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Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2013 Article ID 360935 6 pageshttpdxdoiorg1011552013360935
Research ArticleA Nonlocal Model for Carbon Nanotubes under Axial Loads
Raffaele Barretta and Francesco Marotti de Sciarra
Department of Structures for Engineering and Architecture University of Naples Federico II Via Claudio 21 80125 Naples Italy
Correspondence should be addressed to Raffaele Barretta rabarretuninait
Received 7 July 2013 Revised 8 October 2013 Accepted 10 October 2013
Academic Editor Jun Liu
Copyright copy 2013 R Barretta and F Marotti de Sciarra This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
Various beam theories are formulated in literature using the nonlocal differential constitutive relation proposed by Eringen A newvariational framework is derived in the present paper by following a consistent thermodynamic approach based on a nonlocalconstitutive law of gradient-type Contrary to the results obtained by Eringen the new model exhibits the nonlocality effect alsofor constant axial load distributions The treatment can be adopted to get new benchmarks for numerical analyses
1 Introduction
Carbon nanotubes (CNTs) are a topic of major interest bothfrom theoretical and applicative points of view This subjectis widely investigated in literature to describe small-scaleeffects [1ndash4] vibration and buckling [5ndash13] and nonlocalfinite element analysis [14ndash18] A comprehensive review onapplications of nonlocal elastic models for CNTs is reportedin [19] and therein references Buckling of triple-walled CNTsunder temperature fields is dealt with in [20] An alternativemethodology is based on an atomistic-based approach [21]which predicts the positions of atoms in terms of interactiveforces and boundary conditions The standard approachto analyze CNTs under axial loads consists in solving aninhomogeneous second-order ordinary differential equationproviding the axial displacement field see for example [22]The known term of the differential equation is the sum oftwo contributions The former describes the local effectslinearly depending on the axial load The latter characterizesthe small-scale effects depending linearly on the secondderivative along the rod axis of the axial load This model isthus not able to evaluate small-scale effects due to constantaxial loads per unit lengthThis approach commonly adoptedin literature is based on the following nonlocal linearly elasticconstitutive law proposed by Eringen [23]
120590 minus 1198902
1199001198862120590(2)
= 119864120576 (1)
where 119890119900is a material constant 119886 is the internal length
119864 is the Young modulus 120590 is the normal stress the apex(∙)(2) is second derivative along the rod axis and 120576 is the
axial elongation Indeed integrating on the rod cross sectiondomainΩ and imposing that the axial force119873 is equal to theresultant of normal stress field we get the differential equation
119873 minus 1198902
1199001198862119873(2)
= 119864119860119908(1)
(2)
where 120576 = 119908(1) with 119908
(1) being first derivative along therod axis of the axial displacement field 119908 [0 119871] 997891rarr Rwhere 119871 is the rod length and 119860 denotes the cross sectionarea Since the equilibrium prescribes that the first derivativeof 119873 is opposite to the axial load 119901 we infer the well-knowndifferential equation (see eg [7]) as follows
119908(2)
=
minus119901 + 1198902
1199001198862119901(2)
119864119860
(3)
Note that the nonlocal contribution vanishes for constantloads 119901 In the present paper an alternative nonlocal consti-tutive behavior is adopted to assess small-scale effects in nan-otubes also for constant axial loads The corresponding axialdisplacement field is shown to be governed by a fourth-orderinhomogeneous differential equation Boundary conditionsare naturally inferred by performing a standard localizationprocedure of a variational problem formulated by makingrecourse to thermodynamic restrictions see for example
2 Advances in Materials Science and Engineering
[24ndash26] according to the geometric approach illustrated in[27ndash30] As an example the displacement field of nanotubesunder constant axial loads per unit length is evaluated in theappendix Vibration and buckling effects are not the subjectof this paper and will be addressed in a forthcoming paper
2 Nonlocal Variational Formulation
Let B be the three-dimensional spatial domain of a straightrod subjected to axial loads An apex (∙)
(119899) stands for 119899thderivative along the rod centroidal 119911-axis Kinematic compat-ibility between axial elongations 120576 and axial displacements 119908
is expressed by the differential equation 120576 = 119908(1) Denoting by
a dot the time-rate the following noteworthy relations holdtrue
120576 = (1)
120576(1)
= (2)
(4)
Thedifferential equation of equilibrium turns out to be119873(1) =minus119901 Boundary equilibrium prescribes that at the end crosssections act axial loads equal to 119873(0) for 119911 = 0 and to 119873(119871)
for 119911 = 119871 Let us now consider a nonlocal constitutive modelof gradient-type defined by assigning the following elasticenergy functional per unit volume
120595 (120576 120576(1)
) =
1
2
1198641205762+
1
2
1198641198882120576(1)2
(5)
with 119888 = 119890119900119886 being nonlocal parameter Relation (5) is
similar to the elastic energy density proposed in [31] wherea homogeneous quadratic functional including also mixedterms is assumed The elastic energy time rate is henceexpressed by the formula
(120576 120576(1)
) =
120597120595
120597120576
120576 +
120597120595
120597120576(1)
120576(1)
= 120590119900
120576 + 1205901
120576(1)
(6)
where
120590119900=
120597120595
120597120576
= 119864120576 1205901=
120597120595
120597120576(1)
= 1198641198882120576(1)
(7)
are the static variables conjugating with the kinematic vari-ables 120576 and 120576
(1)The static variable 1205901is the scalar counterpart
of the so-called double stress tensor [31] By imposing thethermodynamic condition (see eg [32ndash34])
int
B
120590 120576119889119881 minus int
B
119889119881 = 0 (8)
where 120590 is the normal stress we infer the relation
int
B
120590 120576119889119881 = int
B
120590119900
120576119889119881 + int
B
1205901
120576(1)
119889119881 (9)
The relevant differential and boundary equations are thusobtained as shown hereafter Substituting the expression of
the rates 120576 and 120576(1) in terms of the axial displacement 119908(119911) of
the cross section at abscissa 119911 we get the formulae
int
B
120590 120576119889119881 = int
B
120590(1)
119889119881 = int
119871
0
(int
Ω
120590119889119860) (1)
119889119911
= int
119871
0
119873(1)
119889119911
int
B
120590119900
120576119889119881 = int
119871
0
119873119900(1)
119889119911
int
B
1205901
120576(1)
119889119881 = int
B
1205901(2)
119889119881 = int
119871
0
(int
Ω
1205901119889119860)
(2)119889119911
= int
119871
0
1198731(2)
119889119911
(10)
with119873 = intΩ
120590119889119860 axial force (static equivalence condition onthe cross sections) and 119873
119894= intΩ
120590119894119889119860 for 119894 isin 0 1 Thermo-
dynamic condition (9) provides the axial contribution
int
119871
0
119873(1)
119889119911 = int
119871
0
119873119900(1)
119889119911 + int
119871
0
1198731(2)
119889119911 (11)
3 Differential and Boundary Equations ofElastic Equilibrium
Resorting to Greenrsquos formula a standard localization proce-dure provides the differential and boundary equations corre-sponding to the variational conditions inferred in Section 2as follows A direct computation gives
int
119871
0
119873(1)
119889119911 = [119873 ]119871
0minus int
119871
0
119873(1)
119889119911
int
119871
0
119873119900(1)
119889119911 = [119873119900]119871
0minus int
119871
0
119873(1)
119900119889119911
int
119871
0
1198731(2)
119889119911 = [1198731(1)
]
119871
0minus int
119871
0
119873(1)
1(1)
119889119911
= [1198731(1)
]
119871
0minus [119873(1)
1]
119871
0+ int
119871
0
119873(2)
1119889119911
(12)
Substituting into the variational condition (11) a suitablelocalization provides the relevant differential equation
119873(1)
= 119873(1)
119900minus 119873(2)
1(13)
and boundary conditions
119873 = 119873119900minus 119873(1)
1 dual of
0 = 1198731 dual of
(1)
(14)
Advances in Materials Science and Engineering 3
These conditions can be conveniently expressed in termsof the axial displacement field 119908 as follows A direct evalua-tion of the scalar functions 119873
119894 [0 119871] 997891rarr R for 119894 isin 0 1 and
of their derivatives gives
119873119900= int
Ω
120590119900119889119860 = int
Ω
119864120576119889119860 = int
Ω
119864119908(1)
119889119860 = 119864119860119908(1)
119873(119895)
119900= 119864119860119908
(1+119895)
1198731= int
Ω
1205901119889119860 = int
Ω
1198641198882120576(1)
119889119860
= int
Ω
1198641198882119908(2)
119889119860 = 1198641198601198882119908(2)
119873(119895)
1= 119864119860119888
2119908(2+119895)
(15)
with 119895 isin 1 2 119899 Accordingly the boundary anddifferential conditions of elastic equilibrium (13) and (14) takethe form
119873(1)
= 119873(1)
119900minus 119873(2)
1= 119864119860119908
(2)minus 119864119860119888
2119908(4)
119873 = 119873119900minus 119873(1)
1= 119864119860119908
(1)minus 119864119860119888
2119908(3)
dual of
0 = 1198731= 119864 119860 119888
2119908(2)
dual of (1)
(16)
4 Example
Let us consider a straight rod subject to a constant axial load119901 as depicted in Figure 1 End cross sections A and B areassumed to be hinged and simply supported respectivelyAs illustrated in Section 3 the computation of the rod axialdisplacement field 119908 involves the following cross sectiongeometric and elastic properties area 119860 Young modulus 119864and nonlocal parameter 119888 By setting 120572 = 119864119860119888
2 and 120573 = 119864119860the differential equation of elastic equilibrium is as follows
120572119908(4)
minus 120573119908(2)
= 119901 (17)
The general integral takes thus the form (see the appendix)
119908 (119911) = 119908119867
(119911) + 119908 (119911) (18)
with
119908119867
(119911) = 1198881+ 1198882119911 + 1198883exp(radic
120573
120572
119911)
+ 1198884exp(minusradic
120573
120572
119911)
119908 (119911) = 11988851199112
(19)
z
L
p
N(0)
A B
Figure 1 Rod under constant axial load
The evaluation of the constants is carried out by imposing thefollowing boundary conditions (see also Section 3)
119908 (0) = 0
119873 (119871) = 119864119860119908(1)
(119871) minus 1198641198601198882119908(3)
(119871) = 0
1198731(0) = 119864119860119888
2119908(2)
(0) = 0
1198731(119871) = 119864119860119888
2119908(2)
(119871) = 0
119908 (0) = 0
119908(1)
(119871) minus 1198882119908(3)
(119871) = 0
119908(2)
(0) = 0
119908(2)
(119871) = 0
(20)
Resorting to the expressions of the derivatives 119908(119895)
119867and 119908
(119895)
for 119895 isin 1 2 3 4
119908(1)
119867(119911) = 119888
2+
1198883
119888
exp(
1
119888
119911) minus
1198884
119888
exp(minus
1
119888
119911)
119908(2)
119867(119911) =
1198883
1198882exp (
1
119888
119911) +
1198884
1198882exp (minus
1
119888
119911)
119908(3)
119867(119911) =
1198883
1198883exp (
1
119888
119911) minus
1198884
1198883exp (minus
1
119888
119911)
119908(4)
119867(119911) =
1198883
1198884exp (
1
119888
119911) +
1198884
1198884exp (minus
1
119888
119911)
119908(1)
(119911) = 21198885119911
119908(2)
(119911) = 21198885
119908(3)
