is assumed to be much smaller thanthe thickness of the cross section ts � D. The relation-ship between Es

1 and Es is stated as Es = Es1t

s [32].Based on the composite beam theory and with the

assumption that ts � D, the effect of surface elasticity istaken into account in the effective flexural rigidity of NWas follows [36].

(EI)∗ = EbIb + Es1I

s =πEbD4

64+π

8EsD3 (6)

Due to the presence of surface stress, the traction vec-tor field across the outer surface of the NW is discon-tinuous and has to verify the Laplace-Young equation asfollows [33–36]

⟨σ+

ij − σ−ij

⟩ninj = τs

αβκαβ (7)

⟨σ+

ij − σ−ij

⟩is the stress jump across the surface, σ+

ij andσ−

ij represent the stress at upmost and bottommost layerof the surface respectively. ni is the unit vector normal tothe surface, τs

αβ is the surface stress and καβ denotes thecurvature tensor. Applying the Timoshenko beam theoryand assuming small deformation, the curvature κ is ap-proximated by the first derivative of the rotation angle ofthe beam cross section due to pure bending as ϕ′(x).

The stress tensor σij is generally acting on the volumeof the body and is three dimensional. Therefore the in-dices of i, j varies as i, j = 1, 2, 3 while the surface stressacting in a two dimensional surface intrinsically and theGreek indices take the values α, β = 1, 2. Assuming smalldeflection of the beam, a distributed transverse load qs(x)is imposed along the longitudinal direction of deformed


NW which is coming from the stress jump described inEquation (7).

qs(x) = Hϕ′(x) (8)The parameter H for circular cross section is definedby [36].

H = 2τs0D (9)

The direction of qs(x) depends on the signs of deformedbeam curvature and residual surface stress of the beam. Itis noted that the sign of the residual surface stress of ma-terials depends on the surface crystallographic directionand can be either positive or negative. In Equations (6)and (8), are captured the surface parameters effects asthe surface Young modulus Es and the residual surfacestress τs

0 in analysis of the NW.By this approach, determining the mechanical prop-

erties and behaviors of the different nanomaterials withdifferent surface crystallographic orientations is straight-forward. Basically, it is one of the important advantagesof the above approach.

In static bending deformation of beams, when dimen-sions of the cross-section are not small compared to thelength, it is needed to consider the effect of shear deforma-tion. This effect is modeled by Timoshenko beam theorywhich is well-known as the thick beam theory. As it isseen in Figure 1

γ = ϕ− ∂v

∂x(10)

where the parameter of γ signifies the shear deformationand v is the transverse displacement of the beam. Thebending moment M and shear force V are functions of vand ϕ given as follows

M(x) = −(EI)∗∂ϕ(x)∂x

, V (x) = αsAG

(∂v(x)∂x

− ϕ(x))

(11)where G is the shear modulus, A is the cross-sectionof area and αs is the shear coefficient which is func-tion of the cross sectional shape. The shear coefficientis αs = 5(1+υ)(6+5υ)−1 for a rectangular cross section,and is αs = 6(1+υ)2(7+12υ+4υ2)−1 for a circular crosssection [47]. Based on the Timoshenko beam theory, thegoverning differential equations for static bending defor-mation of a NW under the total transverse distributedforce p(x) are given by:

(EI)∗∂2ϕ

∂x2− αsAGϕ + αsAG

∂v

∂x= 0 (12)

αsAG∂2v

∂x2− αsAG

∂ϕ

∂x= −p(x) (13)

In presence of surface effect and Winkler-Pasternak elas-tic type substrate, the total transverse distributed forcep(x) acting on NW is as follows

p(x) = qs(x) + qf (x) + q(x) (14)

where qs(x), qf (x) and q(x) are the transverse distributedforces corresponding to residual surface stress and two pa-rameters elastic foundation and external stimulus respec-tively. qf (x) can be written as [48]

qf (x) = −Cwv(x) + Cp∇2v(x) (15)

The Winkler modulus Cw (nN.nm−2) and Pasternakmodulus Cp (nN) are computed by multiplying the di-ameter of the NW by the Winkler and Pasternak stiff-ness constants of Kw (nN.nm−3), Kp (nN.nm−1) respec-tively. This is needed for the consistency of term units ofEquation (14) that they should be nN.nm−1.

