Research Article A Note on Torsion of Nonlocal Composite ... · random composites [ ], and nonlocal...

Research ArticleA Note on Torsion of Nonlocal Composite Nanobeams

Luciano Feo and Rosa Penna

Department of Civil Engineering University of Salerno Via Ponte don Melillo 84084 Fisciano Italy

Correspondence should be addressed to Luciano Feo lfeounisait

Received 4 December 2015 Accepted 5 October 2016

Academic Editor Theodoros C Rousakis

Copyright copy 2016 L Feo and R PennaThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

TheEringen elastic constitutive relation is used in this paper in order to assess small-scale effects in nanobeams Structural behavioris studied for functionally graded materials in the cross-sectional plane and torsional loading conditions The governing boundaryvalue problem has been formulated in a mixed framework Torsional rotations and equilibrated moments are evaluated by solvinga first-order differential equation of elastic equilibrium with boundary conditions of kinematic-type Benchmarks examples arebriefly discussed enlightening thus effectiveness of the proposed methodology

1 Introduction

Assessments of stress and displacement fields in continuousmedia are a subject of special interest in the theory ofstructures Numerous case studies have been examined inthe current literature with reference to beams [1ndash6] half-spaces [7 8] thin plates [9 10] compressible cubes [11]and concrete [12 13] Several methodologies of analysis havebeen developed in the research field of geometric continuummechanics [14 15] limit analysis [16ndash19] homogenization[20] elastodynamics [21ndash25] thermal problems [26ndash28]random composites [29ndash32] and nonlocal and gradientformulations [33ndash38] A comprehensive analysis of classicaland generalized models of elastic structures with specialemphasis on rods can be found in the interesting book byIesan [39] In particular Iesan [40ndash42] formulated amethodfor the solution of Saint-Venant problems in micropolarbeams with arbitrary cross-section Detailed solution of thetorsion problem for an isotropic micropolar beam withcircular cross-section is given in [43 44] Experimentalinvestigations are required for the evaluation of the behaviorof composite structures [45]

In the context of the present research particular attentionis devoted to the investigation of scale effects in nanostruc-tures see for example [46ndash55] and the reviews [56 57]Recent contributions on functionally graded materials havebeen developed for nanobeams under flexure [58 59] andtorsion [60]

Unlike previous treatments on torsion of gradient elasticbars (see eg [61]) in which higher-order boundary condi-tions have to be enforced this paper is concerned with theanalysis of composite nanobeams with nonlocal constitutivebehavior conceived by Eringen in [62] Basic equationsgoverning the Eringen model are preliminarily recalled inSection 2 The corresponding elastic equilibrium problem oftorsion of an Eringen circular nanobeam is then formulatedin Section 3 It is worth noting that only classical boundaryconditions are involved in the present study Small-scaleeffects are detected in Section 4 for two static schemes ofapplicative interest Some concluding remarks are delineatedin Section 5

2 Eringen Nonlocal Elastic Model

Before formulating the elastostatic problem of a nonlocalnanobeam subjected to torsion we shortly recall in thesequel some notions of nonlocal elasticity To this end let usconsider a body B made of a material possibly compositecharacterized by the following integral relation between thestress t119894119895 at a point x and the elastic strain field e119894119895 inB [62]

t119894119895 (x) = intB

119870(10038161003816100381610038161003816x1015840 minus x10038161003816100381610038161003816 120591)E119894119895ℎ119896 (x1015840) eℎ119896 (x1015840) 119889119881 (1)

The fourth-order tensor E119894119895ℎ119896(x1015840) symmetric and positivedefinite describes the material elastic stiffness at the pointx1015840 isin B

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2016 Article ID 5934814 5 pageshttpdxdoiorg10115520165934814

2 Modelling and Simulation in Engineering

The attenuation function 119870 depends on the Euclideandistance |x1015840 minus x| and on a nonlocal dimensionless parameterdefined by

120591 = 1198900119886119897 (2)

where 1198900 is a material constant 119886 is an internal characteristiclength and 119897 stands for external characteristic length