(119911) = 0
119908(4)
(119911) = 0
(21)
and having radic120573120572 = 1119888 a direct computation provides thealgebraic system
1198881+ 1198883+ 1198884= 0
1198882+ 21198711198885= 0
1
11988821198883+
1
11988821198884+ 21198885= 0
1
1198882exp(
1
119888
119871) 1198883+
1
1198882exp (minus
1
119888
119871) 1198884+ 21198885= 0
(22)
4 Advances in Materials Science and Engineering
20
15
10
05
2 4 6 8 10
z (nm)
w(n
m)
Figure 2 Axial displacement field119908 for 119888 = 0 (local solutionmdashblackline) 119888 = 1 nm (blue line) 119888 = 2 nm (red line) 119888 = 3 nm (greenline) 119888 = 4 nm (orange line) 119888 = 5 nm (yellow line) and 119888 = 25 nm(brown line) 119864 = 300GPa 119871 = 10 nm 119860 = 80 sdot 10
minus2 nm2 and119901 = 10
minus8 Nnm
A further condition can be obtained by imposing that thescalar field
119908 (119911) = 11988851199112 (23)
is a particular solution of the differential equation (17)whence it follows that 119888
5= minus1199012119864119860The remaining constants
are given by the formulae
1198881= minus (119888
3+ 1198884)
1198882= minus2119871119888
5
1198883=
21198882(1 minus exp (minus (1119888) 119871))
exp (minus (1119888) 119871) minus exp ((1119888) 119871)
1198885
1198884=
21198882(1 minus exp ((1119888) 119871))
exp ((1119888) 119871) minus exp (minus (1119888) 119871)
1198885
(24)
having 1198885
= minus1199012119864119860 A plot of the rod axial displacementfield 119908 for different values of the nonlocal parameter 119888 isprovided in Figure 2 It is apparent that the rod becomesstiffer if the nonlocal parameter increasesThe evaluated axialdisplacement at the free end of the rod B provides the samevalue independently of the nonlocal parameter Such a valuecoincideswith the displacement of the pointB if a localmodelis considered Moreover the limit of the axial displacementfield for 119888 tending to plus infinity can be evaluated to get thelower bound
119908low
(119911) = lim119888rarr+infin
119908 (119911 119888) = 0208333119911 (25)
Hence large values of the nonlocal parameter provide adisplacement field which tends to a linear one see Figure 2for 119888 = 25 Further the limit value of the axial displacementfor 119911 = 119871 and 119888 rarr +infin obtained by (25) yields 119908
low(119871) =
208333 nm which coincides with the axial displacement at Bfor any value of the nonlocal parameter 119888 see Figure 3 andTable 1
0 5 10 15 20 250
05
1
15
2
25
c (nm)
w(L2)
andw(L)
(nm
)
Figure 3 Axial displacement in terms of the nonlocal parameter 119888
at the abscissa 119911 = 1198712 (blue line) and 119911 = 119871 (red line)
Table 1 Axial displacements 119908(1198712) and 119908(119871) versus the nonlocalparameter 119888
119888 (nm) 119908(1198712) (nm) 119908(119871) (nm)0 15625 2083331 152139 2083332 142301 2083333 132428 2083334 124886 2083335 119589 20833325 105021 208333
The upper bound of the axial displacement is provided bythe local solution (ie 119888 = 0)
119908upp
(119911) = 119908 (119911 0) =
119901 (119911) (2119871 minus 119911) 119911
2119864119860
(26)
The axial displacement evaluated for 119911 = 119871 by (26) yieldsthe value119908
upp(119871) = 2512 nm which coincides with the axial
displacement at B for any value of 119888 see Figure 3 and Table 1For the considered model the upper and lower bounds ofthe axial displacement field are given by (25) and (26) Theaxial displacement 119908(1198712) at the middle point of the rodand the maximum axial displacement 119908(119871) as functionsof the nonlocal parameter 119888 are depicted in Figure 3 Thecorresponding numerical values of119908(1198712) and119908(119871) are listedin Table 1
It is worth noting that equilibrium prescribes that axialforce 119873 must be a linear function confirmed by the bluediagram in Figure 4 obtained as difference between the localcontribution 119873
119900(dashed line) and the nonlocal one 119873
(1)
1
(continuous thin line) according to (14)1for any value of 119888
5 Conclusions
The outcomes of the present paper may be summarized asfollows
(i) Linearly elastic carbon nanotubes under axial loadshave been investigated by a nonlocal variationalapproach based on thermodynamic restrictions Thetreatment provides an effective tool to evaluate small-scale effects in nanotubes subject also to constant
Advances in Materials Science and Engineering 5
1 times 10minus7
2 4 6 8 10
8 times 10minus8
6 times 10minus8
4 times 10minus8
2 times 10minus8
minus2 times 10minus8
minus4 times 10minus8
z (nm)
N (N
)
Figure 4 Axial force 119873 = 119873119900minus 119873(1)
1(blue line) 119873
119900(dashed line)
119873(1)
1(continuous thin line) 119888 = 0 (local solution-black line) 119888 =
1 nm (blue line) 119888 = 2 nm (red line) 119888 = 3 nm (green line) 119888 = 4 nm(orange line) 119888 = 5 nm (yellow line) and 119888 = 25 nm (brown line)
axial loads a goal not achievable by the Eringenmodel commonly adopted in literature as motivatedin Section 1
(ii) Relevant boundary and differential conditions ofelastic equilibrium have been inferred by a standardlocalization procedure Such a procedure providesin a consistent way the relevant class of boundaryconditions for the nonlocal model
(iii) The present approach yields a firm thermodynamicprocedure to derive different nonlocal models forCNTs by suitable specializations of the elastic energy
(iv) Exact solutions of carbon nanotubes subject to aconstant axial load have been obtained An advantageof the proposed procedure consists in providing aneffective tool to be used as a benchmark for numericalanalyses Finally a range to which any nonlocalsolution must belong is analytically evaluated
Appendix
The procedure to solve the ordinary differential equation
120572119908(4)
minus 120573119908(2)
= 119891 (A1)
with 120572 120573 gt 0 being constant coefficients and 119891 119868 sube R 997891rarr Rbeing a continuous function is summarized as follows Let usconsider the homogeneous differential equation
120572119908(4)
minus 120573119908(2)
= 0 (A2)
and the relevant characteristic (algebraic) equation 1205721205824
minus
1205731205822
= 0 The roots of the polynomial 1205721205824 minus 1205731205822 are 120582
1= 0
with multiplicity 2 1205822
= radic120573120572 with multiplicity 1 and 1205823
=
minusradic120573120572 with multiplicity 1 The general integral of (A2) isthus expressed by the formula
119908119867
(119911) = 1198881+ 1198882119911 + 1198883exp(radic
120573
120572
119911)
+ 1198884exp(minusradic
120573
120572
119911)
(A3)
with exp denoting exponential function and 119888119894isin R for 119894 =
1 4 The general integral of (A1) is writen therefore as
119908 (119911) = 119908119867
(119911) + 119908 (119911) (A4)
where 119908 is a particular solution of (A1) It is worth notingthat for 119891 defined by a polynomial 119901
119898of degree 119898 ge 0 the
solution V can be looked for by setting
119908 (119911) = 1199112
(1198600+ 1198601119911 + sdot sdot sdot + 119860
119898119911119898) (A5)
with 119860119894isin R for 119894 = 1 119898
Acknowledgments
The authors were supported by the ldquoPolo delle Scienze e delleTecnologierdquo University of Naples Federico II through theresearch project FARO Useful hints and precious commentsby anonymous reviewers are also gratefully acknowledged
References
[1] Q Wang and K M Liew ldquoApplication of nonlocal continuummechanics to static analysis of micro- and nano-structuresrdquoPhysics Letters A vol 363 no 3 pp 236ndash242 2007
[2] M Aydogdu ldquoA general nonlocal beam theory its applicationto nanobeam bending buckling and vibrationrdquo Physica E vol41 no 9 pp 1651ndash1655 2009
[3] O Civalek and C Demir ldquoBending analysis of microtubulesusing nonlocal Euler-Bernoulli beam theoryrdquo Applied Mathe-matical Modelling vol 35 no 5 pp 2053ndash2067 2011
[4] M A De Rosa and C Franciosi ldquoA simple approach to detectthe nonlocal effects in the static analysis of Euler-Bernoulli andTimoshenko beamsrdquoMechanics Research Communications vol48 pp 66ndash69 2013
[5] M Aydogdu ldquoAxial vibration analysis of nanorods (carbonnanotubes) embedded in an elastic medium using nonlocalelasticityrdquoMechanics Research Communications vol 41 pp 34ndash40 2012
[6] M A Kazemi-Lari S A Fazelzadeh and E Ghavanloo ldquoNon-conservative instability of cantilever carbon nanotubes restingon viscoelastic foundationrdquo Physica E vol 44 no 7-8 pp 1623ndash1630 2012
[7] H-T Thai and T P Vo ldquoA nonlocal sinusoidal shear deforma-tion beam theory with application to bending buckling andvibration of nanobeamsrdquo International Journal of EngineeringScience vol 54 pp 58ndash66 2012
[8] M A Eltaher S A Emam and F F Mahmoud ldquoStatic andstability analysis of nonlocal functionally graded nanobeamsrdquoComposite Structures vol 96 pp 82ndash88 2013
[9] S A Emam ldquoA general nonlocal nonlinear model for bucklingof nanobeamsrdquo Applied Mathematical Modelling vol 37 no 10-11 pp 6929ndash6939 2013
[10] B Fang Y-X Zhen C-P Zhang and Y Tang ldquoNonlinearvibration analysis of double-walled carbon nanotubes based onnonlocal elasticity theoryrdquoAppliedMathematicalModelling vol37 no 3 pp 1096ndash1107 2013
[11] S A M Ghannadpour B Mohammadi and J Fazilati ldquoBend-ing buckling and vibration problems of nonlocal Euler beamsusing Ritz methodrdquo Composite Structures vol 96 pp 584ndash5892013
6 Advances in Materials Science and Engineering
[12] M Simsek and H H Yurtcu ldquoAnalytical solutions for bendingand buckling of functionally graded nanobeams based on thenonlocal Timoshenko beam theoryrdquo Composite Structures vol97 pp 378ndash386 2013
[13] B L Wang and K F Wang ldquoVibration analysis of embeddednanotubes using nonlocal continuum theoryrdquo Composites Bvol 47 pp 96ndash101 2013
[14] J K Phadikar and S C Pradhan ldquoVariational formulationand finite element analysis for nonlocal elastic nanobeams andnanoplatesrdquo Computational Materials Science vol 49 no 3 pp492ndash499 2010
[15] C M C Roque A J M Ferreira and J N Reddy ldquoAnalysisof Timoshenko nanobeams with a nonlocal formulation andmeshless methodrdquo International Journal of Engineering Sciencevol 49 no 9 pp 976ndash984 2011
[16] F F Mahmoud M A Eltaher A E Alshorbagy and E IMeletis ldquoStatic analysis of nanobeams including surface effectsby nonlocal finite elementsrdquo Journal of Mechanical Science andTechnology vol 26 no 11 pp 3555ndash3563 2012
[17] S C Pradhan ldquoNonlocal finite element analysis and small scaleeffects of CNTs with Timoshenko beam theoryrdquo Finite Elementsin Analysis and Design vol 50 pp 8ndash20 2012
[18] M A Eltaher A E Alshorbagy and F F Mahmoud ldquoVibrationanalysis of Euler-Bernoulli nanobeams by using finite elementmethodrdquo Applied Mathematical Modelling vol 37 no 7 pp4787ndash4797 2013
[19] B Arash and Q Wang ldquoA review on the application ofnonlocal elastic models in modeling of carbon nanotubes andgraphenesrdquo Computational Materials Science vol 51 no 1 pp303ndash313 2012
[20] Y YanW QWang and L X Zhang ldquoNonlocal effect on axiallycompressed buckling of triple-walled carbon nanotubes undertemperature fieldrdquo Applied Mathematical Modelling vol 34 no11 pp 3422ndash3429 2010
[21] R Rafiee and R M Moghadam ldquoOn the modeling of carbonnanotubes a critical reviewrdquo Composites B vol 56 pp 435ndash4490 2014
[22] J N Reddy ldquoNonlocal theories for bending buckling and vibra-tion of beamsrdquo International Journal of Engineering Science vol45 no 2-8 pp 288ndash307 2007