By substituting Equations (8) and (15) into Equa-tion (14), p(x) is determined by

p(x) = Hϕ′(x) − Cwv(x) + Cp∇2v(x) + q(x) (16)

It must be mentioned that in the present study q(x) iszero because the external load is imposed to the NWas a concentrated force, so, the effect of concentratedexternal force is accounted in the boundary conditions.The substrate medium of NWs is assumed to be madeup from polymer. According to the values of Winklerand Pasternak stiffness utilized by Liew [49], the non-dimensional modulus parameters are defined as [50]

Cw =CwL

4

(EI)∗, Cp =

CpL2

(EI)∗(17)

where Cw and Cp represent the non-dimensional Winklerand Pasternak modulus parameters respectively.

In order to decouple Equations (12) and (13), we useEquation (13) and its second derivative to determine ϕ′and ϕ′′′ in terms of derivatives of v; then with substitutingϕ′ and ϕ′′′ in the first derivative of Equation (12), thefollowing differential equation is derived:[(EI)∗(αsAG+ Cp)

d4v(x)dx4

−[(EI)∗Cw+(αsAGH + Cp)]]

× d2v(x)dx2

+ [αsAGCw ] v(x) = 0 (18)

The general solution for Equation (16) can be written as

v(x) = Aemx (19)

Substituting Equation (19) into Equation (18) yield thefollowing characteristic equation

[(EI)∗(αsAG+ Cp)] m4 − [(EI)∗Cw + αsAG(H + Cp)]

×m2 + [αsAGCw] m = 0 (20)

The roots of above equation can be obtained as

m1,2 = ±ξ1m3,4 = ±ξ2

ξ1 =√

22α

√α

[β +

√λ]

ξ1 =√

22α

√α

[β −

√λ]

(21)

where

α = (EI)∗ (αsAG+ Cp)β = (EI)∗Cw + αsAG(H + Cp)

λ = β2 − 4α(αsAGCw) (22)


In the present study, the substrate medium assumedto be polymer and the range of non-dimensional Win-kler and Pasternak modulus parameter values are con-sidered as the same as those used by Liew et al. [49].Consequently, two circumstances may occur in the so-lution process according to the roots of characteristicEquation (20). The values of ξ1 and ξ2 are complex con-jugate if λ < 0, and they are real if λ > 0. When the rootsof characteristic Equation (20) are complex conjugate asm1,2,3,4 = ±p± iq, the general solution for the governingequation of a NW lying on two parameters elastic typesubstrate medium based on the Timoshenko beam theoryis obtained as

v(x) = epx [C1 cos(qx) + C2 sin(qx)] + e−px

× [C3 cos(qx) + C4 sin(qx)] (23)

when the roots of the characteristic equation (ξ1 andξ2) are real, the general solution of the governing Equa-tion (18) becomes,

v(x) = C1eξ1x + C2e

ξ2x + C3eξ3x + C4e

ξ4x (24)

v(x) is the transverse displacement of NW. Determiningϕ′′ from the first derivative of Equation (13) and substi-tuting in Equation (12), ϕ(x) can be written in terms ofv(x) as follows

ϕ(x) =(

1η

) [α

d3v(x)dx3

+ μdv(x)dx

](25)

where

η = αsAG(αsAG−H)μ = η − (EI)∗Cw (26)

when the roots of the characteristic equation to be com-plex conjugate, the parameters p and q in Equation (23)can be obtained from a mathematical procedure as below

z1 = αβ; z2 = α√−λ

θ = tan−1

(z2z1

)= tan−1

(√−λβ

)

z =√z21 + z2

2 = α√β2 + λ (27)

p =√

22α

(z)1/2 cos(

12θ

)=

√2

2α

(α√β2 + λ

)1/2

× cos(

12

tan−1

(√−λβ

))(28a)

q =√

22α

(z)1/2 sin(

12θ

)=

√2

2α

(α√β2 + λ

)1/2

× sin(

12

tan−1

(√−λβ

))(28b)

In Equations (23) and (24), C1 − C4 are the unknowncoefficients that will be determined from the boundaryconditions. In the present study the unknown coefficientshave been determined for doubly simply supported (S-S), doubly clamped (C-C) and cantilevered (C-F) NWsexplicitly. In appendices A and B the coefficients C1−C4

are given for (S-S) NWs.The boundary conditions for NWs with two ends sim-

ply supported (S-S) and subjected to a concentrated forceat the midpoint of the NW are given as

v(0) = 0 (29a)ϕ(L/2) = 0 (29b)

