A Nonlocal Model for Carbon Nanotubes Under Axial Loads

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)


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


1 times 10minus7

2 4 6 8 10

8 times 10minus8

6 times 10minus8

4 times 10minus8

2 times 10minus8

minus2 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)


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


[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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CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of

BioMed Research International



Volume 2014

MaterialsJournal of

Nano

materials


Journal ofNanomaterials


[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)





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)


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



120572119908(4)

minus 120573119908(2)

= 119901 (17)


119908 (119911) = 119908119867

(119911) + 119908 (119911) (18)

with

119908119867

(119911) = 1198881+ 1198882119911 + 1198883exp(radic

120573

120572

119911)


120573

120572

119911)

119908 (119911) = 11988851199112

(19)

z

L

p

N(0)

A B



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)


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)


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)


20

15

10

05

2 4 6 8 10

z (nm)

w(n

m)



minus8 Nnm


119908 (119911) = 11988851199112 (23)




1198881= minus (119888

3+ 1198884)

1198882= minus2119871119888

5

1198883=

21198882(1 minus exp (minus (1119888) 119871))


1198885

1198884=

21198882(1 minus exp ((1119888) 119871))


1198885

(24)

having 1198885


119908low


119908 (119911 119888) = 0208333119911 (25)


low(119871) =


0 5 10 15 20 250

05

1

15

2

25

c (nm)

w(L2)

andw(L)

(nm

)




119888 (nm) 119908(1198712) (nm) 119908(119871) (nm)0 15625 2083331 152139 2083332 142301 2083333 132428 2083334 124886 2083335 119589 20833325 105021 208333


119908upp

(119911) = 119908 (119911 0) =

119901 (119911) (2119871 minus 119911) 119911

2119864119860

(26)






(1)

1


5 Conclusions




1 times 10minus7

2 4 6 8 10

8 times 10minus8

6 times 10minus8

4 times 10minus8

2 times 10minus8



z (nm)

N (N

)


1(blue line) 119873

119900(dashed line)

119873(1)







Appendix


120572119908(4)

minus 120573119908(2)

= 119891 (A1)


120572119908(4)

minus 120573119908(2)

= 0 (A2)


minus

1205731205822


1= 0



=


119908119867

(119911) = 1198881+ 1198882119911 + 1198883exp(radic

120573

120572

119911)


120573

120572

119911)

(A3)



119908 (119911) = 119908119867

(119911) + 119908 (119911) (A4)




119908 (119911) = 1199112

(1198600+ 1198601119911 + sdot sdot sdot + 119860

119898119911119898) (A5)

with 119860119894isin R for 119894 = 1 119898

Acknowledgments


References











































CeramicsJournal of










Journal of


Journal of


Volume 2014




CoatingsJournal of

Advances in









Volume 2014

MaterialsJournal of

Nano

materials







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)


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



120572119908(4)

minus 120573119908(2)

= 119901 (17)


119908 (119911) = 119908119867

(119911) + 119908 (119911) (18)

with

119908119867

(119911) = 1198881+ 1198882119911 + 1198883exp(radic

120573

120572

119911)


120573

120572

119911)

119908 (119911) = 11988851199112

(19)

z

L

p

N(0)

A B



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)


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)


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)


20

15

10

05

2 4 6 8 10

z (nm)

w(n

m)



minus8 Nnm


119908 (119911) = 11988851199112 (23)




1198881= minus (119888

3+ 1198884)

1198882= minus2119871119888

5

1198883=

21198882(1 minus exp (minus (1119888) 119871))


1198885

1198884=

21198882(1 minus exp ((1119888) 119871))


1198885

(24)

having 1198885


119908low


119908 (119911 119888) = 0208333119911 (25)


low(119871) =


0 5 10 15 20 250

05

1

15

2

25

c (nm)

w(L2)

andw(L)

(nm

)




119888 (nm) 119908(1198712) (nm) 119908(119871) (nm)0 15625 2083331 152139 2083332 142301 2083333 132428 2083334 124886 2083335 119589 20833325 105021 208333


119908upp

(119911) = 119908 (119911 0) =

119901 (119911) (2119871 minus 119911) 119911

2119864119860

(26)






(1)

1


5 Conclusions




1 times 10minus7

2 4 6 8 10

8 times 10minus8

6 times 10minus8

4 times 10minus8

2 times 10minus8



z (nm)

N (N

)


1(blue line) 119873

119900(dashed line)

119873(1)







Appendix


120572119908(4)

minus 120573119908(2)

= 119891 (A1)


120572119908(4)

minus 120573119908(2)

= 0 (A2)


minus

1205731205822


1= 0



=


119908119867

(119911) = 1198881+ 1198882119911 + 1198883exp(radic

120573

120572

119911)


120573

120572

119911)

(A3)



119908 (119911) = 119908119867

(119911) + 119908 (119911) (A4)




119908 (119911) = 1199112

(1198600+ 1198601119911 + sdot sdot sdot + 119860

119898119911119898) (A5)

with 119860119894isin R for 119894 = 1 119898

Acknowledgments


References











































CeramicsJournal of










Journal of


Journal of


Volume 2014




CoatingsJournal of

Advances in









Volume 2014

MaterialsJournal of

Nano

materials




20

15

10

05

2 4 6 8 10

z (nm)

w(n

m)



minus8 Nnm


119908 (119911) = 11988851199112 (23)




1198881= minus (119888

3+ 1198884)

1198882= minus2119871119888

5

1198883=

21198882(1 minus exp (minus (1119888) 119871))


1198885

1198884=

21198882(1 minus exp ((1119888) 119871))


1198885

(24)

having 1198885


119908low


119908 (119911 119888) = 0208333119911 (25)


low(119871) =


0 5 10 15 20 250

05

1

15

2

25

c (nm)

w(L2)

andw(L)

(nm

)




119888 (nm) 119908(1198712) (nm) 119908(119871) (nm)0 15625 2083331 152139 2083332 142301 2083333 132428 2083334 124886 2083335 119589 20833325 105021 208333


119908upp

(119911) = 119908 (119911 0) =

119901 (119911) (2119871 minus 119911) 119911

2119864119860

(26)






(1)

1


5 Conclusions




1 times 10minus7

2 4 6 8 10

8 times 10minus8

6 times 10minus8

4 times 10minus8

2 times 10minus8



z (nm)

N (N

)


1(blue line) 119873

119900(dashed line)

119873(1)







Appendix


120572119908(4)

minus 120573119908(2)

= 119891 (A1)


120572119908(4)

minus 120573119908(2)

= 0 (A2)


minus

1205731205822


1= 0



=


119908119867

(119911) = 1198881+ 1198882119911 + 1198883exp(radic

120573

120572

119911)


120573

120572

119911)

(A3)



119908 (119911) = 119908119867

(119911) + 119908 (119911) (A4)




119908 (119911) = 1199112

(1198600+ 1198601119911 + sdot sdot sdot + 119860

119898119911119898) (A5)

with 119860119894isin R for 119894 = 1 119898

Acknowledgments


References











































CeramicsJournal of










Journal of


Journal of


Volume 2014




CoatingsJournal of

Advances in









Volume 2014

MaterialsJournal of

Nano

materials




1 times 10minus7

2 4 6 8 10

8 times 10minus8

6 times 10minus8

4 times 10minus8

2 times 10minus8



z (nm)

N (N

)


1(blue line) 119873

119900(dashed line)

119873(1)







Appendix


120572119908(4)

minus 120573119908(2)

= 119891 (A1)


120572119908(4)

minus 120573119908(2)

= 0 (A2)


minus

1205731205822


1= 0



=


119908119867

(119911) = 1198881+ 1198882119911 + 1198883exp(radic

120573

120572

119911)


120573

120572

119911)

(A3)



119908 (119911) = 119908119867

(119911) + 119908 (119911) (A4)




119908 (119911) = 1199112

(1198600+ 1198601119911 + sdot sdot sdot + 119860

119898119911119898) (A5)

with 119860119894isin R for 119894 = 1 119898

Acknowledgments


References











































CeramicsJournal of










Journal of


Journal of


Volume 2014




CoatingsJournal of

Advances in









Volume 2014

MaterialsJournal of

Nano

materials


































CeramicsJournal of










Journal of


Journal of


Volume 2014




CoatingsJournal of

Advances in









Volume 2014

MaterialsJournal of

Nano

materials



A Nonlocal Model for Carbon Nanotubes Under Axial Loads

Documents

Transcript of A Nonlocal Model for Carbon Nanotubes Under Axial Loads