[23] A C Eringen ldquoOn differential equations of nonlocal elasticityand solutions of screw dislocation and surface wavesrdquo Journalof Applied Physics vol 54 no 9 pp 4703ndash4710 1983
[24] F Marotti de Sciarra ldquoVariational formulations convergenceand stability properties in nonlocal elastoplasticityrdquo Interna-tional Journal of Solids and Structures vol 45 no 7-8 pp 2322ndash2354 2008
[25] G RomanoMDiaco andR Barretta ldquoVariational formulationof the first principle of continuum thermodynamicsrdquo Contin-uumMechanics andThermodynamics vol 22 no 3 pp 177ndash1872010
[26] FMarotti De Sciarra ldquoHardening plasticitywith nonlocal straindamagerdquo International Journal of Plasticity vol 34 pp 114ndash1382012
[27] G Romano and R Barretta ldquoCovariant hypo-elasticityrdquo Euro-pean Journal of Mechanics A vol 30 no 6 pp 1012ndash1023 2011
[28] G Romano and R Barretta ldquoOn Eulerrsquos stretching formula incontinuummechanicsrdquo Acta Mechanica vol 224 no 1 pp 211ndash230 2013
[29] G Romano and R Barretta ldquoGeometric constitutive theoryand frame invariancerdquo International Journal of Non-LinearMechanics vol 51 pp 75ndash86 2013
[30] G Romano R Barretta and M Diaco ldquoGeometric continuummechanicsrdquoMeccanica 2013
[31] R D Mindlin ldquoMicro-structure in linear elasticityrdquo Archive forRational Mechanics and Analysis vol 16 no 1 pp 51ndash78 1964
[32] F Marotti de Sciarra ldquoNovel variational formulations fornonlocal plasticityrdquo International Journal of Plasticity vol 25no 2 pp 302ndash331 2009
[33] F Marotti de Sciarra ldquoOn non-local and non-homogeneouselastic continuardquo International Journal of Solids and Structuresvol 46 no 3-4 pp 651ndash676 2009
[34] F Marotti de Sciarra ldquoA nonlocal model with strain-baseddamagerdquo International Journal of Solids and Structures vol 46no 22-23 pp 4107ndash4122 2009
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thinspInternationalthinspJournalthinspof
BiomaterialsHindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NaNoscieNceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Volume 2014
CrystallographyJournal of
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
BioMed Research International
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Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
MaterialsJournal of
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
2 Advances in Materials Science and Engineering
[24ndash26] according to the geometric approach illustrated in[27ndash30] As an example the displacement field of nanotubesunder constant axial loads per unit length is evaluated in theappendix Vibration and buckling effects are not the subjectof this paper and will be addressed in a forthcoming paper
2 Nonlocal Variational Formulation
Let B be the three-dimensional spatial domain of a straightrod subjected to axial loads An apex (∙)
(119899) stands for 119899thderivative along the rod centroidal 119911-axis Kinematic compat-ibility between axial elongations 120576 and axial displacements 119908
is expressed by the differential equation 120576 = 119908(1) Denoting by
a dot the time-rate the following noteworthy relations holdtrue
120576 = (1)
120576(1)
= (2)
(4)
Thedifferential equation of equilibrium turns out to be119873(1) =minus119901 Boundary equilibrium prescribes that at the end crosssections act axial loads equal to 119873(0) for 119911 = 0 and to 119873(119871)
for 119911 = 119871 Let us now consider a nonlocal constitutive modelof gradient-type defined by assigning the following elasticenergy functional per unit volume
120595 (120576 120576(1)
) =
1
2
1198641205762+
1
2
1198641198882120576(1)2
(5)
with 119888 = 119890119900119886 being nonlocal parameter Relation (5) is
similar to the elastic energy density proposed in [31] wherea homogeneous quadratic functional including also mixedterms is assumed The elastic energy time rate is henceexpressed by the formula
(120576 120576(1)
) =
120597120595
120597120576
120576 +
120597120595
120597120576(1)
120576(1)
= 120590119900
120576 + 1205901
120576(1)
(6)
where
120590119900=
120597120595
120597120576
= 119864120576 1205901=
120597120595
120597120576(1)
= 1198641198882120576(1)
(7)
are the static variables conjugating with the kinematic vari-ables 120576 and 120576
(1)The static variable 1205901is the scalar counterpart
of the so-called double stress tensor [31] By imposing thethermodynamic condition (see eg [32ndash34])
int
B
120590 120576119889119881 minus int
B
119889119881 = 0 (8)
where 120590 is the normal stress we infer the relation
int
B
120590 120576119889119881 = int
B
120590119900
120576119889119881 + int
B
1205901
120576(1)
119889119881 (9)
The relevant differential and boundary equations are thusobtained as shown hereafter Substituting the expression of
the rates 120576 and 120576(1) in terms of the axial displacement 119908(119911) of
the cross section at abscissa 119911 we get the formulae
int
B
120590 120576119889119881 = int
B
120590(1)
119889119881 = int
119871
0
(int
Ω
120590119889119860) (1)
119889119911
= int
119871
0
119873(1)
119889119911
int
B
120590119900
120576119889119881 = int
119871
0
119873119900(1)
119889119911
int
B
1205901
120576(1)
119889119881 = int
B
1205901(2)
119889119881 = int
119871
0
(int
Ω
1205901119889119860)
(2)119889119911
= int
119871
0
1198731(2)
119889119911
(10)
with119873 = intΩ
120590119889119860 axial force (static equivalence condition onthe cross sections) and 119873
119894= intΩ
120590119894119889119860 for 119894 isin 0 1 Thermo-
dynamic condition (9) provides the axial contribution
int
119871
0
119873(1)
119889119911 = int
119871
0
119873119900(1)
119889119911 + int
119871
0
1198731(2)
119889119911 (11)
3 Differential and Boundary Equations ofElastic Equilibrium
Resorting to Greenrsquos formula a standard localization proce-dure provides the differential and boundary equations corre-sponding to the variational conditions inferred in Section 2as follows A direct computation gives
int
119871
0
119873(1)
119889119911 = [119873 ]119871
0minus int
119871
0
119873(1)
119889119911
int
119871
0
119873119900(1)
119889119911 = [119873119900]119871
0minus int
119871
0
119873(1)
119900119889119911
int
119871
0
1198731(2)
119889119911 = [1198731(1)
]
119871
0minus int
119871
0
119873(1)
1(1)
119889119911
= [1198731(1)
]
119871
0minus [119873(1)
1]
119871
0+ int
119871
0
119873(2)
1119889119911
(12)
Substituting into the variational condition (11) a suitablelocalization provides the relevant differential equation
119873(1)
= 119873(1)
119900minus 119873(2)
1(13)
and boundary conditions
119873 = 119873119900minus 119873(1)
1 dual of
0 = 1198731 dual of
(1)
(14)
Advances in Materials Science and Engineering 3
These conditions can be conveniently expressed in termsof the axial displacement field 119908 as follows A direct evalua-tion of the scalar functions 119873
119894 [0 119871] 997891rarr R for 119894 isin 0 1 and
of their derivatives gives
119873119900= int
Ω
120590119900119889119860 = int
Ω
119864120576119889119860 = int
Ω
119864119908(1)
119889119860 = 119864119860119908(1)
119873(119895)
119900= 119864119860119908
(1+119895)
1198731= int
Ω
1205901119889119860 = int
Ω
1198641198882120576(1)
119889119860
= int
Ω
1198641198882119908(2)
119889119860 = 1198641198601198882119908(2)
119873(119895)
1= 119864119860119888
2119908(2+119895)
(15)
with 119895 isin 1 2 119899 Accordingly the boundary anddifferential conditions of elastic equilibrium (13) and (14) takethe form
119873(1)
= 119873(1)
119900minus 119873(2)
1= 119864119860119908
(2)minus 119864119860119888
2119908(4)
119873 = 119873119900minus 119873(1)
1= 119864119860119908
(1)minus 119864119860119888
2119908(3)
dual of
0 = 1198731= 119864 119860 119888
2119908(2)
dual of (1)
(16)
4 Example
Let us consider a straight rod subject to a constant axial load119901 as depicted in Figure 1 End cross sections A and B areassumed to be hinged and simply supported respectivelyAs illustrated in Section 3 the computation of the rod axialdisplacement field 119908 involves the following cross sectiongeometric and elastic properties area 119860 Young modulus 119864and nonlocal parameter 119888 By setting 120572 = 119864119860119888
2 and 120573 = 119864119860the differential equation of elastic equilibrium is as follows
120572119908(4)
minus 120573119908(2)
= 119901 (17)
The general integral takes thus the form (see the appendix)
119908 (119911) = 119908119867
(119911) + 119908 (119911) (18)
with
119908119867
(119911) = 1198881+ 1198882119911 + 1198883exp(radic
120573
120572
119911)
+ 1198884exp(minusradic
120573
120572
119911)
119908 (119911) = 11988851199112
(19)
z
L
p
N(0)
A B
Figure 1 Rod under constant axial load
The evaluation of the constants is carried out by imposing thefollowing boundary conditions (see also Section 3)
119908 (0) = 0
119873 (119871) = 119864119860119908(1)
(119871) minus 1198641198601198882119908(3)
(119871) = 0
1198731(0) = 119864119860119888
2119908(2)
(0) = 0
1198731(119871) = 119864119860119888
2119908(2)
(119871) = 0
119908 (0) = 0
119908(1)
(119871) minus 1198882119908(3)
(119871) = 0
119908(2)
(0) = 0
119908(2)
(119871) = 0
(20)
Resorting to the expressions of the derivatives 119908(119895)
119867and 119908
(119895)
for 119895 isin 1 2 3 4
119908(1)
119867(119911) = 119888
2+
1198883
119888
exp(
1
119888
119911) minus
1198884
119888
exp(minus
1
119888
119911)
119908(2)
119867(119911) =
1198883
1198882exp (
1
119888
119911) +
1198884
1198882exp (minus
1
119888
119911)
119908(3)
119867(119911) =
1198883
1198883exp (
1
119888
119911) minus
1198884
1198883exp (minus
1
119888
119911)
119908(4)
119867(119911) =
1198883
1198884exp (
1
119888
119911) +
1198884
1198884exp (minus
1
119888
119911)
119908(1)
(119911) = 21198885119911
119908(2)
(119911) = 21198885
119908(3)
(119911) = 0
119908(4)
(119911) = 0
(21)
and having radic120573120572 = 1119888 a direct computation provides thealgebraic system
1198881+ 1198883+ 1198884= 0
1198882+ 21198711198885= 0
1
11988821198883+
1
11988821198884+ 21198885= 0
1
1198882exp(
1
119888
119871) 1198883+
1
1198882exp (minus
1
119888
119871) 1198884+ 21198885= 0
(22)
4 Advances in Materials Science and Engineering
20
15
10
05
2 4 6 8 10
z (nm)
w(n
m)
Figure 2 Axial displacement field119908 for 119888 = 0 (local solutionmdashblackline) 119888 = 1 nm (blue line) 119888 = 2 nm (red line) 119888 = 3 nm (greenline) 119888 = 4 nm (orange line) 119888 = 5 nm (yellow line) and 119888 = 25 nm(brown line) 119864 = 300GPa 119871 = 10 nm 119860 = 80 sdot 10
minus2 nm2 and119901 = 10
minus8 Nnm
A further condition can be obtained by imposing that thescalar field
119908 (119911) = 11988851199112 (23)
is a particular solution of the differential equation (17)whence it follows that 119888
5= minus1199012119864119860The remaining constants
are given by the formulae
1198881= minus (119888
3+ 1198884)
1198882= minus2119871119888
5
1198883=
21198882(1 minus exp (minus (1119888) 119871))
exp (minus (1119888) 119871) minus exp ((1119888) 119871)
1198885
1198884=
21198882(1 minus exp ((1119888) 119871))
exp ((1119888) 119871) minus exp (minus (1119888) 119871)
1198885
(24)
having 1198885
= minus1199012119864119860 A plot of the rod axial displacementfield 119908 for different values of the nonlocal parameter 119888 isprovided in Figure 2 It is apparent that the rod becomesstiffer if the nonlocal parameter increasesThe evaluated axialdisplacement at the free end of the rod B provides the samevalue independently of the nonlocal parameter Such a valuecoincideswith the displacement of the pointB if a localmodelis considered Moreover the limit of the axial displacementfield for 119888 tending to plus infinity can be evaluated to get thelower bound