M(0) = −(EI)∗ϕ′(0) = 0 (29c)

V (0) = αsAG(v′(0) − ϕ(0)) − F

2+ Cw

×L/2∫0

v(x)d(x) − Cp(v′(L/2)− v′(0)) +Hϕ(0)

(29d)

The boundary conditions for two end clamped (C-C) NWswith a concentrated force F at its mid-point are

v(0) = 0 (30a)

ϕ(0) = 0 (30b)

ϕ(L/2) = 0 (30c)

V (0) = αsAG(v′(0) − ϕ(0)) − F

2+ Cw

×L/2∫0

v(x)d(x) − Cp(v′(L/2)− v′(0)) (30d)

Due to the symmetry, deflection of the NW for twoaforementioned boundary conditions are investigated forx ∈ [0, L/2].

The boundary conditions for cantilevered (C-F) NWscan be written as follows

v(0) = 0 (31a)

ϕ(0) = 0 (31b)

V (0) = αsAG(v′(0) − ϕ(0)) − F + Cw

L∫0

v(x)d(x)

− Cp(v′(L) − v′(0)) −Hϕ(L) (31c)

−M(0) = (EI)∗ϕ′(0) − FL+ Cw

L∫0

xv(x)dx

− Cp(Lv′(L) − v(L)) −H

⎛⎝Lϕ(L)−

L∫0

ϕ(x)dx

⎞⎠

(31d)


3 Results and discussion

3.1 Effects of size and shear deformation on thenormalized contact stiffness of NWs lyingon Winkler-Pasternak elastic type substrate

The contact stiffness which is basically determined un-der static bending of NW, is defined as

Ki =Fi

vi(32)

where Fi is a force prescribed at point i inducing a deflec-tion vi. Here, it is assumed that the point i is the centralpoint for (C-C) and (S-S) NWs and is at the free end forthe (C-F) NWs.

To represent the effects of size (surface) and sheardeformation on the contact stiffness of NWs lying onWinkler-Pasternak substrate, we define normalized stiff-ness of Ks as below.

Ks =Ksf

KEf

(33)

Where KEf denotes the contact stiffness of the conven-

tional E.B.B with two parameters elastic foundation. Ksf

represents the contact stiffness of a NW modeled by eitherE.B.B (Euler Bernoulli beam) or T.B (Timoshenko beam)theories lying on the elastic foundation with consideringthe surface effects.

In other words, Ksf is the contact stiffness of aNW with considering both the size-dependence and elas-tic foundation effects. According to the above defini-tions, when Ksf corresponds to the E.B.B theory, then,Ks characterizes the attribution of the size-dependency,while, when Ksf corresponds to the T.B theory, then,Ks describes the attribution of both the size-dependencyand shear deformation on the stiffness of NWs lying onWinkler-Pasternak elastic substrate.

Variation of the normalized stiffness of a silver NW,Ks, corresponds to the E.B.B and T.B theories versusthe aspect ratio (L/D) of the NW are shown in Figure 2for two diameter values of 20 and 80 nm. The mechanicalproperties of the silver NW are considered as the followingvalues: Eb = 76 GPa, G = 30 GPa and υ = 0.26 whereEb, G and υ are the Young modulus, shear modulus andPoisson ratio of the bulk of silver NW respectively [37,51].The residual surface stress and the elastic modulus of thesurface with crystallographic direction of [001] for a silverNW are: τs

0 = 0.89 N.m−1, Es = 1.22 N.m−1 [10]. Thenon-dimensional Winkler and Pasternak modulus param-eters are considered as Cw = 30 and Cp = 3 respectively.Figure 2 illustrates the variation of the normalized contactstiffness Ks and corresponds to a (S-S) NW with respectto the aspect ratio for the two different NW diameters andbased on E.B.B and T.B theories. With increasing the as-pect ratio (L/D), the NW becomes thinner and the valueof surface per volume ratio increases, therefore the effectof surface on the mechanical properties of NW increases.Consequently, the normal stiffness grows in all cases due

Fig. 2. Normalized contact stiffness of Ks respect to aspectratio (L/D) for (S-S) NWs using by the E.B.B and T.B as-sumptions.

to the surface effects as shown in Figure 2. This growth ismore dominant for the thinner NW (D = 20 nm). Com-paring the normal stiffness for NWs with low aspect ratios(small L/D), obtained by the E.B.B and T.B assump-tions in Figure 2, substantial difference is observed. Thisdifference is due to the shear deformation. The shear de-formation effect is taken to account in T.B theory whiledisregarded in E.B.B theory.