Assuming a suitable expression of the nonlocal modulus119870 in terms of a variant of the Bessel function we get theinverse differential relationship of (1) between the nonlocalstress and the elastic deformation

(1 minus (1198900119886)2 nabla2) t119894119895 = E119894119895ℎ119896eℎ119896 (3)

with nabla2 the Laplacian Note that (3) can be conveniently usedin order to describe the law between the nonlocal shear stressfield 120591119894 on the cross-section of a nanobeam and the elasticshear strain 120574119894 as follows

120591119894 minus (1198900119886)2 1198892120591119894

1198891199092 = 120583120574119894 (4)

where 119909 is the axial direction and 120583 is the shear modulus

3 Torsion of Nonlocal Circular Nanobeams

Let Ω be the cross-section of a circular nanobeam of length119871 subjected to the following loading conditions depicted inFigure 1119898119905distributed couples per unit length in the interval [0 119871] M119905

concentrated couples at the end cross-sections 0 119871

(5)

The triplet (119909 119910 119911) describes a set of Cartesian axes originat-ing at the left cross-section centreO

Equilibrium equations are expressed by119889119872119905119889119909 = minus119898119905 in [0 119871] (6a)

119872119905 = M119905 at 0 119871 (6b)

where119872119905 is the twisting momentComponents of the displacement field up to a rigid body

motion of a circular nanobeam under torsion write as119904119909 (119909 119910119911) = 0119904119910 (119909 119910119911) = minus120579 (119909) 119911119904119911 (119909 119910119911) = 120579 (119909) 119910

(7)

where 120579(119909) is the torsional rotation of the cross-section at theabscissa 119909 Shear strains compatible with the displacementfield equation (7) are given by

120574119910119909 (119909 119910 119911) = minus119889120579119889119909119909119911

120574119911119909 (119909 119910 119911) = 119889120579119889119909119909119910

(8)

where

120594119905 (119909) = 119889120579119889119909 (119909) (9)

is the twisting curvature The shear modulus 120583 is assumed tobe functionally graded only in the cross-sectional plane (119910 119911)

The elastic twisting stiffness is provided by

119896119905 = intΩ120583 (1199102 + 1199112) 119889119860 (10)

where the symbol sdot is the inner product between vectorsThe differential equation of nonlocal elastic equilibrium

of a nanobeam under torsion is formulated as follows Let uspreliminarily multiply (4) by Rr fl minus119911 119910 and integrate onΩ

intΩRr sdot 120591119889119860 minus (1198900119886)2 int

ΩRr sdot 119889120591

1198891199092 119889119860 = intΩ120583Rr sdot 120574119889119860 (11)

with the vector 120574 fl 120574119910119909 120574119911119909 given by (8)Enforcing (6a) and imposing the static equivalence con-

dition

119872119905 = intΩ(120591119911119909119910 minus 120591119910119909119911) 119889119860 (12)

we get the relation

119872119905 (119909) + (1198900119886)2 119889119898119905119889119909 (119909) = 119896119905 119889120579119889119909 (119909) (13)

This equation can be interpreted as decomposition formula ofthe twisting curvature 120594119905 into elastic (120594119905)el and inelastic (120594119905)inparts

120594119905 = (120594119905)el + (120594119905)in (14)

with

(120594119905)el = 119872119905119896119905 (15a)

(120594119905)in = (1198900119886)2119896119905

119889119898119905119889119909 (15b)

Accordingly the scale effect exhibited by the torsional rota-tion function of a nonlocal nanobeam can be evaluated bysolving a corresponding linearly elastic beam subjected to thetwisting curvature distortion (120594119905)in (15b)

4 Benchmark Examples

Let us consider a nanocantilever and a fully clampednanobeam of length 119871 subjected to the following quadraticdistribution of couples per unit length

119898119905 = 1198981198712 1199092 (16)

The cross-sectional torsional rotation is evaluated by fol-lowing the methodology illustrated in the previous section