119908low
(119911) = lim119888rarr+infin
119908 (119911 119888) = 0208333119911 (25)
Hence large values of the nonlocal parameter provide adisplacement field which tends to a linear one see Figure 2for 119888 = 25 Further the limit value of the axial displacementfor 119911 = 119871 and 119888 rarr +infin obtained by (25) yields 119908
low(119871) =
208333 nm which coincides with the axial displacement at Bfor any value of the nonlocal parameter 119888 see Figure 3 andTable 1
0 5 10 15 20 250
05
1
15
2
25
c (nm)
w(L2)
andw(L)
(nm
)
Figure 3 Axial displacement in terms of the nonlocal parameter 119888
at the abscissa 119911 = 1198712 (blue line) and 119911 = 119871 (red line)
Table 1 Axial displacements 119908(1198712) and 119908(119871) versus the nonlocalparameter 119888
119888 (nm) 119908(1198712) (nm) 119908(119871) (nm)0 15625 2083331 152139 2083332 142301 2083333 132428 2083334 124886 2083335 119589 20833325 105021 208333
The upper bound of the axial displacement is provided bythe local solution (ie 119888 = 0)
119908upp
(119911) = 119908 (119911 0) =
119901 (119911) (2119871 minus 119911) 119911
2119864119860
(26)
The axial displacement evaluated for 119911 = 119871 by (26) yieldsthe value119908
upp(119871) = 2512 nm which coincides with the axial
displacement at B for any value of 119888 see Figure 3 and Table 1For the considered model the upper and lower bounds ofthe axial displacement field are given by (25) and (26) Theaxial displacement 119908(1198712) at the middle point of the rodand the maximum axial displacement 119908(119871) as functionsof the nonlocal parameter 119888 are depicted in Figure 3 Thecorresponding numerical values of119908(1198712) and119908(119871) are listedin Table 1
It is worth noting that equilibrium prescribes that axialforce 119873 must be a linear function confirmed by the bluediagram in Figure 4 obtained as difference between the localcontribution 119873
119900(dashed line) and the nonlocal one 119873
(1)
1
(continuous thin line) according to (14)1for any value of 119888
5 Conclusions
The outcomes of the present paper may be summarized asfollows
(i) Linearly elastic carbon nanotubes under axial loadshave been investigated by a nonlocal variationalapproach based on thermodynamic restrictions Thetreatment provides an effective tool to evaluate small-scale effects in nanotubes subject also to constant
Advances in Materials Science and Engineering 5
1 times 10minus7
2 4 6 8 10
8 times 10minus8
6 times 10minus8
4 times 10minus8
2 times 10minus8
minus2 times 10minus8
minus4 times 10minus8
z (nm)
N (N
)
Figure 4 Axial force 119873 = 119873119900minus 119873(1)
1(blue line) 119873
119900(dashed line)
119873(1)
1(continuous thin line) 119888 = 0 (local solution-black line) 119888 =
1 nm (blue line) 119888 = 2 nm (red line) 119888 = 3 nm (green line) 119888 = 4 nm(orange line) 119888 = 5 nm (yellow line) and 119888 = 25 nm (brown line)
axial loads a goal not achievable by the Eringenmodel commonly adopted in literature as motivatedin Section 1
(ii) Relevant boundary and differential conditions ofelastic equilibrium have been inferred by a standardlocalization procedure Such a procedure providesin a consistent way the relevant class of boundaryconditions for the nonlocal model
(iii) The present approach yields a firm thermodynamicprocedure to derive different nonlocal models forCNTs by suitable specializations of the elastic energy
(iv) Exact solutions of carbon nanotubes subject to aconstant axial load have been obtained An advantageof the proposed procedure consists in providing aneffective tool to be used as a benchmark for numericalanalyses Finally a range to which any nonlocalsolution must belong is analytically evaluated
Appendix
The procedure to solve the ordinary differential equation
120572119908(4)
minus 120573119908(2)
= 119891 (A1)
with 120572 120573 gt 0 being constant coefficients and 119891 119868 sube R 997891rarr Rbeing a continuous function is summarized as follows Let usconsider the homogeneous differential equation
120572119908(4)
minus 120573119908(2)
= 0 (A2)
and the relevant characteristic (algebraic) equation 1205721205824
minus
1205731205822
= 0 The roots of the polynomial 1205721205824 minus 1205731205822 are 120582
1= 0
with multiplicity 2 1205822
= radic120573120572 with multiplicity 1 and 1205823
=
minusradic120573120572 with multiplicity 1 The general integral of (A2) isthus expressed by the formula
119908119867
(119911) = 1198881+ 1198882119911 + 1198883exp(radic
120573
120572
119911)
+ 1198884exp(minusradic
120573
120572
119911)
(A3)
with exp denoting exponential function and 119888119894isin R for 119894 =
1 4 The general integral of (A1) is writen therefore as
119908 (119911) = 119908119867
(119911) + 119908 (119911) (A4)
where 119908 is a particular solution of (A1) It is worth notingthat for 119891 defined by a polynomial 119901
119898of degree 119898 ge 0 the
solution V can be looked for by setting
119908 (119911) = 1199112
(1198600+ 1198601119911 + sdot sdot sdot + 119860
119898119911119898) (A5)
with 119860119894isin R for 119894 = 1 119898
Acknowledgments
The authors were supported by the ldquoPolo delle Scienze e delleTecnologierdquo University of Naples Federico II through theresearch project FARO Useful hints and precious commentsby anonymous reviewers are also gratefully acknowledged
References
[1] Q Wang and K M Liew ldquoApplication of nonlocal continuummechanics to static analysis of micro- and nano-structuresrdquoPhysics Letters A vol 363 no 3 pp 236ndash242 2007
[2] M Aydogdu ldquoA general nonlocal beam theory its applicationto nanobeam bending buckling and vibrationrdquo Physica E vol41 no 9 pp 1651ndash1655 2009
[3] O Civalek and C Demir ldquoBending analysis of microtubulesusing nonlocal Euler-Bernoulli beam theoryrdquo Applied Mathe-matical Modelling vol 35 no 5 pp 2053ndash2067 2011
[4] M A De Rosa and C Franciosi ldquoA simple approach to detectthe nonlocal effects in the static analysis of Euler-Bernoulli andTimoshenko beamsrdquoMechanics Research Communications vol48 pp 66ndash69 2013
[5] M Aydogdu ldquoAxial vibration analysis of nanorods (carbonnanotubes) embedded in an elastic medium using nonlocalelasticityrdquoMechanics Research Communications vol 41 pp 34ndash40 2012
[6] M A Kazemi-Lari S A Fazelzadeh and E Ghavanloo ldquoNon-conservative instability of cantilever carbon nanotubes restingon viscoelastic foundationrdquo Physica E vol 44 no 7-8 pp 1623ndash1630 2012
[7] H-T Thai and T P Vo ldquoA nonlocal sinusoidal shear deforma-tion beam theory with application to bending buckling andvibration of nanobeamsrdquo International Journal of EngineeringScience vol 54 pp 58ndash66 2012
[8] M A Eltaher S A Emam and F F Mahmoud ldquoStatic andstability analysis of nonlocal functionally graded nanobeamsrdquoComposite Structures vol 96 pp 82ndash88 2013
[9] S A Emam ldquoA general nonlocal nonlinear model for bucklingof nanobeamsrdquo Applied Mathematical Modelling vol 37 no 10-11 pp 6929ndash6939 2013
[10] B Fang Y-X Zhen C-P Zhang and Y Tang ldquoNonlinearvibration analysis of double-walled carbon nanotubes based onnonlocal elasticity theoryrdquoAppliedMathematicalModelling vol37 no 3 pp 1096ndash1107 2013
[11] S A M Ghannadpour B Mohammadi and J Fazilati ldquoBend-ing buckling and vibration problems of nonlocal Euler beamsusing Ritz methodrdquo Composite Structures vol 96 pp 584ndash5892013
6 Advances in Materials Science and Engineering
[12] M Simsek and H H Yurtcu ldquoAnalytical solutions for bendingand buckling of functionally graded nanobeams based on thenonlocal Timoshenko beam theoryrdquo Composite Structures vol97 pp 378ndash386 2013
[13] B L Wang and K F Wang ldquoVibration analysis of embeddednanotubes using nonlocal continuum theoryrdquo Composites Bvol 47 pp 96ndash101 2013
[14] J K Phadikar and S C Pradhan ldquoVariational formulationand finite element analysis for nonlocal elastic nanobeams andnanoplatesrdquo Computational Materials Science vol 49 no 3 pp492ndash499 2010
[15] C M C Roque A J M Ferreira and J N Reddy ldquoAnalysisof Timoshenko nanobeams with a nonlocal formulation andmeshless methodrdquo International Journal of Engineering Sciencevol 49 no 9 pp 976ndash984 2011
[16] F F Mahmoud M A Eltaher A E Alshorbagy and E IMeletis ldquoStatic analysis of nanobeams including surface effectsby nonlocal finite elementsrdquo Journal of Mechanical Science andTechnology vol 26 no 11 pp 3555ndash3563 2012
[17] S C Pradhan ldquoNonlocal finite element analysis and small scaleeffects of CNTs with Timoshenko beam theoryrdquo Finite Elementsin Analysis and Design vol 50 pp 8ndash20 2012
[18] M A Eltaher A E Alshorbagy and F F Mahmoud ldquoVibrationanalysis of Euler-Bernoulli nanobeams by using finite elementmethodrdquo Applied Mathematical Modelling vol 37 no 7 pp4787ndash4797 2013
[19] B Arash and Q Wang ldquoA review on the application ofnonlocal elastic models in modeling of carbon nanotubes andgraphenesrdquo Computational Materials Science vol 51 no 1 pp303ndash313 2012
[20] Y YanW QWang and L X Zhang ldquoNonlocal effect on axiallycompressed buckling of triple-walled carbon nanotubes undertemperature fieldrdquo Applied Mathematical Modelling vol 34 no11 pp 3422ndash3429 2010
[21] R Rafiee and R M Moghadam ldquoOn the modeling of carbonnanotubes a critical reviewrdquo Composites B vol 56 pp 435ndash4490 2014
[22] J N Reddy ldquoNonlocal theories for bending buckling and vibra-tion of beamsrdquo International Journal of Engineering Science vol45 no 2-8 pp 288ndash307 2007
[23] A C Eringen ldquoOn differential equations of nonlocal elasticityand solutions of screw dislocation and surface wavesrdquo Journalof Applied Physics vol 54 no 9 pp 4703ndash4710 1983
[24] F Marotti de Sciarra ldquoVariational formulations convergenceand stability properties in nonlocal elastoplasticityrdquo Interna-tional Journal of Solids and Structures vol 45 no 7-8 pp 2322ndash2354 2008
[25] G RomanoMDiaco andR Barretta ldquoVariational formulationof the first principle of continuum thermodynamicsrdquo Contin-uumMechanics andThermodynamics vol 22 no 3 pp 177ndash1872010
[26] FMarotti De Sciarra ldquoHardening plasticitywith nonlocal straindamagerdquo International Journal of Plasticity vol 34 pp 114ndash1382012
[27] G Romano and R Barretta ldquoCovariant hypo-elasticityrdquo Euro-pean Journal of Mechanics A vol 30 no 6 pp 1012ndash1023 2011
[28] G Romano and R Barretta ldquoOn Eulerrsquos stretching formula incontinuummechanicsrdquo Acta Mechanica vol 224 no 1 pp 211ndash230 2013
[29] G Romano and R Barretta ldquoGeometric constitutive theoryand frame invariancerdquo International Journal of Non-LinearMechanics vol 51 pp 75ndash86 2013
[30] G Romano R Barretta and M Diaco ldquoGeometric continuummechanicsrdquoMeccanica 2013
[31] R D Mindlin ldquoMicro-structure in linear elasticityrdquo Archive forRational Mechanics and Analysis vol 16 no 1 pp 51ndash78 1964
[32] F Marotti de Sciarra ldquoNovel variational formulations fornonlocal plasticityrdquo International Journal of Plasticity vol 25no 2 pp 302ndash331 2009
[33] F Marotti de Sciarra ldquoOn non-local and non-homogeneouselastic continuardquo International Journal of Solids and Structuresvol 46 no 3-4 pp 651ndash676 2009