By increasing the aspect ratio of NWs, the shear de-formation effect decreases and the stiffness of thin NWsresulted by the both theories becomes closed together.

To illustrate the combined effect of size and elasticfoundation on the contact stiffness of NWs lying on asubstrate, a normal stiffness Ksf has been defined as

Ksf =Ksf

KE(34)

where KE , represents the contact stiffness of the con-ventional E.B.B contact stiffness of the (without surfaceeffect and elastic foundation) and Ksf is defined as be-fore. Variation of the normalized stiffness Ksf based onE.B.B and T.B theories in terms of aspect ratio (L/D) isshown in Figures 3a–c The results are plotted for the threeboundary conditions of (S-S), (C-C) and (C-F). For thecases of (S-S) and (C-C) NWs, the normal stiffness (Ksf )increases by adding the aspect ratio of NWs; whereas, thenormal stiffness for the (C-F) NWs slightly increases atfirst and, then uniformly decreases. This trend cannot berecognized based on E.B.B theory. By the assumption ofE.B.B, the normal stiffness for (C-F) NWs monotonicallydecreases as the aspect ratio is added.

Again, because of shear deformation, the normal stiff-ness (Ksf ) based on T.B theory is smaller than the


Fig. 3. Normalized contact stiffness of Ksf respect to as-pect ratio (L/D) obtained by the EBB and TB theories. (a)(S-S) NWs, (b) (C-C) NWs and (c) (C-F) NWs.

corresponding normal stiffness based on E.B.B theory forsmall values of aspect ratios.

It is interesting that for the (C-C) NW (Fig. 3b), thenormalized stiffness Ksf turn out to be unity for a specialvalues of L/D. It means that the shear deformation effectovercomes the surface effect and effect of elastic substrateand the NW behaves as simple E.B.B.

3.2 Effects of elastic foundation and sheardeformation on the normalized contact stiffnessof NWs lying on Winkler-Pasternak elastic typesubstrate

We define normal stiffness of Kf as follows to describethe foundation effects on the contact stiffness of NWs.

Kf =Ksf

KEs

(35)

where,KEs is the contact stiffness of a NW without elastic

substrate by considering surface effect based on E.B.Bassumption and Ksf is defined as before, i.e., Ksf is thecontact stiffness of a NW with considering both the size-dependency and elastic foundation effects based on eitherE.B.B or T.B assumption. When Ksf corresponds to theE.B.B theory, then, Kf characterizes the attribution ofthe elastic substrate, while, as Ksf corresponds to theT.B theory, then,Kf describes the attribution of both thefoundation and shear deformation effects on the stiffnessof NWs lying on Winkler-Pasternak elastic substrate.

The non-dimensional Winkler and Pasternak modulusparameters for the substrate polymer medium are takento be in the range of 0 to 300 and 0 to 10 respectively ac-cording to Liew et al. [49], since they have used the simi-lar values for modulus of Winkler and Pasternak founda-tions in vibration analysis of grapheme sheets embeddedin polymer matrix. In Figures 4a–c and 5a–c, the influ-ence of the non-dimensional Winkler Cw and PasternakCp modulus parameters on the local contact stiffness ofNWs lying on elastic foundation is investigated using theE.B.B and T.B theories.

3.2.1 Effect of Winkler modulus parameter Cw

on the normalized contact stiffness of NWs lyingon Winkler elastic type foundation (Cp = 0)

The influence of non-dimensional Winkler modulusparameter Cw (while Cp = 0) on the normalized contactstiffness of Kf has been investigated for two sizes of NWswith geometric dimensions of L = 600 nm, D1 = 40 nm(L/D = 15) and D2 = 100 nm (L/D = 6) and the samemechanical properties are considered as the previous ex-ample. The results for the three types of boundary condi-tions of (S-S), (C-C) and (C-F) are shown in Figures 4a–c.The surface crystallographic direction is [001] for the allcases. The results based on both E.B.B and T.B theoriesare presented in Figures 4a–c. As depicted from the fig-ures, the normal stiffness of NWs increases linearly when


Fig. 4. EBB and TB solutions of normalized stiffness of Kf

due to increasing the Winkler elastic modulus Cw of NWs lying

on Winkler elastic type substrate(Cp = 0

)and two different

aspect ratios, (a) (S-S) NW, (b) (C-C) NW and (c) (C-F) NW.