Modelling and Simulation in Engineering 3

ℳt ℳt

m yy

z

z L

x

t

Ω

O

Figure 1 Sketch of a nanobeam under torsion

120591 = 02120591 = 01

120591 = 03

120591 = 04

120591 = 05

120591

120591

= 0

120579lowast

01

02

03

04

05

04 06 08 1002120585

Figure 2 Torsional rotation 120579lowast for the nanocantilever versus 120585 forincreasing values of 120591

120591 = 015

120591 = 01

120591 = 02

120591 = 025

120591 = 03

120591

120591

= 0

120579lowast

001

002

003

004

04 06 08 1002120585

Figure 3 Dimensionless torsional rotation 120579lowast for the fully clampednanobeam versus 120585 for increasing values of 120591

The nonlocality effect is assessed by prescribing the twistingcurvature distortion equation (15b)

(120594119905)in = (1198900119886)2119896119905

21198981198712 119909 (17)

on corresponding (cantilever and fully clamped) localnanobeams Let us set 120585 = 119909119871 and 120579lowast(120585) = 119896119905(1198981198712)120579(120585)Torsional rotations 120579lowast versus 120585 of both the nanobeams aredisplayed in Figures 2 and 3 for selected values of the nonlocalparameter 120591 fl 1198900119886119871

5 Concluding Remarks

Thebasic outcomes contributed in the present paper are listedas follows

(1) Size-effects in nanobeams under torsion have beenevaluated by resorting to the nonlocal theory ofelasticity

(2) Exact torsional rotations solutions of cross-sections offunctionally gradednanobeamshave been establishedfor nanocantilevers and fully clamped nanobeamsunder a quadratic distribution of couples per unitlength

(3) It has been observed that the stiffness of a nanobeamunder torsional loadings is affected by the scaleparameter and depends on the boundary kinematicconstraints Indeed as shown in Figures 2 and 3contrary to the nanocantilever structural behaviorthe fully clamped nanobeam becomes stiffer forincreasing values of the nonlocal parameter

Competing Interests

There is no conflict of interests related to this paper

References

[1] R Barretta ldquoOn Cesaro-Volterra method in orthotropic Saint-Venant beamrdquo Journal of Elasticity vol 112 no 2 pp 233ndash2532013

[2] R Barretta and M Diaco ldquoOn the shear centre in Saint-Venantbeam theoryrdquoMechanics Research Communications vol 52 pp52ndash56 2013

[3] G Romano A Barretta and R Barretta ldquoOn torsion andshear of Saint-Venant beamsrdquo European Journal of MechanicsA Solids vol 35 pp 47ndash60 2012

[4] R Barretta ldquoOn stress function in Saint-Venant beamsrdquoMecca-nica vol 48 no 7 pp 1811ndash1816 2013

[5] F Marmo and L Rosati ldquoAnalytical integration of elasto-plasticuniaxial constitutive laws over arbitrary sectionsrdquo InternationalJournal for Numerical Methods in Engineering vol 91 no 9 pp990ndash1022 2012

[6] F Marmo and L Rosati ldquoThe fiber-free approach in the eval-uation of the tangent stiffness matrix for elastoplastic uniaxialconstitutive lawsrdquo International Journal for Numerical Methodsin Engineering vol 94 no 9 pp 868ndash894 2013

[7] L Rosati and F Marmo ldquoClosed-form expressions of thethermo-mechanical fields induced by a uniform heat sourceacting over an isotropic half-spacerdquo International Journal ofHeat and Mass Transfer vol 75 pp 272ndash283 2014

[8] F Marmo and L Rosati ldquoA general approach to the solutionof Boussinesqrsquos problem for polynomial pressures acting overpolygonal domainsrdquo Journal of Elasticity vol 122 no 1 pp 75ndash112 2016


[9] R Barretta ldquoAnalogies between Kirchhoff plates and Saint-Venant beams under torsionrdquo Acta Mechanica vol 224 no 12pp 2955ndash2964 2013

[10] R Barretta ldquoAnalogies between Kirchhoff plates and Saint-Venant beams under flexurerdquo Acta Mechanica vol 225 no 7pp 2075ndash2083 2014