[34] F Marotti de Sciarra ldquoA nonlocal model with strain-baseddamagerdquo International Journal of Solids and Structures vol 46no 22-23 pp 4107ndash4122 2009
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
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CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
thinspInternationalthinspJournalthinspof
BiomaterialsHindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NaNoscieNceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
CrystallographyJournal of
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
BioMed Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
MaterialsJournal of
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Advances in Materials Science and Engineering 3
These conditions can be conveniently expressed in termsof the axial displacement field 119908 as follows A direct evalua-tion of the scalar functions 119873
119894 [0 119871] 997891rarr R for 119894 isin 0 1 and
of their derivatives gives
119873119900= int
Ω
120590119900119889119860 = int
Ω
119864120576119889119860 = int
Ω
119864119908(1)
119889119860 = 119864119860119908(1)
119873(119895)
119900= 119864119860119908
(1+119895)
1198731= int
Ω
1205901119889119860 = int
Ω
1198641198882120576(1)
119889119860
= int
Ω
1198641198882119908(2)
119889119860 = 1198641198601198882119908(2)
119873(119895)
1= 119864119860119888
2119908(2+119895)
(15)
with 119895 isin 1 2 119899 Accordingly the boundary anddifferential conditions of elastic equilibrium (13) and (14) takethe form
119873(1)
= 119873(1)
119900minus 119873(2)
1= 119864119860119908
(2)minus 119864119860119888
2119908(4)
119873 = 119873119900minus 119873(1)
1= 119864119860119908
(1)minus 119864119860119888
2119908(3)
dual of
0 = 1198731= 119864 119860 119888
2119908(2)
dual of (1)
(16)
4 Example
Let us consider a straight rod subject to a constant axial load119901 as depicted in Figure 1 End cross sections A and B areassumed to be hinged and simply supported respectivelyAs illustrated in Section 3 the computation of the rod axialdisplacement field 119908 involves the following cross sectiongeometric and elastic properties area 119860 Young modulus 119864and nonlocal parameter 119888 By setting 120572 = 119864119860119888
2 and 120573 = 119864119860the differential equation of elastic equilibrium is as follows
120572119908(4)
minus 120573119908(2)
= 119901 (17)
The general integral takes thus the form (see the appendix)
119908 (119911) = 119908119867
(119911) + 119908 (119911) (18)
with
119908119867
(119911) = 1198881+ 1198882119911 + 1198883exp(radic
120573
120572
119911)
+ 1198884exp(minusradic
120573
120572
119911)
119908 (119911) = 11988851199112
(19)
z
L
p
N(0)
A B
Figure 1 Rod under constant axial load
The evaluation of the constants is carried out by imposing thefollowing boundary conditions (see also Section 3)
119908 (0) = 0
119873 (119871) = 119864119860119908(1)
(119871) minus 1198641198601198882119908(3)
(119871) = 0
1198731(0) = 119864119860119888
2119908(2)
(0) = 0
1198731(119871) = 119864119860119888
2119908(2)
(119871) = 0
119908 (0) = 0
119908(1)
(119871) minus 1198882119908(3)
(119871) = 0
119908(2)
(0) = 0
119908(2)
(119871) = 0
(20)
Resorting to the expressions of the derivatives 119908(119895)
119867and 119908
(119895)
for 119895 isin 1 2 3 4
119908(1)
119867(119911) = 119888
2+
1198883
119888
exp(
1
119888
119911) minus
1198884
119888
exp(minus
1
119888
119911)
119908(2)
119867(119911) =
1198883
1198882exp (
1
119888
119911) +
1198884
1198882exp (minus
1
119888
119911)
119908(3)
119867(119911) =
1198883
1198883exp (
1
119888
119911) minus
1198884
1198883exp (minus
1
119888
119911)
119908(4)
119867(119911) =
1198883
1198884exp (
1
119888
119911) +
1198884
1198884exp (minus
1
119888
119911)
119908(1)
(119911) = 21198885119911
119908(2)
(119911) = 21198885
119908(3)
(119911) = 0
119908(4)
(119911) = 0
(21)
and having radic120573120572 = 1119888 a direct computation provides thealgebraic system
1198881+ 1198883+ 1198884= 0
1198882+ 21198711198885= 0
1
11988821198883+
1
11988821198884+ 21198885= 0
1
1198882exp(
1
119888
119871) 1198883+
1
1198882exp (minus
1
119888
119871) 1198884+ 21198885= 0
(22)
4 Advances in Materials Science and Engineering
20
15
10
05
2 4 6 8 10
z (nm)
w(n
m)
Figure 2 Axial displacement field119908 for 119888 = 0 (local solutionmdashblackline) 119888 = 1 nm (blue line) 119888 = 2 nm (red line) 119888 = 3 nm (greenline) 119888 = 4 nm (orange line) 119888 = 5 nm (yellow line) and 119888 = 25 nm(brown line) 119864 = 300GPa 119871 = 10 nm 119860 = 80 sdot 10
minus2 nm2 and119901 = 10
minus8 Nnm
A further condition can be obtained by imposing that thescalar field
119908 (119911) = 11988851199112 (23)
is a particular solution of the differential equation (17)whence it follows that 119888
5= minus1199012119864119860The remaining constants
are given by the formulae
1198881= minus (119888
3+ 1198884)
1198882= minus2119871119888
5
1198883=
21198882(1 minus exp (minus (1119888) 119871))
exp (minus (1119888) 119871) minus exp ((1119888) 119871)
1198885
1198884=
21198882(1 minus exp ((1119888) 119871))
exp ((1119888) 119871) minus exp (minus (1119888) 119871)
1198885
(24)
having 1198885
= minus1199012119864119860 A plot of the rod axial displacementfield 119908 for different values of the nonlocal parameter 119888 isprovided in Figure 2 It is apparent that the rod becomesstiffer if the nonlocal parameter increasesThe evaluated axialdisplacement at the free end of the rod B provides the samevalue independently of the nonlocal parameter Such a valuecoincideswith the displacement of the pointB if a localmodelis considered Moreover the limit of the axial displacementfield for 119888 tending to plus infinity can be evaluated to get thelower bound
119908low
(119911) = lim119888rarr+infin
119908 (119911 119888) = 0208333119911 (25)
Hence large values of the nonlocal parameter provide adisplacement field which tends to a linear one see Figure 2for 119888 = 25 Further the limit value of the axial displacementfor 119911 = 119871 and 119888 rarr +infin obtained by (25) yields 119908
low(119871) =
208333 nm which coincides with the axial displacement at Bfor any value of the nonlocal parameter 119888 see Figure 3 andTable 1
0 5 10 15 20 250
05
1
15
2
25
c (nm)
w(L2)
andw(L)
(nm
)
Figure 3 Axial displacement in terms of the nonlocal parameter 119888
at the abscissa 119911 = 1198712 (blue line) and 119911 = 119871 (red line)
Table 1 Axial displacements 119908(1198712) and 119908(119871) versus the nonlocalparameter 119888
119888 (nm) 119908(1198712) (nm) 119908(119871) (nm)0 15625 2083331 152139 2083332 142301 2083333 132428 2083334 124886 2083335 119589 20833325 105021 208333
The upper bound of the axial displacement is provided bythe local solution (ie 119888 = 0)
119908upp
(119911) = 119908 (119911 0) =
119901 (119911) (2119871 minus 119911) 119911
2119864119860
(26)
The axial displacement evaluated for 119911 = 119871 by (26) yieldsthe value119908
upp(119871) = 2512 nm which coincides with the axial
displacement at B for any value of 119888 see Figure 3 and Table 1For the considered model the upper and lower bounds ofthe axial displacement field are given by (25) and (26) Theaxial displacement 119908(1198712) at the middle point of the rodand the maximum axial displacement 119908(119871) as functionsof the nonlocal parameter 119888 are depicted in Figure 3 Thecorresponding numerical values of119908(1198712) and119908(119871) are listedin Table 1
It is worth noting that equilibrium prescribes that axialforce 119873 must be a linear function confirmed by the bluediagram in Figure 4 obtained as difference between the localcontribution 119873
119900(dashed line) and the nonlocal one 119873
(1)
1
(continuous thin line) according to (14)1for any value of 119888
5 Conclusions
The outcomes of the present paper may be summarized asfollows
(i) Linearly elastic carbon nanotubes under axial loadshave been investigated by a nonlocal variationalapproach based on thermodynamic restrictions Thetreatment provides an effective tool to evaluate small-scale effects in nanotubes subject also to constant
Advances in Materials Science and Engineering 5
1 times 10minus7
2 4 6 8 10
8 times 10minus8
6 times 10minus8
4 times 10minus8
2 times 10minus8
minus2 times 10minus8
minus4 times 10minus8
z (nm)
N (N
)
Figure 4 Axial force 119873 = 119873119900minus 119873(1)
1(blue line) 119873
119900(dashed line)
119873(1)
1(continuous thin line) 119888 = 0 (local solution-black line) 119888 =
1 nm (blue line) 119888 = 2 nm (red line) 119888 = 3 nm (green line) 119888 = 4 nm(orange line) 119888 = 5 nm (yellow line) and 119888 = 25 nm (brown line)
axial loads a goal not achievable by the Eringenmodel commonly adopted in literature as motivatedin Section 1
(ii) Relevant boundary and differential conditions ofelastic equilibrium have been inferred by a standardlocalization procedure Such a procedure providesin a consistent way the relevant class of boundaryconditions for the nonlocal model
(iii) The present approach yields a firm thermodynamicprocedure to derive different nonlocal models forCNTs by suitable specializations of the elastic energy
(iv) Exact solutions of carbon nanotubes subject to aconstant axial load have been obtained An advantageof the proposed procedure consists in providing aneffective tool to be used as a benchmark for numericalanalyses Finally a range to which any nonlocalsolution must belong is analytically evaluated
Appendix
The procedure to solve the ordinary differential equation
120572119908(4)
minus 120573119908(2)
= 119891 (A1)
with 120572 120573 gt 0 being constant coefficients and 119891 119868 sube R 997891rarr Rbeing a continuous function is summarized as follows Let usconsider the homogeneous differential equation
120572119908(4)
minus 120573119908(2)
= 0 (A2)
and the relevant characteristic (algebraic) equation 1205721205824
minus
1205731205822
= 0 The roots of the polynomial 1205721205824 minus 1205731205822 are 120582
1= 0
with multiplicity 2 1205822
= radic120573120572 with multiplicity 1 and 1205823
=
minusradic120573120572 with multiplicity 1 The general integral of (A2) isthus expressed by the formula
119908119867
(119911) = 1198881+ 1198882119911 + 1198883exp(radic
120573
120572
119911)
+ 1198884exp(minusradic
120573
120572
119911)
(A3)
with exp denoting exponential function and 119888119894isin R for 119894 =
1 4 The general integral of (A1) is writen therefore as
119908 (119911) = 119908119867
(119911) + 119908 (119911) (A4)
where 119908 is a particular solution of (A1) It is worth notingthat for 119891 defined by a polynomial 119901
119898of degree 119898 ge 0 the
solution V can be looked for by setting
119908 (119911) = 1199112
(1198600+ 1198601119911 + sdot sdot sdot + 119860
119898119911119898) (A5)
with 119860119894isin R for 119894 = 1 119898
Acknowledgments
The authors were supported by the ldquoPolo delle Scienze e delleTecnologierdquo University of Naples Federico II through theresearch project FARO Useful hints and precious commentsby anonymous reviewers are also gratefully acknowledged
References
[1] Q Wang and K M Liew ldquoApplication of nonlocal continuummechanics to static analysis of micro- and nano-structuresrdquoPhysics Letters A vol 363 no 3 pp 236ndash242 2007
[2] M Aydogdu ldquoA general nonlocal beam theory its applicationto nanobeam bending buckling and vibrationrdquo Physica E vol41 no 9 pp 1651ndash1655 2009