Fig. 5. EBB and TB solutions of normalized stiffness ofKf due to increasing the Pasternak elastic modulus Cp ofNWs lying on two parameters Winkler-Pasternak elastic sub-

strate model(Cw = 30

)and two different aspect ratios, (a)

(S-S) NW, (b) (C-C) NW, (c) (C-F) NW.


the Winkler modulus parameter is added. According tothe figures, among the three boundary conditions, the (C-F) NW is the most affected by the Winkler modulus. Itis also depicted from the figures, holding the length ofthe NW as constant (600 nm), for the (S-S) and (C-C)boundary conditions, the contact stiffness decreases as theaspect ratio L/D increases. While, for the (C-F) NW thistrend is reversed. Comparing the results corresponds tothe E.B.B and T.B assumptions, again, clear that theshear deformation is more considerable for shorter NWs(L/D = 6) and for large values of Winkler parameter.

3.2.2 Effect of non-dimensional Pasternak modulusparameter Cp on the normalized contact stiffnessof NWs lying on Winkler elastic type foundation

The effect of non-dimensional Pasternak modulus pa-rameter Cp on the normalized stiffness Kf for the silverNWs is investigated in Figures 5a–c. It is assumed thatthe Winkler modulus parameter is constant (Cw = 30).The curves are drawn for two values of aspect ratiosL/D = 6 and L/D = 15, and the three types of bound-ary conditions are considered as before. As depicted fromthe figures, the overall trends for the variations of Kf

in terms of Cp are the same as Kf in terms of Cw for(S-S) and (C-C) NWs. However, for the (C-F) NWs thecontact stiffness Kf decreases as the Cp is increased. Inother words, by increasing the non-dimensional Pasternakmodulus parameter Cp with two values of aspect ratio asL/D = 6 and L/D = 15 and the three types of bound-ary conditions, the (S-S) and (C-C) NWs become stiffer,while, (C-F) NWs become softer.

3.3 Determining the effective Young’s modulusof NWs and size-dependent behavior

To show the size-dependent mechanical properties ofNWs and verifying the presented results, the effectiveYoung’s modulus of a clamped-clamped silver NW havebeen calculated by equating the deflection of the middlepoints of the present NW and the analytical solution ofthe classical E.B.B model as follows [36]

vNW (c−c)

∣∣at x=L/2 (from present modeling)

=F (3L− 4x)x2

48(EeffI)

∣∣∣∣at x=L/2

⇒ Eeff (36)

For determining the effective Young’s modulus of a sil-ver NW with the surface crystallographic direction of[001], the parameters are considered as τs

0 = 0.89 N.m−1,Es = 1.22 N.m−1, Eb = 76 GPa, G = 30 GPa, υ = 0.26and L = 800 nm. Figure 6 shows variation of the effectiveYoung’s modulus obtained by the above method, versusNW diameter. In this figure, the results obtained by thecontinuum modeling of NWs based on the E.B.B. andT.B. theories associated with the surface effect are com-pared with the experimental results of Jing et al. [17].

Fig. 6. Effective Young’s modulus due to the diameter of silverNW [001] with clamped-clamped boundary condition.

A good agreement between the present approach andthe experimental results for a silver NW [001] can beseen in Figure 6. With increasing NW diameter, the ef-fective Young’s modulus decreases and when the diame-ter reaches to a limit value, the size dependency of theYoung’s modulus is eliminated. For the large values ofthe aspect ratio (L/D) (thinner NWs), a small alterationin diameter strongly affects the Young’s modulus. It isseen in Figure 6 that the T.B theory comparing with theE.B.B theory gives the better prediction for the values ofthe Young’s modulus in the region with the lower valuesof the aspect ratio. Therefore it is found that for thickerNWs considering the shear deformation is needed.