[11] A M Tarantino ldquoHomogeneous equilibrium configurations ofa hyperelastic compressible cube under equitriaxial dead-loadtractionsrdquo Journal of Elasticity vol 92 no 3 pp 227ndash254 2008

[12] V A Salomoni GMazzucco C Pellegrino andC EMajoranaldquoThree-dimensional modelling of bond behaviour betweenconcrete and FRP reinforcementrdquo Engineering Computationsvol 28 no 1 pp 5ndash29 2011

[13] V A Salomoni C E Majorana B Pomaro G Xotta andF Gramegna ldquoMacroscale and mesoscale analysis of concreteas a multiphase material for biological shields against nuclearradiationrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 38 no 5 pp 518ndash535 2014

[14] G Romano R Barretta and M Diaco ldquoThe geometry ofnonlinear elasticityrdquo Acta Mechanica vol 225 no 11 pp 3199ndash3235 2014

[15] G Romano R Barretta and M Diaco ldquoGeometric continuummechanicsrdquoMeccanica vol 49 no 1 pp 111ndash133 2014

[16] A Caporale R Luciano and L Rosati ldquoLimit analysis ofmasonry arches with externally bonded FRP reinforcementsrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 1ndash3 pp 247ndash260 2006

[17] A Caporale and R Luciano ldquoLimit analysis of masonry archeswith finite compressive strength and externally bonded rein-forcementrdquo Composites Part B Engineering vol 43 no 8 pp3131ndash3145 2012

[18] A Caporale L Feo R Luciano and R Penna ldquoNumericalcollapse load of multi-span masonry arch structures with FRPreinforcementrdquo Composites Part B Engineering vol 54 no 1pp 71ndash84 2013

[19] A Caporale L Feo D Hui and R Luciano ldquoDebonding ofFRP in multi-span masonry arch structures via limit analysisrdquoComposite Structures vol 108 no 1 pp 856ndash865 2014

[20] F Greco and R Luciano ldquoA theoretical and numerical stabilityanalysis for composite micro-structures by using homogeniza-tion theoryrdquo Composites Part B Engineering vol 42 no 3 pp382ndash401 2011

[21] A M Tarantino ldquoNonlinear fracture mechanics for an elasticBell materialrdquo The Quarterly Journal of Mechanics and AppliedMathematics vol 50 no 3 pp 435ndash456 1997

[22] A M Tarantino ldquoThe singular equilibrium field at the notch-tip of a compressible material in finite elastostaticsrdquo Journal ofApplied Mathematics and Physics vol 48 no 3 pp 370ndash3881997

[23] AM Tarantino ldquoOn extreme thinning at the notch tip of a neo-Hookean sheetrdquoTheQuarterly Journal ofMechanics andAppliedMathematics vol 51 no 2 pp 179ndash190 1998

[24] A M Tarantino ldquoOn the finite motions generated by a modeI propagating crackrdquo Journal of Elasticity vol 57 no 2 pp 85ndash103 1999

[25] A M Tarantino ldquoCrack propagation in finite elastodynamicsrdquoMathematics andMechanics of Solids vol 10 no 6 pp 577ndash6012005

[26] V A Salomoni C E Majorana G M Giannuzzi and AMiliozzi ldquoThermal-fluid flow within innovative heat storageconcrete systems for solar power plantsrdquo International Journal

of Numerical Methods for Heat and Fluid Flow vol 18 no 7-8pp 969ndash999 2008

[27] FMarotti de Sciarra andM Salerno ldquoOn thermodynamic func-tions in thermoelasticity without energy dissipationrdquo EuropeanJournal of Mechanics A Solids vol 46 pp 84ndash95 2014

[28] G Xotta GMazzucco V A Salomoni C EMajorana andK JWillam ldquoComposite behavior of concrete materials under hightemperaturesrdquo International Journal of Solids and Structures vol64 pp 86ndash99 2015

[29] R Luciano and J R Willis ldquoBounds on non-local effectiverelations for random composites loaded by configuration-dependent body forcerdquo Journal of the Mechanics and Physics ofSolids vol 48 no 9 pp 1827ndash1849 2000

[30] R Luciano and J R Willis ldquoNon-local effective relationsfor fibre-reinforced composites loaded by configuration-dependent body forcesrdquo Journal of the Mechanics and Physics ofSolids vol 49 no 11 pp 2705ndash2717 2001

[31] R Luciano and J R Willis ldquoBoundary-layer corrections forstress and strain fields in randomly heterogeneous materialsrdquoJournal of the Mechanics and Physics of Solids vol 51 no 6 pp1075ndash1088 2003

[32] R Luciano and J R Willis ldquoHashin-Shtrikman based FEanalysis of the elastic behaviour of finite random compositebodiesrdquo International Journal of Fracture vol 137 no 1ndash4 pp261ndash273 2006

[33] R De Borst and H-B Muehlhaus ldquoGradient-dependent plas-ticity formulation and algorithmic aspectsrdquo International Jour-nal for Numerical Methods in Engineering vol 35 no 3 pp 521ndash539 1992

[34] E C Aifantis ldquoStrain gradient interpretation of size effectsrdquoInternational Journal of Fracture vol 95 no 1ndash4 pp 299ndash3141999

[35] E C Aifantis ldquoGradient deformation models at nano microand macro scalesrdquo Journal of Engineering Materials and Tech-nology Transactions of the ASME vol 121 no 2 pp 189ndash2021999

[36] R H J Peerlings R De Borst W A M Brekelmans and JH P De Vree ldquoGradient enhanced damage for quasi-brittlematerialsrdquo International Journal for Numerical Methods inEngineering vol 39 no 19 pp 3391ndash3403 1996

[37] R H J Peerlings M G D Geers R de Borst and W A MBrekelmans ldquoA critical comparison of nonlocal and gradient-enhanced softening continuardquo International Journal of Solidsand Structures vol 38 no 44-45 pp 7723ndash7746 2001

[38] H Askes and E C Aifantis ldquoGradient elasticity in statics anddynamics an overview of formulations length scale identi-fication procedures finite element implementations and newresultsrdquo International Journal of Solids and Structures vol 48no 13 pp 1962ndash1990 2011

[39] D Iesan ldquoClassical and generalized models of elastic rodsrdquo inModern Mechanics and Mathematics D Gao and R W OgdenEds CRC Series CRC Press 2008

[40] D Iesan ldquoSaint-Venantrsquos problem in micropolar elasticityrdquo inMechanics of Micropolar Media O Brulin and R K T HsiehEds pp 281ndash390 World Scientific Singapore 1982

[41] D Iesan ldquoGeneralized twist for the torsion of micropolarcylindersrdquoMeccanica vol 21 no 2 pp 94ndash96 1986

[42] D Iesan ldquoGeneralized torsion of elastic cylinders withmicrostructurerdquo Note di Matematica vol 27 no 2 pp 121ndash1302007


[43] R D Gauthier andW E Jahsman ldquoQuest formicropolar elasticconstantsrdquo Journal of Applied Mechanics Transactions ASMEvol 42 no 2 pp 369ndash374 1975

[44] G V K Reddy and N K Venkatasubramanian ldquoSaint-Venantrsquosproblem for amicropolar elastic circular cylinderrdquo InternationalJournal of Engineering Science vol 14 no 11 pp 1047ndash1057 1976

[45] G Dinelli G Belz C E Majorana and B A Schrefler ldquoExperi-mental investigation on the use of fly ash for lightweight precaststructural elementsrdquo Materials and StructuresMateriaux etConstructions vol 29 no 194 pp 632ndash638 1996

[46] J Peddieson G R Buchanan and R P McNitt ldquoApplication ofnonlocal continuum models to nanotechnologyrdquo InternationalJournal of Engineering Science vol 41 no 3ndash5 pp 305ndash3122003

[47] J N Reddy ldquoNonlocal theories for bending buckling and vibra-tion of beamsrdquo International Journal of Engineering Science vol45 no 2ndash8 pp 288ndash307 2007

[48] Q Wang and K M Liew ldquoApplication of nonlocal continuummechanics to static analysis of micro- and nano-structuresrdquoPhysics Letters Section A General Atomic and Solid StatePhysics vol 363 no 3 pp 236ndash242 2007

[49] M Aydogdu ldquoA general nonlocal beam theory its applicationto nanobeam bending buckling and vibrationrdquo Physica E Low-Dimensional Systems andNanostructures vol 41 no 9 pp 1651ndash1655 2009

[50] 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

[51] 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

[52] R Barretta and F Marotti De Sciarra ldquoA nonlocal model forcarbon nanotubes under axial loadsrdquo Advances in MaterialsScience and Engineering vol 2013 Article ID 360935 6 pages2013

[53] F Marotti de Sciarra and R Barretta ldquoA gradient model forTimoshenko nanobeamsrdquo Physica E Low-Dimensional Systemsand Nanostructures vol 62 pp 1ndash9 2014

[54] R Barretta L Feo R Luciano and F Marotti de SciarraldquoApplication of an enhanced version of the Eringen differentialmodel to nanotechnologyrdquo Composites B vol 96 pp 274ndash2802016

[55] R Barretta L Feo R Luciano F Marotti de Sciarra and RPenna ldquoFunctionally graded Timoshenko nanobeams a novelnonlocal gradient formulationrdquoComposites B vol 100 pp 208ndash219 2016

[56] 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

[57] R Rafiee and R M Moghadam ldquoOn the modeling of carbonnanotubes a critical reviewrdquo Composites Part B Engineeringvol 56 pp 435ndash449 2014

[58] R Barretta R Luciano and F M de Sciarra ldquoA fully gradientmodel for Euler-Bernoulli nanobeamsrdquoMathematical Problemsin Engineering vol 2015 Article ID 495095 8 pages 2015

[59] M A Eltaher S A Emam and F F Mahmoud ldquoStatic andstability analysis of nonlocal functionally graded nanobeamsrdquoComposite Structures vol 96 pp 82ndash88 2013

[60] F Marotti de Sciarra M Canadija and R Barretta ldquoA gradientmodel for torsion of nanobeamsrdquoComptes RendusmdashMecaniquevol 343 no 4 pp 289ndash300 2015

[61] K A Lazopoulos and A K Lazopoulos ldquoOn the torsionproblem of strain gradient elastic barsrdquo Mechanics ResearchCommunications vol 45 pp 42ndash47 2012

[62] 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

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Control Scienceand Engineering

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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



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Advances inOptoElectronics


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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



The attenuation function 119870 depends on the Euclideandistance |x1015840 minus x| and on a nonlocal dimensionless parameterdefined by

120591 = 1198900119886119897 (2)

where 1198900 is a material constant 119886 is an internal characteristiclength and 119897 stands for external characteristic length

Assuming a suitable expression of the nonlocal modulus119870 in terms of a variant of the Bessel function we get theinverse differential relationship of (1) between the nonlocalstress and the elastic deformation

(1 minus (1198900119886)2 nabla2) t119894119895 = E119894119895ℎ119896eℎ119896 (3)

with nabla2 the Laplacian Note that (3) can be conveniently usedin order to describe the law between the nonlocal shear stressfield 120591119894 on the cross-section of a nanobeam and the elasticshear strain 120574119894 as follows

120591119894 minus (1198900119886)2 1198892120591119894

1198891199092 = 120583120574119894 (4)

where 119909 is the axial direction and 120583 is the shear modulus

3 Torsion of Nonlocal Circular Nanobeams

Let Ω be the cross-section of a circular nanobeam of length119871 subjected to the following loading conditions depicted inFigure 1119898119905distributed couples per unit length in the interval [0 119871] M119905

concentrated couples at the end cross-sections 0 119871

(5)

The triplet (119909 119910 119911) describes a set of Cartesian axes originat-ing at the left cross-section centreO

Equilibrium equations are expressed by119889119872119905119889119909 = minus119898119905 in [0 119871] (6a)

119872119905 = M119905 at 0 119871 (6b)

where119872119905 is the twisting momentComponents of the displacement field up to a rigid body

motion of a circular nanobeam under torsion write as119904119909 (119909 119910119911) = 0119904119910 (119909 119910119911) = minus120579 (119909) 119911119904119911 (119909 119910119911) = 120579 (119909) 119910

(7)

where 120579(119909) is the torsional rotation of the cross-section at theabscissa 119909 Shear strains compatible with the displacementfield equation (7) are given by

120574119910119909 (119909 119910 119911) = minus119889120579119889119909119909119911

120574119911119909 (119909 119910 119911) = 119889120579119889119909119909119910

(8)

where

120594119905 (119909) = 119889120579119889119909 (119909) (9)

is the twisting curvature The shear modulus 120583 is assumed tobe functionally graded only in the cross-sectional plane (119910 119911)

The elastic twisting stiffness is provided by

119896119905 = intΩ120583 (1199102 + 1199112) 119889119860 (10)

where the symbol sdot is the inner product between vectorsThe differential equation of nonlocal elastic equilibrium

of a nanobeam under torsion is formulated as follows Let uspreliminarily multiply (4) by Rr fl minus119911 119910 and integrate onΩ

intΩRr sdot 120591119889119860 minus (1198900119886)2 int

ΩRr sdot 119889120591

1198891199092 119889119860 = intΩ120583Rr sdot 120574119889119860 (11)

with the vector 120574 fl 120574119910119909 120574119911119909 given by (8)Enforcing (6a) and imposing the static equivalence con-

dition

119872119905 = intΩ(120591119911119909119910 minus 120591119910119909119911) 119889119860 (12)

we get the relation

119872119905 (119909) + (1198900119886)2 119889119898119905119889119909 (119909) = 119896119905 119889120579119889119909 (119909) (13)

This equation can be interpreted as decomposition formula ofthe twisting curvature 120594119905 into elastic (120594119905)el and inelastic (120594119905)inparts

120594119905 = (120594119905)el + (120594119905)in (14)

with

(120594119905)el = 119872119905119896119905 (15a)

(120594119905)in = (1198900119886)2119896119905

119889119898119905119889119909 (15b)

Accordingly the scale effect exhibited by the torsional rota-tion function of a nonlocal nanobeam can be evaluated bysolving a corresponding linearly elastic beam subjected to thetwisting curvature distortion (120594119905)in (15b)

4 Benchmark Examples

Let us consider a nanocantilever and a fully clampednanobeam of length 119871 subjected to the following quadraticdistribution of couples per unit length

119898119905 = 1198981198712 1199092 (16)

The cross-sectional torsional rotation is evaluated by fol-lowing the methodology illustrated in the previous section


ℳt ℳt

m yy

z

z L

x

t

Ω

O


120591 = 02120591 = 01

120591 = 03

120591 = 04

120591 = 05

120591

120591

= 0

120579lowast

01

02

03

04

05

04 06 08 1002120585


120591 = 015

120591 = 01

120591 = 02

120591 = 025

120591 = 03

120591

120591

= 0

120579lowast

001

002

003

004

04 06 08 1002120585



(120594119905)in = (1198900119886)2119896119905

21198981198712 119909 (17)







Competing Interests


References




































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










ℳt ℳt

m yy

z

z L

x

t

Ω

O


120591 = 02120591 = 01

120591 = 03

120591 = 04

120591 = 05

120591

120591

= 0

120579lowast

01

02

03

04

05

04 06 08 1002120585


120591 = 015

120591 = 01

120591 = 02

120591 = 025

120591 = 03

120591

120591

= 0

120579lowast

001

002

003

004

04 06 08 1002120585



(120594119905)in = (1198900119886)2119896119905

21198981198712 119909 (17)







Competing Interests


References




































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article A Note on Torsion of Nonlocal Composite ... · random composites [ ], and nonlocal...

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