[3] O Civalek and C Demir ldquoBending analysis of microtubulesusing nonlocal Euler-Bernoulli beam theoryrdquo Applied Mathe-matical Modelling vol 35 no 5 pp 2053ndash2067 2011
[4] M A De Rosa and C Franciosi ldquoA simple approach to detectthe nonlocal effects in the static analysis of Euler-Bernoulli andTimoshenko beamsrdquoMechanics Research Communications vol48 pp 66ndash69 2013
[5] M Aydogdu ldquoAxial vibration analysis of nanorods (carbonnanotubes) embedded in an elastic medium using nonlocalelasticityrdquoMechanics Research Communications vol 41 pp 34ndash40 2012
[6] M A Kazemi-Lari S A Fazelzadeh and E Ghavanloo ldquoNon-conservative instability of cantilever carbon nanotubes restingon viscoelastic foundationrdquo Physica E vol 44 no 7-8 pp 1623ndash1630 2012
[7] H-T Thai and T P Vo ldquoA nonlocal sinusoidal shear deforma-tion beam theory with application to bending buckling andvibration of nanobeamsrdquo International Journal of EngineeringScience vol 54 pp 58ndash66 2012
[8] M A Eltaher S A Emam and F F Mahmoud ldquoStatic andstability analysis of nonlocal functionally graded nanobeamsrdquoComposite Structures vol 96 pp 82ndash88 2013
[9] S A Emam ldquoA general nonlocal nonlinear model for bucklingof nanobeamsrdquo Applied Mathematical Modelling vol 37 no 10-11 pp 6929ndash6939 2013
[10] B Fang Y-X Zhen C-P Zhang and Y Tang ldquoNonlinearvibration analysis of double-walled carbon nanotubes based onnonlocal elasticity theoryrdquoAppliedMathematicalModelling vol37 no 3 pp 1096ndash1107 2013
[11] S A M Ghannadpour B Mohammadi and J Fazilati ldquoBend-ing buckling and vibration problems of nonlocal Euler beamsusing Ritz methodrdquo Composite Structures vol 96 pp 584ndash5892013
6 Advances in Materials Science and Engineering
[12] M Simsek and H H Yurtcu ldquoAnalytical solutions for bendingand buckling of functionally graded nanobeams based on thenonlocal Timoshenko beam theoryrdquo Composite Structures vol97 pp 378ndash386 2013
[13] B L Wang and K F Wang ldquoVibration analysis of embeddednanotubes using nonlocal continuum theoryrdquo Composites Bvol 47 pp 96ndash101 2013
[14] J K Phadikar and S C Pradhan ldquoVariational formulationand finite element analysis for nonlocal elastic nanobeams andnanoplatesrdquo Computational Materials Science vol 49 no 3 pp492ndash499 2010
[15] C M C Roque A J M Ferreira and J N Reddy ldquoAnalysisof Timoshenko nanobeams with a nonlocal formulation andmeshless methodrdquo International Journal of Engineering Sciencevol 49 no 9 pp 976ndash984 2011
[16] F F Mahmoud M A Eltaher A E Alshorbagy and E IMeletis ldquoStatic analysis of nanobeams including surface effectsby nonlocal finite elementsrdquo Journal of Mechanical Science andTechnology vol 26 no 11 pp 3555ndash3563 2012
[17] S C Pradhan ldquoNonlocal finite element analysis and small scaleeffects of CNTs with Timoshenko beam theoryrdquo Finite Elementsin Analysis and Design vol 50 pp 8ndash20 2012
[18] M A Eltaher A E Alshorbagy and F F Mahmoud ldquoVibrationanalysis of Euler-Bernoulli nanobeams by using finite elementmethodrdquo Applied Mathematical Modelling vol 37 no 7 pp4787ndash4797 2013
[19] B Arash and Q Wang ldquoA review on the application ofnonlocal elastic models in modeling of carbon nanotubes andgraphenesrdquo Computational Materials Science vol 51 no 1 pp303ndash313 2012
[20] Y YanW QWang and L X Zhang ldquoNonlocal effect on axiallycompressed buckling of triple-walled carbon nanotubes undertemperature fieldrdquo Applied Mathematical Modelling vol 34 no11 pp 3422ndash3429 2010
[21] R Rafiee and R M Moghadam ldquoOn the modeling of carbonnanotubes a critical reviewrdquo Composites B vol 56 pp 435ndash4490 2014
[22] J N Reddy ldquoNonlocal theories for bending buckling and vibra-tion of beamsrdquo International Journal of Engineering Science vol45 no 2-8 pp 288ndash307 2007
[23] A C Eringen ldquoOn differential equations of nonlocal elasticityand solutions of screw dislocation and surface wavesrdquo Journalof Applied Physics vol 54 no 9 pp 4703ndash4710 1983
[24] F Marotti de Sciarra ldquoVariational formulations convergenceand stability properties in nonlocal elastoplasticityrdquo Interna-tional Journal of Solids and Structures vol 45 no 7-8 pp 2322ndash2354 2008
[25] G RomanoMDiaco andR Barretta ldquoVariational formulationof the first principle of continuum thermodynamicsrdquo Contin-uumMechanics andThermodynamics vol 22 no 3 pp 177ndash1872010
[26] FMarotti De Sciarra ldquoHardening plasticitywith nonlocal straindamagerdquo International Journal of Plasticity vol 34 pp 114ndash1382012
[27] G Romano and R Barretta ldquoCovariant hypo-elasticityrdquo Euro-pean Journal of Mechanics A vol 30 no 6 pp 1012ndash1023 2011
[28] G Romano and R Barretta ldquoOn Eulerrsquos stretching formula incontinuummechanicsrdquo Acta Mechanica vol 224 no 1 pp 211ndash230 2013
[29] G Romano and R Barretta ldquoGeometric constitutive theoryand frame invariancerdquo International Journal of Non-LinearMechanics vol 51 pp 75ndash86 2013
[30] G Romano R Barretta and M Diaco ldquoGeometric continuummechanicsrdquoMeccanica 2013
[31] R D Mindlin ldquoMicro-structure in linear elasticityrdquo Archive forRational Mechanics and Analysis vol 16 no 1 pp 51ndash78 1964
[32] F Marotti de Sciarra ldquoNovel variational formulations fornonlocal plasticityrdquo International Journal of Plasticity vol 25no 2 pp 302ndash331 2009
[33] F Marotti de Sciarra ldquoOn non-local and non-homogeneouselastic continuardquo International Journal of Solids and Structuresvol 46 no 3-4 pp 651ndash676 2009
[34] F Marotti de Sciarra ldquoA nonlocal model with strain-baseddamagerdquo International Journal of Solids and Structures vol 46no 22-23 pp 4107ndash4122 2009
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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CeramicsJournal of
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CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
thinspInternationalthinspJournalthinspof
BiomaterialsHindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
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NaNoscieNceJournal of
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Journal of
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Volume 2014
CrystallographyJournal of
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Advances in
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Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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BioMed Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
MaterialsJournal of
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
4 Advances in Materials Science and Engineering
20
15
10
05
2 4 6 8 10
z (nm)
w(n
m)
Figure 2 Axial displacement field119908 for 119888 = 0 (local solutionmdashblackline) 119888 = 1 nm (blue line) 119888 = 2 nm (red line) 119888 = 3 nm (greenline) 119888 = 4 nm (orange line) 119888 = 5 nm (yellow line) and 119888 = 25 nm(brown line) 119864 = 300GPa 119871 = 10 nm 119860 = 80 sdot 10
minus2 nm2 and119901 = 10
minus8 Nnm
A further condition can be obtained by imposing that thescalar field
119908 (119911) = 11988851199112 (23)
is a particular solution of the differential equation (17)whence it follows that 119888
5= minus1199012119864119860The remaining constants
are given by the formulae
1198881= minus (119888
3+ 1198884)
1198882= minus2119871119888
5
1198883=
21198882(1 minus exp (minus (1119888) 119871))
exp (minus (1119888) 119871) minus exp ((1119888) 119871)
1198885
1198884=
21198882(1 minus exp ((1119888) 119871))
exp ((1119888) 119871) minus exp (minus (1119888) 119871)
1198885
(24)
having 1198885
= minus1199012119864119860 A plot of the rod axial displacementfield 119908 for different values of the nonlocal parameter 119888 isprovided in Figure 2 It is apparent that the rod becomesstiffer if the nonlocal parameter increasesThe evaluated axialdisplacement at the free end of the rod B provides the samevalue independently of the nonlocal parameter Such a valuecoincideswith the displacement of the pointB if a localmodelis considered Moreover the limit of the axial displacementfield for 119888 tending to plus infinity can be evaluated to get thelower bound
119908low
(119911) = lim119888rarr+infin
119908 (119911 119888) = 0208333119911 (25)
Hence large values of the nonlocal parameter provide adisplacement field which tends to a linear one see Figure 2for 119888 = 25 Further the limit value of the axial displacementfor 119911 = 119871 and 119888 rarr +infin obtained by (25) yields 119908
low(119871) =
208333 nm which coincides with the axial displacement at Bfor any value of the nonlocal parameter 119888 see Figure 3 andTable 1
0 5 10 15 20 250
05
1
15
2
25
c (nm)
w(L2)
andw(L)
(nm
)
Figure 3 Axial displacement in terms of the nonlocal parameter 119888
at the abscissa 119911 = 1198712 (blue line) and 119911 = 119871 (red line)
Table 1 Axial displacements 119908(1198712) and 119908(119871) versus the nonlocalparameter 119888
119888 (nm) 119908(1198712) (nm) 119908(119871) (nm)0 15625 2083331 152139 2083332 142301 2083333 132428 2083334 124886 2083335 119589 20833325 105021 208333
The upper bound of the axial displacement is provided bythe local solution (ie 119888 = 0)
119908upp
(119911) = 119908 (119911 0) =
119901 (119911) (2119871 minus 119911) 119911
2119864119860
(26)
The axial displacement evaluated for 119911 = 119871 by (26) yieldsthe value119908
upp(119871) = 2512 nm which coincides with the axial
displacement at B for any value of 119888 see Figure 3 and Table 1For the considered model the upper and lower bounds ofthe axial displacement field are given by (25) and (26) Theaxial displacement 119908(1198712) at the middle point of the rodand the maximum axial displacement 119908(119871) as functionsof the nonlocal parameter 119888 are depicted in Figure 3 Thecorresponding numerical values of119908(1198712) and119908(119871) are listedin Table 1
It is worth noting that equilibrium prescribes that axialforce 119873 must be a linear function confirmed by the bluediagram in Figure 4 obtained as difference between the localcontribution 119873
119900(dashed line) and the nonlocal one 119873
(1)
1
(continuous thin line) according to (14)1for any value of 119888
5 Conclusions
The outcomes of the present paper may be summarized asfollows
(i) Linearly elastic carbon nanotubes under axial loadshave been investigated by a nonlocal variationalapproach based on thermodynamic restrictions Thetreatment provides an effective tool to evaluate small-scale effects in nanotubes subject also to constant
Advances in Materials Science and Engineering 5
1 times 10minus7
2 4 6 8 10
8 times 10minus8
6 times 10minus8
4 times 10minus8
2 times 10minus8
minus2 times 10minus8
minus4 times 10minus8
z (nm)
N (N
)
Figure 4 Axial force 119873 = 119873119900minus 119873(1)
1(blue line) 119873
119900(dashed line)
119873(1)
1(continuous thin line) 119888 = 0 (local solution-black line) 119888 =
1 nm (blue line) 119888 = 2 nm (red line) 119888 = 3 nm (green line) 119888 = 4 nm(orange line) 119888 = 5 nm (yellow line) and 119888 = 25 nm (brown line)
axial loads a goal not achievable by the Eringenmodel commonly adopted in literature as motivatedin Section 1
(ii) Relevant boundary and differential conditions ofelastic equilibrium have been inferred by a standardlocalization procedure Such a procedure providesin a consistent way the relevant class of boundaryconditions for the nonlocal model