4 Conclusion

In this work, explicit solutions were obtained for staticbending response of nanowires (NWs) lying on elasticsubstrate subjected to a concentrated load. The surfacecrystallographic direction was considered to be [001]. Theinfluences of different parameters including size, elasticfoundation, shear deformation and their combinationswere investigated. We used the Timoshenko beam (T.B)theory in tandem with the Laplace-Young equation forthe surface effect and Winkler-Pasternak elastic founda-tion for modeling of the substrate medium. The solutionswere obtained for three kinds of boundary conditions: can-tilever (C-F), simply supported (S-S) and clamped (C-C).The main findings can be summarized as follows.

1. Based on T.B theory, size-dependency (the effect ofsurface) of the mechanical properties increases withincreasing the aspect ratio of NWs. Accordingly, therate of variations of stiffness increases when the NWsbecome thinner. The trend of stiffness variations is dif-ferent for various boundary conditions. For (S-S) and


(C-C) NWs, the contact stiffness of NWs increases byincreasing the aspect ratio of NWs, while, the contactstiffness for the (C-F) NWs initially increases slightly,and then decreases uniformly. This behavior for (C-F)NWs cannot be seen based on Euler-Bernoulli beam(E.B.B) theory.

2. For NWs with low aspect ratios (small L/D), the sheardeformation is considerable and cannot be neglected,But for the thinner NWs (large L/D), the shear de-formation effect is negligible. The shear deformationeffect is more prominent in the case of (C-C) bound-ary conditions, and is relatively small for the case of(C-F) boundary conditions.

3. With increasing the Winkler modulus parameter, thecontact stiffness of NWs increases approximately aslinear for all boundary conditions. Although, therate of increasing the stiffness is more prominent for(C-F) NWs.

4. With increasing the Pasternak modulus parame-ter, (S-S) and(C-C) NWs become stiffer, while,(C-F) NWs turn out to be softer.

5. The effect of aspect ratio on stiffness of NWs lyingon elastic substrate is more significant in (C-F) typeof boundary condition compared to the NWs withoutfoundation.

6. Shear deformation is more considerable for higher val-ues of elastic foundation parameters.

Appendix A. (S-S) NW with complexconjugate roots of characteristic equation

By defining p and q present in Equations (28a)and (28b) as p = Ω/2 and q = ψ/2, the constant coeffi-cients (C1 − C4) of Equation (23) are obtained as follows

C1 =X1

Y; C2 =

X2

Y; C3 = −C1; C4 = −C2 (A.1)

X1 = 4Fη(Ω2 + ψ2

) {Ω

[4αΩ2 + μ− 12α

]× sin(ψL) sinh(ΩL) + ψ

[−4αψ2 + 12αΩ2 + μ]

× cos(ψL) cosh(ΩL)}

(A.2)

X2 = −4Fη(Ω2 + ψ2

) {ψ

[4αψ2 − μ− 12αΩ2

]

× sin(ψL) sinh(ΩL) +Ω[−4αΩ2 + 12αψ2 + μ

]

× cos(ψL) cosh(ΩL)}

(A.3)

Y = αsAG{ [

A8ψ8 +A6ψ

6 +A4ψ4 +A2ψ

2 +A0

]× sin(ψL) +

[A5ψ

5 +A3ψ3 +A1ψ

]× cos(ψL)

}+ Cw

{ [B4ψ

4 +B2ψ2 +B0

]sin(ψL)

+[B3ψ

3 +B1ψ]cos(ψL) +

[B3ψ

3 + B1ψ]

× cos2(ψL)+[B3ψ

3 + B1ψ]+Cp

{[C6ψ

6 +C4ψ4 +C2ψ

2

+ C0

]sin(ψL) +

[C5ψ

5 + C3ψ3 + C1ψ

]cos(ψL)

+[C5ψ

5 + C3ψ3 + C1ψ

]cos2(ψL)+

[C5ψ

5 + C3ψ3 + C1ψ

]}

+H{D8ψ

8 +D6ψ6 +D4ψ

4+D2ψ2 +D0

}sin(ψL)

(A.4)

A8 = −512α2 sinh(ΩL);

A6 = 128α(2μ− 16αΩ2 − η

)sinh(ΩL)

A4 = 32(ημ+ 8αΩ2μ− 96

(αΩ2

)2 − 4ηαΩ2)

sinh(ΩL)

A2 = −64Ω2(4αΩ2μ+ μ2 + 32

(αΩ2

)2 − ημ− 2ηαΩ2)

× sinh(ΩL)