(iii) The present approach yields a firm thermodynamicprocedure to derive different nonlocal models forCNTs by suitable specializations of the elastic energy
(iv) Exact solutions of carbon nanotubes subject to aconstant axial load have been obtained An advantageof the proposed procedure consists in providing aneffective tool to be used as a benchmark for numericalanalyses Finally a range to which any nonlocalsolution must belong is analytically evaluated
Appendix
The procedure to solve the ordinary differential equation
120572119908(4)
minus 120573119908(2)
= 119891 (A1)
with 120572 120573 gt 0 being constant coefficients and 119891 119868 sube R 997891rarr Rbeing a continuous function is summarized as follows Let usconsider the homogeneous differential equation
120572119908(4)
minus 120573119908(2)
= 0 (A2)
and the relevant characteristic (algebraic) equation 1205721205824
minus
1205731205822
= 0 The roots of the polynomial 1205721205824 minus 1205731205822 are 120582
1= 0
with multiplicity 2 1205822
= radic120573120572 with multiplicity 1 and 1205823
=
minusradic120573120572 with multiplicity 1 The general integral of (A2) isthus expressed by the formula
119908119867
(119911) = 1198881+ 1198882119911 + 1198883exp(radic
120573
120572
119911)
+ 1198884exp(minusradic
120573
120572
119911)
(A3)
with exp denoting exponential function and 119888119894isin R for 119894 =
1 4 The general integral of (A1) is writen therefore as
119908 (119911) = 119908119867
(119911) + 119908 (119911) (A4)
where 119908 is a particular solution of (A1) It is worth notingthat for 119891 defined by a polynomial 119901
119898of degree 119898 ge 0 the
solution V can be looked for by setting
119908 (119911) = 1199112
(1198600+ 1198601119911 + sdot sdot sdot + 119860
119898119911119898) (A5)
with 119860119894isin R for 119894 = 1 119898
Acknowledgments
The authors were supported by the ldquoPolo delle Scienze e delleTecnologierdquo University of Naples Federico II through theresearch project FARO Useful hints and precious commentsby anonymous reviewers are also gratefully acknowledged
References
[1] Q Wang and K M Liew ldquoApplication of nonlocal continuummechanics to static analysis of micro- and nano-structuresrdquoPhysics Letters A vol 363 no 3 pp 236ndash242 2007
[2] M Aydogdu ldquoA general nonlocal beam theory its applicationto nanobeam bending buckling and vibrationrdquo Physica E vol41 no 9 pp 1651ndash1655 2009
[3] O Civalek and C Demir ldquoBending analysis of microtubulesusing nonlocal Euler-Bernoulli beam theoryrdquo Applied Mathe-matical Modelling vol 35 no 5 pp 2053ndash2067 2011
[4] M A De Rosa and C Franciosi ldquoA simple approach to detectthe nonlocal effects in the static analysis of Euler-Bernoulli andTimoshenko beamsrdquoMechanics Research Communications vol48 pp 66ndash69 2013
[5] M Aydogdu ldquoAxial vibration analysis of nanorods (carbonnanotubes) embedded in an elastic medium using nonlocalelasticityrdquoMechanics Research Communications vol 41 pp 34ndash40 2012
[6] M A Kazemi-Lari S A Fazelzadeh and E Ghavanloo ldquoNon-conservative instability of cantilever carbon nanotubes restingon viscoelastic foundationrdquo Physica E vol 44 no 7-8 pp 1623ndash1630 2012
[7] H-T Thai and T P Vo ldquoA nonlocal sinusoidal shear deforma-tion beam theory with application to bending buckling andvibration of nanobeamsrdquo International Journal of EngineeringScience vol 54 pp 58ndash66 2012
[8] M A Eltaher S A Emam and F F Mahmoud ldquoStatic andstability analysis of nonlocal functionally graded nanobeamsrdquoComposite Structures vol 96 pp 82ndash88 2013
[9] S A Emam ldquoA general nonlocal nonlinear model for bucklingof nanobeamsrdquo Applied Mathematical Modelling vol 37 no 10-11 pp 6929ndash6939 2013
[10] B Fang Y-X Zhen C-P Zhang and Y Tang ldquoNonlinearvibration analysis of double-walled carbon nanotubes based onnonlocal elasticity theoryrdquoAppliedMathematicalModelling vol37 no 3 pp 1096ndash1107 2013
[11] S A M Ghannadpour B Mohammadi and J Fazilati ldquoBend-ing buckling and vibration problems of nonlocal Euler beamsusing Ritz methodrdquo Composite Structures vol 96 pp 584ndash5892013
6 Advances in Materials Science and Engineering
[12] M Simsek and H H Yurtcu ldquoAnalytical solutions for bendingand buckling of functionally graded nanobeams based on thenonlocal Timoshenko beam theoryrdquo Composite Structures vol97 pp 378ndash386 2013
[13] B L Wang and K F Wang ldquoVibration analysis of embeddednanotubes using nonlocal continuum theoryrdquo Composites Bvol 47 pp 96ndash101 2013
[14] J K Phadikar and S C Pradhan ldquoVariational formulationand finite element analysis for nonlocal elastic nanobeams andnanoplatesrdquo Computational Materials Science vol 49 no 3 pp492ndash499 2010
[15] C M C Roque A J M Ferreira and J N Reddy ldquoAnalysisof Timoshenko nanobeams with a nonlocal formulation andmeshless methodrdquo International Journal of Engineering Sciencevol 49 no 9 pp 976ndash984 2011
[16] F F Mahmoud M A Eltaher A E Alshorbagy and E IMeletis ldquoStatic analysis of nanobeams including surface effectsby nonlocal finite elementsrdquo Journal of Mechanical Science andTechnology vol 26 no 11 pp 3555ndash3563 2012
[17] S C Pradhan ldquoNonlocal finite element analysis and small scaleeffects of CNTs with Timoshenko beam theoryrdquo Finite Elementsin Analysis and Design vol 50 pp 8ndash20 2012
[18] M A Eltaher A E Alshorbagy and F F Mahmoud ldquoVibrationanalysis of Euler-Bernoulli nanobeams by using finite elementmethodrdquo Applied Mathematical Modelling vol 37 no 7 pp4787ndash4797 2013
[19] B Arash and Q Wang ldquoA review on the application ofnonlocal elastic models in modeling of carbon nanotubes andgraphenesrdquo Computational Materials Science vol 51 no 1 pp303ndash313 2012
[20] Y YanW QWang and L X Zhang ldquoNonlocal effect on axiallycompressed buckling of triple-walled carbon nanotubes undertemperature fieldrdquo Applied Mathematical Modelling vol 34 no11 pp 3422ndash3429 2010
[21] R Rafiee and R M Moghadam ldquoOn the modeling of carbonnanotubes a critical reviewrdquo Composites B vol 56 pp 435ndash4490 2014
[22] J N Reddy ldquoNonlocal theories for bending buckling and vibra-tion of beamsrdquo International Journal of Engineering Science vol45 no 2-8 pp 288ndash307 2007
[23] A C Eringen ldquoOn differential equations of nonlocal elasticityand solutions of screw dislocation and surface wavesrdquo Journalof Applied Physics vol 54 no 9 pp 4703ndash4710 1983
[24] F Marotti de Sciarra ldquoVariational formulations convergenceand stability properties in nonlocal elastoplasticityrdquo Interna-tional Journal of Solids and Structures vol 45 no 7-8 pp 2322ndash2354 2008
[25] G RomanoMDiaco andR Barretta ldquoVariational formulationof the first principle of continuum thermodynamicsrdquo Contin-uumMechanics andThermodynamics vol 22 no 3 pp 177ndash1872010
[26] FMarotti De Sciarra ldquoHardening plasticitywith nonlocal straindamagerdquo International Journal of Plasticity vol 34 pp 114ndash1382012
[27] G Romano and R Barretta ldquoCovariant hypo-elasticityrdquo Euro-pean Journal of Mechanics A vol 30 no 6 pp 1012ndash1023 2011
[28] G Romano and R Barretta ldquoOn Eulerrsquos stretching formula incontinuummechanicsrdquo Acta Mechanica vol 224 no 1 pp 211ndash230 2013
[29] G Romano and R Barretta ldquoGeometric constitutive theoryand frame invariancerdquo International Journal of Non-LinearMechanics vol 51 pp 75ndash86 2013
[30] G Romano R Barretta and M Diaco ldquoGeometric continuummechanicsrdquoMeccanica 2013
[31] R D Mindlin ldquoMicro-structure in linear elasticityrdquo Archive forRational Mechanics and Analysis vol 16 no 1 pp 51ndash78 1964
[32] F Marotti de Sciarra ldquoNovel variational formulations fornonlocal plasticityrdquo International Journal of Plasticity vol 25no 2 pp 302ndash331 2009
[33] F Marotti de Sciarra ldquoOn non-local and non-homogeneouselastic continuardquo International Journal of Solids and Structuresvol 46 no 3-4 pp 651ndash676 2009
[34] F Marotti de Sciarra ldquoA nonlocal model with strain-baseddamagerdquo International Journal of Solids and Structures vol 46no 22-23 pp 4107ndash4122 2009
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
thinspInternationalthinspJournalthinspof
BiomaterialsHindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NaNoscieNceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
CrystallographyJournal of
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
BioMed Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
MaterialsJournal of
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Advances in Materials Science and Engineering 5
1 times 10minus7
2 4 6 8 10
8 times 10minus8
6 times 10minus8
4 times 10minus8
2 times 10minus8
minus2 times 10minus8
minus4 times 10minus8
z (nm)
N (N
)
Figure 4 Axial force 119873 = 119873119900minus 119873(1)
1(blue line) 119873
119900(dashed line)
119873(1)
1(continuous thin line) 119888 = 0 (local solution-black line) 119888 =
1 nm (blue line) 119888 = 2 nm (red line) 119888 = 3 nm (green line) 119888 = 4 nm(orange line) 119888 = 5 nm (yellow line) and 119888 = 25 nm (brown line)
axial loads a goal not achievable by the Eringenmodel commonly adopted in literature as motivatedin Section 1
(ii) Relevant boundary and differential conditions ofelastic equilibrium have been inferred by a standardlocalization procedure Such a procedure providesin a consistent way the relevant class of boundaryconditions for the nonlocal model
(iii) The present approach yields a firm thermodynamicprocedure to derive different nonlocal models forCNTs by suitable specializations of the elastic energy
(iv) Exact solutions of carbon nanotubes subject to aconstant axial load have been obtained An advantageof the proposed procedure consists in providing aneffective tool to be used as a benchmark for numericalanalyses Finally a range to which any nonlocalsolution must belong is analytically evaluated
Appendix
The procedure to solve the ordinary differential equation
120572119908(4)
minus 120573119908(2)
= 119891 (A1)
with 120572 120573 gt 0 being constant coefficients and 119891 119868 sube R 997891rarr Rbeing a continuous function is summarized as follows Let usconsider the homogeneous differential equation
120572119908(4)
minus 120573119908(2)
= 0 (A2)
and the relevant characteristic (algebraic) equation 1205721205824
minus
1205731205822
= 0 The roots of the polynomial 1205721205824 minus 1205731205822 are 120582
1= 0
with multiplicity 2 1205822
= radic120573120572 with multiplicity 1 and 1205823
=
minusradic120573120572 with multiplicity 1 The general integral of (A2) isthus expressed by the formula
119908119867
(119911) = 1198881+ 1198882119911 + 1198883exp(radic
120573
120572
119911)
+ 1198884exp(minusradic
120573
120572
119911)
(A3)
with exp denoting exponential function and 119888119894isin R for 119894 =
1 4 The general integral of (A1) is writen therefore as
119908 (119911) = 119908119867
(119911) + 119908 (119911) (A4)
where 119908 is a particular solution of (A1) It is worth notingthat for 119891 defined by a polynomial 119901
119898of degree 119898 ge 0 the
solution V can be looked for by setting
119908 (119911) = 1199112
(1198600+ 1198601119911 + sdot sdot sdot + 119860
119898119911119898) (A5)
with 119860119894isin R for 119894 = 1 119898
Acknowledgments