A0 = 32Ω4(ημ− 8αΩ2μ+ 4ηαΩ2 − 16

(αΩ2

)2+ μ2

)× sinh(ΩL)

A5 = 256ηαΩ cosh(ΩL) ; A3 = 512ηΩ3 cosh(ΩL) ; A1

= 256ηαΩ5 cosh(ΩL) (A.5)

B4 = −32ηα sinh(ΩL);

B2 =(192ηαΩ2 + 4ημ

)sinh(ΩL)

B0 = − (32ηαΩ4 + ημΩ2

)sinh(ΩL);

B3 = 128ηαΩ cosh(ΩL)

B1 = −16ηΩ(8αΩ2 + μ) cosh(ΩL); B3 = −128ηαΩ;

B1 = 16η(8αΩ3 + μΩ

)B3 = 64ηαΩ (1 − cosh(2ΩL)) ;

B1 = 8η Ω((μ+ 8αΩ2

)cosh(2ΩL) − 8αΩ2 − μ

)(A.6)

C6 = −128ηα sinh(ΩL) ; C4 = 32η(μ− 4αΩ2

)sinh(ΩL)

C2 = 64ηΩ2 (μ+ 2α) sinh(ΩL) ; C0 = 32ηΩ4(μ+ 4αΩ2

)× sinh(ΩL)

C5 = 256ηΩα cosh(ΩL) ; C3 = 512 ηΩ3α cosh(ΩL)

C1 = 256ηΩ5α cosh(ΩL) ; C5 = −256ηαΩ ;

C3 = −512αΩ3η ; C1 = 256αΩ5η

C5 = −128ηαΩ (cosh(2ΩL) − 1) ;

C3 = −256ηαΩ3 (cosh(2ΩL) − 1)

C1 = −128ηαΩ5 (cosh(2ΩL) − 1) (A.7)


D8 = 512α2 sinh(ΩL) ; D6 = 256α(8αΩ2 − μ

)sinh(ΩL)

D4 = 32(96

(αΩ2

)2+ μ2 − 8αΩ2μ

)sinh(ΩL)

D2 = 64Ω2(μ2 + 32

(αΩ2

)2+ 4αΩ2μ

)sinh(ΩL)

D0 = 32Ω4(16

(αΩ2

)2+ 16μ2 + 8αΩ2μ

)sinh(ΩL)

(A.8)

Appendix B. (S-S) NW with real rootsof characteristic equation

The constant coefficients (C1 − C4) of Equation (24)can be written as follows

C1 =X11

Y1; C2 =

X22

Y1; C3 = −C1 ; C4 = C2 (B.1)

X11 = −8FηΩψ2(4αψ2 + μ

)cosh(ψL) (B.2)

X22 = 8FηΩ2ψ(4αΩ2 + μ

)cosh(ΩL) (B.3)

Y1 =αsAG[A4ψ

4+A2ψ2+A0

]+Cw

[B4ψ

4+B2ψ2 +B0

]+ Cp

[C4ψ

4+C2ψ2]+H

[D4ψ

4 +D2ψ2]

(B.4)

A4 = 256αΩ2[(

4αΩ2 + μ− η)cosh(ψL)

− (4αΩ2 + μ

)cosh(ΩL)

]A2 = 64Ω2

[(4αΩ2η + μη − μ2 − 4αΩ2μ

)cosh(ΩL)

+(4αΩ2μ+ μ2 − μη

)cosh(ψL)

](B.5)

B4 = 64ηα (1 − cosh(ΩL)) cosh(ψL);B2 = 16μη (1 − cosh(ΩL)) cosh(ψL)

B0 = 16Ω2[4αΩ2η cosh(ΩL) cosh(ψL) + ημ cosh(ΩL)

× cosh(ψL) − 4ηαΩ2 cosh(ΩL) − μη cosh(ΩL)]

(B.6)

C4 = 256αΩ2η [(cosh(ΩL) − 1)] cosh(ψL)

C2 = 64Ω2[(ημ+ 4αΩ2η

)cosh(ΩL)

−4αΩ2η cosh(ΩL) cosh(ψL) − μ cosh(ψL)]

(B.7)

D4 = 256αΩ2[(

4αΩ2 + μ)(cosh(ΩL) − cosh(ψL))

]

D2 = 64Ω2μ[(

4αΩ2 + μ)(cosh(ΩL) − cosh(ψL))

](B.8)

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surface effect investigation for static bending of

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