The authors were supported by the ldquoPolo delle Scienze e delleTecnologierdquo University of Naples Federico II through theresearch project FARO Useful hints and precious commentsby anonymous reviewers are also gratefully acknowledged
References
[1] Q Wang and K M Liew ldquoApplication of nonlocal continuummechanics to static analysis of micro- and nano-structuresrdquoPhysics Letters A vol 363 no 3 pp 236ndash242 2007
[2] M Aydogdu ldquoA general nonlocal beam theory its applicationto nanobeam bending buckling and vibrationrdquo Physica E vol41 no 9 pp 1651ndash1655 2009
[3] O Civalek and C Demir ldquoBending analysis of microtubulesusing nonlocal Euler-Bernoulli beam theoryrdquo Applied Mathe-matical Modelling vol 35 no 5 pp 2053ndash2067 2011
[4] M A De Rosa and C Franciosi ldquoA simple approach to detectthe nonlocal effects in the static analysis of Euler-Bernoulli andTimoshenko beamsrdquoMechanics Research Communications vol48 pp 66ndash69 2013
[5] M Aydogdu ldquoAxial vibration analysis of nanorods (carbonnanotubes) embedded in an elastic medium using nonlocalelasticityrdquoMechanics Research Communications vol 41 pp 34ndash40 2012
[6] M A Kazemi-Lari S A Fazelzadeh and E Ghavanloo ldquoNon-conservative instability of cantilever carbon nanotubes restingon viscoelastic foundationrdquo Physica E vol 44 no 7-8 pp 1623ndash1630 2012
[7] H-T Thai and T P Vo ldquoA nonlocal sinusoidal shear deforma-tion beam theory with application to bending buckling andvibration of nanobeamsrdquo International Journal of EngineeringScience vol 54 pp 58ndash66 2012
[8] M A Eltaher S A Emam and F F Mahmoud ldquoStatic andstability analysis of nonlocal functionally graded nanobeamsrdquoComposite Structures vol 96 pp 82ndash88 2013
[9] S A Emam ldquoA general nonlocal nonlinear model for bucklingof nanobeamsrdquo Applied Mathematical Modelling vol 37 no 10-11 pp 6929ndash6939 2013
[10] B Fang Y-X Zhen C-P Zhang and Y Tang ldquoNonlinearvibration analysis of double-walled carbon nanotubes based onnonlocal elasticity theoryrdquoAppliedMathematicalModelling vol37 no 3 pp 1096ndash1107 2013
[11] S A M Ghannadpour B Mohammadi and J Fazilati ldquoBend-ing buckling and vibration problems of nonlocal Euler beamsusing Ritz methodrdquo Composite Structures vol 96 pp 584ndash5892013
6 Advances in Materials Science and Engineering
[12] M Simsek and H H Yurtcu ldquoAnalytical solutions for bendingand buckling of functionally graded nanobeams based on thenonlocal Timoshenko beam theoryrdquo Composite Structures vol97 pp 378ndash386 2013
[13] B L Wang and K F Wang ldquoVibration analysis of embeddednanotubes using nonlocal continuum theoryrdquo Composites Bvol 47 pp 96ndash101 2013
[14] J K Phadikar and S C Pradhan ldquoVariational formulationand finite element analysis for nonlocal elastic nanobeams andnanoplatesrdquo Computational Materials Science vol 49 no 3 pp492ndash499 2010
[15] C M C Roque A J M Ferreira and J N Reddy ldquoAnalysisof Timoshenko nanobeams with a nonlocal formulation andmeshless methodrdquo International Journal of Engineering Sciencevol 49 no 9 pp 976ndash984 2011
[16] F F Mahmoud M A Eltaher A E Alshorbagy and E IMeletis ldquoStatic analysis of nanobeams including surface effectsby nonlocal finite elementsrdquo Journal of Mechanical Science andTechnology vol 26 no 11 pp 3555ndash3563 2012
[17] S C Pradhan ldquoNonlocal finite element analysis and small scaleeffects of CNTs with Timoshenko beam theoryrdquo Finite Elementsin Analysis and Design vol 50 pp 8ndash20 2012
[18] M A Eltaher A E Alshorbagy and F F Mahmoud ldquoVibrationanalysis of Euler-Bernoulli nanobeams by using finite elementmethodrdquo Applied Mathematical Modelling vol 37 no 7 pp4787ndash4797 2013
[19] B Arash and Q Wang ldquoA review on the application ofnonlocal elastic models in modeling of carbon nanotubes andgraphenesrdquo Computational Materials Science vol 51 no 1 pp303ndash313 2012
[20] Y YanW QWang and L X Zhang ldquoNonlocal effect on axiallycompressed buckling of triple-walled carbon nanotubes undertemperature fieldrdquo Applied Mathematical Modelling vol 34 no11 pp 3422ndash3429 2010
[21] R Rafiee and R M Moghadam ldquoOn the modeling of carbonnanotubes a critical reviewrdquo Composites B vol 56 pp 435ndash4490 2014
[22] J N Reddy ldquoNonlocal theories for bending buckling and vibra-tion of beamsrdquo International Journal of Engineering Science vol45 no 2-8 pp 288ndash307 2007
[23] A C Eringen ldquoOn differential equations of nonlocal elasticityand solutions of screw dislocation and surface wavesrdquo Journalof Applied Physics vol 54 no 9 pp 4703ndash4710 1983
[24] F Marotti de Sciarra ldquoVariational formulations convergenceand stability properties in nonlocal elastoplasticityrdquo Interna-tional Journal of Solids and Structures vol 45 no 7-8 pp 2322ndash2354 2008
[25] G RomanoMDiaco andR Barretta ldquoVariational formulationof the first principle of continuum thermodynamicsrdquo Contin-uumMechanics andThermodynamics vol 22 no 3 pp 177ndash1872010
[26] FMarotti De Sciarra ldquoHardening plasticitywith nonlocal straindamagerdquo International Journal of Plasticity vol 34 pp 114ndash1382012
[27] G Romano and R Barretta ldquoCovariant hypo-elasticityrdquo Euro-pean Journal of Mechanics A vol 30 no 6 pp 1012ndash1023 2011
[28] G Romano and R Barretta ldquoOn Eulerrsquos stretching formula incontinuummechanicsrdquo Acta Mechanica vol 224 no 1 pp 211ndash230 2013
[29] G Romano and R Barretta ldquoGeometric constitutive theoryand frame invariancerdquo International Journal of Non-LinearMechanics vol 51 pp 75ndash86 2013
[30] G Romano R Barretta and M Diaco ldquoGeometric continuummechanicsrdquoMeccanica 2013
[31] R D Mindlin ldquoMicro-structure in linear elasticityrdquo Archive forRational Mechanics and Analysis vol 16 no 1 pp 51ndash78 1964
[32] F Marotti de Sciarra ldquoNovel variational formulations fornonlocal plasticityrdquo International Journal of Plasticity vol 25no 2 pp 302ndash331 2009
[33] F Marotti de Sciarra ldquoOn non-local and non-homogeneouselastic continuardquo International Journal of Solids and Structuresvol 46 no 3-4 pp 651ndash676 2009
[34] F Marotti de Sciarra ldquoA nonlocal model with strain-baseddamagerdquo International Journal of Solids and Structures vol 46no 22-23 pp 4107ndash4122 2009
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
thinspInternationalthinspJournalthinspof
BiomaterialsHindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NaNoscieNceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
CrystallographyJournal of
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
BioMed Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
MaterialsJournal of
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
6 Advances in Materials Science and Engineering
[12] M Simsek and H H Yurtcu ldquoAnalytical solutions for bendingand buckling of functionally graded nanobeams based on thenonlocal Timoshenko beam theoryrdquo Composite Structures vol97 pp 378ndash386 2013
[13] B L Wang and K F Wang ldquoVibration analysis of embeddednanotubes using nonlocal continuum theoryrdquo Composites Bvol 47 pp 96ndash101 2013
[14] J K Phadikar and S C Pradhan ldquoVariational formulationand finite element analysis for nonlocal elastic nanobeams andnanoplatesrdquo Computational Materials Science vol 49 no 3 pp492ndash499 2010
[15] C M C Roque A J M Ferreira and J N Reddy ldquoAnalysisof Timoshenko nanobeams with a nonlocal formulation andmeshless methodrdquo International Journal of Engineering Sciencevol 49 no 9 pp 976ndash984 2011
[16] F F Mahmoud M A Eltaher A E Alshorbagy and E IMeletis ldquoStatic analysis of nanobeams including surface effectsby nonlocal finite elementsrdquo Journal of Mechanical Science andTechnology vol 26 no 11 pp 3555ndash3563 2012
[17] S C Pradhan ldquoNonlocal finite element analysis and small scaleeffects of CNTs with Timoshenko beam theoryrdquo Finite Elementsin Analysis and Design vol 50 pp 8ndash20 2012
[18] M A Eltaher A E Alshorbagy and F F Mahmoud ldquoVibrationanalysis of Euler-Bernoulli nanobeams by using finite elementmethodrdquo Applied Mathematical Modelling vol 37 no 7 pp4787ndash4797 2013
[19] B Arash and Q Wang ldquoA review on the application ofnonlocal elastic models in modeling of carbon nanotubes andgraphenesrdquo Computational Materials Science vol 51 no 1 pp303ndash313 2012
[20] Y YanW QWang and L X Zhang ldquoNonlocal effect on axiallycompressed buckling of triple-walled carbon nanotubes undertemperature fieldrdquo Applied Mathematical Modelling vol 34 no11 pp 3422ndash3429 2010
[21] R Rafiee and R M Moghadam ldquoOn the modeling of carbonnanotubes a critical reviewrdquo Composites B vol 56 pp 435ndash4490 2014
[22] J N Reddy ldquoNonlocal theories for bending buckling and vibra-tion of beamsrdquo International Journal of Engineering Science vol45 no 2-8 pp 288ndash307 2007
[23] A C Eringen ldquoOn differential equations of nonlocal elasticityand solutions of screw dislocation and surface wavesrdquo Journalof Applied Physics vol 54 no 9 pp 4703ndash4710 1983
[24] F Marotti de Sciarra ldquoVariational formulations convergenceand stability properties in nonlocal elastoplasticityrdquo Interna-tional Journal of Solids and Structures vol 45 no 7-8 pp 2322ndash2354 2008
[25] G RomanoMDiaco andR Barretta ldquoVariational formulationof the first principle of continuum thermodynamicsrdquo Contin-uumMechanics andThermodynamics vol 22 no 3 pp 177ndash1872010
[26] FMarotti De Sciarra ldquoHardening plasticitywith nonlocal straindamagerdquo International Journal of Plasticity vol 34 pp 114ndash1382012
[27] G Romano and R Barretta ldquoCovariant hypo-elasticityrdquo Euro-pean Journal of Mechanics A vol 30 no 6 pp 1012ndash1023 2011
[28] G Romano and R Barretta ldquoOn Eulerrsquos stretching formula incontinuummechanicsrdquo Acta Mechanica vol 224 no 1 pp 211ndash230 2013
[29] G Romano and R Barretta ldquoGeometric constitutive theoryand frame invariancerdquo International Journal of Non-LinearMechanics vol 51 pp 75ndash86 2013
[30] G Romano R Barretta and M Diaco ldquoGeometric continuummechanicsrdquoMeccanica 2013
[31] R D Mindlin ldquoMicro-structure in linear elasticityrdquo Archive forRational Mechanics and Analysis vol 16 no 1 pp 51ndash78 1964
[32] F Marotti de Sciarra ldquoNovel variational formulations fornonlocal plasticityrdquo International Journal of Plasticity vol 25no 2 pp 302ndash331 2009
[33] F Marotti de Sciarra ldquoOn non-local and non-homogeneouselastic continuardquo International Journal of Solids and Structuresvol 46 no 3-4 pp 651ndash676 2009
[34] F Marotti de Sciarra ldquoA nonlocal model with strain-baseddamagerdquo International Journal of Solids and Structures vol 46no 22-23 pp 4107ndash4122 2009
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
thinspInternationalthinspJournalthinspof
BiomaterialsHindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NaNoscieNceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
CrystallographyJournal of
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
BioMed Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
MaterialsJournal of
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
thinspInternationalthinspJournalthinspof
BiomaterialsHindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NaNoscieNceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
CrystallographyJournal of
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
BioMed Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
MaterialsJournal of
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials