Research Article On Bending of Bernoulli-Euler Nanobeams...
6
Research Article On Bending of Bernoulli-Euler Nanobeams for Nonlocal Composite Materials Luciano Feo and Rosa Penna Department of Civil Engineering, University of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, Italy Correspondence should be addressed to Luciano Feo; [email protected] Received 4 December 2015; Accepted 24 April 2016 Academic Editor: eodoros C. Rousakis Copyright © 2016 L. Feo and R. Penna. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Evaluation of size effects in functionally graded elastic nanobeams is carried out by making recourse to the nonlocal continuum mechanics. e Bernoulli-Euler kinematic assumption and the Eringen nonlocal constitutive law are assumed in the formulation of the elastic equilibrium problem. An innovative methodology, characterized by a lowering in the order of governing differential equation, is adopted in the present manuscript in order to solve the boundary value problem of a nanobeam under flexure. Unlike standard treatments, a second-order differential equation of nonlocal equilibrium elastic is integrated in terms of transverse displacements and equilibrated bending moments. Benchmark examples are developed, thus providing the nonlocality effect in nanocantilever and clampled-simply supported nanobeams for selected values of the Eringen scale parameter. 1. Introduction Analysis of nanodevices is a subject of special interest in the current literature. Particular attention is given to the static behavior of beam-like components of nanoelectromechanical systems (NEMS). Nonlocal constitutive behaviors are ade- quate in order to evaluate the size phenomenon in nanostruc- tures; see, for example, [1–9]. Investigations on random elastic structures have been carried out in [10–13]. Many research efforts have been devoted to theoretical and computational advances about specific structural models [14–19]. Recent variational formulations of nonlocal continua have been developed in [20–22]. Noteworthy theoretical results on functionally graded nanobeams have been contributed in [23–25]. Nevertheless, exact solutions are not always available so that finite element strategies are needful; see, for example, [26]. Micromechanical approaches are broadly used in order to analyze the effective behavior of composite structures [27]. Innovative applications of engineering interest are pro- posed in [28–30]. Numerical and experimental method- ologies for composite structures are developed in [31, 32]. Effective applications of tensionless models concerning crack propagation are reported in [33–37]. A skillful analysis of equilibrium configurations of hyperelastic cylindrical bodies and compressible cubes is carried out in [38]. e present paper deals with one-dimensional nanos- tructure by making recourse to the tools of nonlocal contin- uum mechanics. Small-scale effects exhibited by functionally graded nanobeams under flexure are analyzed in Section 2. 2. Nonlocal Elasticity In local linear elasticity for isotropic materials, stress and strain at a point x of a Cauchy continuum are functionally related by the following classical law: (x) = 2e (x)+e (x) , (1) with and Lam´ e constants. Such a constitutive behavior is not adequate to evaluate size effects in nanostructures. An effective law able to capture scale phenomena was developed by Eringen in [39] who defined the following nonlocal integral operator: t (x)=∫ ( x − x , ) (x ) , (2) where (1) t is the nonlocal stress, (2) is the macroscopic stress given by (5), Hindawi Publishing Corporation Modelling and Simulation in Engineering Volume 2016, Article ID 6369029, 5 pages http://dx.doi.org/10.1155/2016/6369029
Transcript of Research Article On Bending of Bernoulli-Euler Nanobeams...
Research ArticleOn Bending of Bernoulli-Euler Nanobeams forNonlocal Composite Materials
Luciano Feo and Rosa Penna
Department of Civil Engineering University of Salerno Via Giovanni Paolo II 132 84084 Fisciano Italy
Correspondence should be addressed to Luciano Feo lfeounisait
Received 4 December 2015 Accepted 24 April 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
Evaluation of size effects in functionally graded elastic nanobeams is carried out by making recourse to the nonlocal continuummechanics The Bernoulli-Euler kinematic assumption and the Eringen nonlocal constitutive law are assumed in the formulationof the elastic equilibrium problem An innovative methodology characterized by a lowering in the order of governing differentialequation is adopted in the present manuscript in order to solve the boundary value problem of a nanobeam under flexureUnlike standard treatments a second-order differential equation of nonlocal equilibrium elastic is integrated in terms of transversedisplacements and equilibrated bending moments Benchmark examples are developed thus providing the nonlocality effect innanocantilever and clampled-simply supported nanobeams for selected values of the Eringen scale parameter
1 Introduction
Analysis of nanodevices is a subject of special interest in thecurrent literature Particular attention is given to the staticbehavior of beam-like components of nanoelectromechanicalsystems (NEMS) Nonlocal constitutive behaviors are ade-quate in order to evaluate the size phenomenon in nanostruc-tures see for example [1ndash9] Investigations on randomelasticstructures have been carried out in [10ndash13] Many researchefforts have been devoted to theoretical and computationaladvances about specific structural models [14ndash19] Recentvariational formulations of nonlocal continua have beendeveloped in [20ndash22] Noteworthy theoretical results onfunctionally graded nanobeams have been contributed in[23ndash25] Nevertheless exact solutions are not always availableso that finite element strategies are needful see for example[26] Micromechanical approaches are broadly used in orderto analyze the effective behavior of composite structures [27]
Innovative applications of engineering interest are pro-posed in [28ndash30] Numerical and experimental method-ologies for composite structures are developed in [31 32]Effective applications of tensionless models concerning crackpropagation are reported in [33ndash37] A skillful analysis ofequilibrium configurations of hyperelastic cylindrical bodiesand compressible cubes is carried out in [38]
The present paper deals with one-dimensional nanos-tructure by making recourse to the tools of nonlocal contin-uummechanics Small-scale effects exhibited by functionallygraded nanobeams under flexure are analyzed in Section 2
2 Nonlocal Elasticity
In local linear elasticity for isotropic materials stress andstrain at a point x of a Cauchy continuum are functionallyrelated by the following classical law
120590119894119895 (x) = 2120583e119894119895 (x) + 120582e119903119903 (x) 120575119894119895 (1)
with 120583 and 120582 Lame constantsSuch a constitutive behavior is not adequate to evaluate
size effects in nanostructures An effective law able to capturescale phenomena was developed by Eringen in [39] whodefined the following nonlocal integral operator
t119894119895 (x) = int119881119870(10038161003816100381610038161003816x1015840 minus x10038161003816100381610038161003816 120591)120590119894119895 (x1015840) 119889119881 (2)
where
(1) t119894119895 is the nonlocal stress(2) 120590119894119895 is the macroscopic stress given by (5)
Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2016 Article ID 6369029 5 pageshttpdxdoiorg10115520166369029
2 Modelling and Simulation in Engineering
(3) 119870 is the influence function(4) 120591 = 1198900119886119897 a dimensionless nonlocal parameter defined
in terms of the material constant 1198900 and of theinternal and external characteristic lengths 119886 and 119897respectively
In agreement with the Eringen proposal in choosing thefollowing influence function 1minus1198882nabla2 the nonlocal elastic law(2) rewrites as
(1 minus (1198900119886)2 nabla2) t119894119895 = 120590119894119895 (3)
where nabla2 denotes the Laplace operator The differential formadopted for bending of nanobeams analogous to (3) isprovided by
120590 minus 1198882 11988921205901198891199092 = 120590 (4)
where 120590 is the nonlocal normal stress and 120590 is the macro-scopic normal stress on cross sections Note that the stress 120590is expressed in terms of elastic axial strains by
120590 = 119864120576 (5)
with 119864 Young modulus
3 Bending of Nonlocal Nanobeams
Let us consider a bent nanobeam of length 119871 with Youngmodulus 119864 functionally graded in the cross section Ω anduniform along the beam axis 119909 The cross-sectional elasticcentre and the principal axes of elastic inertia associated withthe scalar field 119864 are respectively denoted by G and by thepair (119910 119911)
The nanobeam is assumed to be subjected in the plane(119909 119910) to the following loading conditions119902119905 distributed load per unit length in the interval[0 119871]F119905 concentrated forces at the end cross sections0 119871M concentrated couples at the end cross sections0 119871
The bending stiffness is defined by
119896119887 fl intΩ1198641199102119889119860 (6)
Differential and boundary conditions of equilibrium areexpressed by
11988921198721198871198891199092 = 119902119905 in [0 119871] 119889119872119887119889119909 = plusmnF119905 at 0 119871 119872119887 = ∓M at 0 119871
(7)
where119872119887 is the bending moment
The bending curvature corresponding to the transversedisplacement V is given by
120594 (119909) = 1198892V1198891199092 (119909) (8)
The differential equation of nonlocal elastic equilibrium ofa nanobeam under flexure is formulated as follows Let uspreliminarily multiply (4) by the coordinate 119910 along thebending axis and integrate on the cross sectionΩ
intΩ120590119910119889119860 minus (1198900119886)2 int
Ω
11988921205901198891199092119910119889119860 = int
Ω119864120576119910 119889119860 (9)
with the axial dilation provided by the known formula 120576(119909) =minus120594(119909)119910Enforcing (8) and (7)1 and imposing the static equiva-
lence condition
119872119887 = minusintΩ120590119910119889119860 (10)
we obtain the relation
119872119887 minus (1198900119886)2 119902119905 = 119896119887 1198892V
1198891199092 (11)
This equation can be interpreted as decomposition formulaof the bending curvature 120594119887 = 1198892V1198891199092 into elastic 120594EL andinelastic 120594IN parts
120594119887 = 120594EL + 120594IN (12)
with
120594EL = 119872119887119896119887
120594IN = minus(1198900119886)2
119896119887 119902119905(13)
Accordingly the scale effect exhibited by bending momentsand displacements of a FG nonlocal nanobeam can beevaluated by solving a corresponding linearly elastic beamsubjected to the bending curvature distortion 120594IN (13)2
4 Examples
The solutionmethodology of the nonlocal elastic equilibriumproblem of a nanobeam enlightened in the previous sectionis here adopted in order to assess small-scale effects innanocantilever and clamped-simply supported nanobeamsunder a uniformly distributed load 119902119905 The nonlocality effecton the transverse displacement is thus due to the uniformbending curvature distortion formulated in (13)2 Graphicalevidences of the elastic displacements are provided in Figures1 and 2 in terms of the following dimensionless parameters120585 = 119909119871 and Vlowast(120585) = (100119896119887)(1199021199051198714)V(120585) for selected values ofthe nonlocal parameter 120591 fl 1198900119886119871 Details of the calculationsand some comments are reported below
41 Cantilever Nanobeam The bending moment is given by
119872119887 (119909) = 1199021199052 1199092 minus 119902119905119871119909 +11990211990511987122 (14)
Modelling and Simulation in Engineering 3
2
4
6
8
10
12
02 04 06 08 10
120591 = 0
120591
lowast
120585
Figure 1 Dimensionless transverse displacements Vlowast for thenanocantilever versus 120585 for 120591 isin 0 005 01 015 02 03 05
02
04
06
08
10
12
14
02 04 06 08 10
120591 = 0
120591
120585
lowast
Figure 2 Dimensionless transverse displacement Vlowast for theclamped-simply supported nanobeam versus 120585 for 120591 isin 0 005 01015 02 03 05
The lhs of (11) is hence known so that the differentialcondition of nonlocal elastic equilibrium to be integratedwrites explicitly as
119896119887 1198892V
1198891199092 (119909) = 119872119887 (119909) minus 12059121198712119902119905 (15)
The general integral of (15) takes the form
V (119909) = 119886119909 + 119887 + V (119909) (16)
where
V (119909) = minus 11990211990524119896119887 1199094 + 1199021199051198716119896119887 119909
3 minus 11990211990511987124 1199092 minus 120591211987121199021199052119896119887 1199092 (17)
is a particular solution of (15) The evaluation of the inte-gration constants 119886 and 119887 is carried out by prescribing theboundary conditions
V (0) = 0119889V119889119909 (0) = 0
(18)
The transversal deflection follows by a direct computation
V (119909) = 11990211990524119896119887 1199094 minus 1199021199051198716119896119887 119909
3 + 11990211990511987124119896119887 1199092 minus 120591211990211990511987122119896119887 1199092 (19)
The maximum displacement is given by
V119898 = V (119871) = (18 minus12059122 )
1199021199051198714119896119887 (20)
Nanocantilever becomes stiffer with increasing the nonlocalparameter 120591 Indeed according to the analysis proposed inSection 3 the sign of the prescribed distortion120594IN describingthe nonlocality effect is opposite to the one of the elasticcurvature 120594EL In particular the displacement of the free endvanishes if 120591 = 05 (see Figure 1) according to (20)
It is worth noting that with the structure being staticallydeterminate the bending moment (14) is not affected by thescale phenomenon
42 Clamped-Simply Supported Nanobeam Equilibratedbending moments are provided by the relation
119872119887 (119909) = 1198861 (119909 minus 119871) + 11990211990511987122 minus 1199021199052 1199092 (21)
with 1198861 isin R being an arbitrary constant The differentialcondition of nonlocal elastic equilibrium (11) to be integratedtakes thus the form
minus119896119887 1198892V
1198891199092 (119909) = 1198861 (119909 minus 119871) +11990211990511987122 minus 1199021199052 1199092 + 12059121198712119902119905 (22)
Equation (22) is solved by imposing the following kinematicboundary conditions
V (0) = 0119889V119889119909 (0) = 0V (119871) = 0
(23)
A direct computation gives the transversal displacement field
V (119909) = 11990211990524119896119887 1199094 minus 511990211990511987148119896119887 119909
3 + 119902119905119871216119896119887 1199092 + 12059121199021199051198714119896119887 119909
2 (119871 minus 119909) (24)
and the bending moment
119872119887 (119909) = 1199021199052 1199092 minus1199021199051198712 (54 + 31205912)119909 +
11990211990511987122 (31205912 + 14) (25)
The maximum displacement is given by
V119898 = V (119909) (26)
with 119909 = 3119871((32)(524+12059122)minusradic(916)1205914 + 11768 + (1396)1205912)solution of the equation (the known local maximum point119909 ≃ 0578119871 is recovered by setting 120591 = 0) (119889V119889119909)(119909) = 0
In agreement with the equivalence method exposedin Section 3 (24) and (25) provide the deflection and
4 Modelling and Simulation in Engineering
120591 = 0
120591
Mlowast
120585
02 04 06 08 10
minus10
minus20
minus30
minus40
minus50
Figure 3 Dimensionless bending moment 119872lowast versus dimension-less abscissa 120585 of a clamped-simply supported nanobeam
bending moment of a corresponding local nanobeam underthe transversal load distribution 119902119905 and the distortion 120594INequivalent to the nonlocality effect
A plot of the dimensionless bending moment 119872lowast =minus(1001199021199051198712)119872 versus the dimensionless parameter 120585 = 119909119871is given in Figure 3 for increasing values of the nonlocalparameter 1205915 Conclusion
TheEringen nonlocal law has been used in order to assess sizeeffects in nanobeams formulated according to the Bernoulli-Euler kinematics The treatment extends to functionallygraded materials the analysis carried out in [24] under thespecial assumption of elastically homogeneous nanobeamsTransverse deflections of cantilever and clamped-simply sup-ported nanobeams have been established for different valuesof the nonlocal parameter Such analytical solutions could beconveniently adopted by other scholars as simple referenceexamples for numerical evaluations in nonlocal compositemechanics
Competing Interests
The authors declare that they have no competing interests
References
[1] R de Borst andH-BMuehlhaus ldquoGradient-dependent plastic-ity formulation and algorithmic aspectsrdquo International Journalfor NumericalMethods in Engineering vol 35 no 3 pp 521ndash5391992
[2] E C Aifantis ldquoGradient deformation models at nano microand macro scalesrdquo Journal of Engineering Materials and Tech-nology vol 121 pp 189ndash202 1999
[3] 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
[4] R Rafiee and R M Moghadam ldquoOn the modeling of carbonnanotubes a critical reviewrdquo Composites Part B Engineeringvol 56 pp 435ndash449 2014
[5] J N Reddy ldquoNonlocal theories for bending buckling and vibra-tion of beamsrdquo International Journal of Engineering Science vol45 no 2ndash8 pp 288ndash307 2007
[6] 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
[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] 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
[9] R Barretta L Feo R Luciano and F Marotti de SciarraldquoApplication of an enhanced version of the Eringen differentialmodel to nanotechnologyrdquo Composites Part B Engineering vol96 pp 274ndash280 2016
[10] 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
[11] R Luciano and J R Willis ldquoNon-local constitutive response ofa random laminate subjected to configuration-dependent bodyforcerdquo Journal of the Mechanics and Physics of Solids vol 49 no2 pp 431ndash444 2001
[12] 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
[13] 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
[14] 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
[15] 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
[16] 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
[17] R Barretta ldquoAnalogies between Kirchhoff plates and Saint-Venant beams under flexurerdquo Acta Mechanica vol 225 no 7pp 2075ndash2083 2014
[18] 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
[19] 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
[20] FMarotti de Sciarra andM Salerno ldquoOn thermodynamic func-tions in thermoelasticity without energy dissipationrdquo EuropeanJournal of MechanicsmdashASolids vol 46 pp 84ndash95 2014
[21] F Marotti de Sciarra ldquoFinite element modelling of nonlocalbeamsrdquo Physica E vol 59 pp 144ndash149 2014
Modelling and Simulation in Engineering 5
[22] F Marotti de Sciarra and R Barretta ldquoA new nonlocal bendingmodel for Euler-Bernoulli nanobeamsrdquo Mechanics ResearchCommunications vol 62 no 1 pp 25ndash30 2014
[23] R Barretta R Luciano and F Marotti de Sciarra ldquoA fullygradient model for Euler-Bernoulli nanobeamsrdquo MathematicalProblems in Engineering vol 2015 Article ID 495095 8 pages2015
[24] R Barretta and F Marotti de Sciarra ldquoAnalogies betweennonlocal and local BernoullindashEuler nanobeamsrdquo Archive ofApplied Mechanics vol 85 no 1 pp 89ndash99 2015
[25] R Barretta L Feo R Luciano F Marotti de Sciarra and RPenna ldquoFunctionally graded Timoshenko nanobeams a novelnonlocal gradient formulationrdquoComposites Part B Engineeringvol 100 pp 208ndash219 2016
[26] 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
[27] 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
[28] V A Salomoni C Majorana G M Giannuzzi and A MiliozzildquoThermal-fluid flow within innovative heat storage concretesystems for solar power plantsrdquo International Journal of Numer-ical Methods for Heat amp Fluid Flow vol 18 pp 969ndash999 2008
[29] 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
[30] 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
[31] G Dinelli G Belz C E Majorana and B A SchreflerldquoExperimental investigation on the use of fly ash for lightweightprecast structural elementsrdquo Materials and Structures vol 29no 194 pp 632ndash638 1996
[32] 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
[33] A M Tarantino ldquoNonlinear fracture mechanics for an elasticBell materialrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 50 part 3 pp 435ndash456 1997
[34] 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
[35] AMTarantino ldquoOn extreme thinning at the notch-tip of a neo-Hookean sheetrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 51 part 2 pp 179ndash190 1998
[36] A M Tarantino ldquoOn the finite motions generated by a modeI propagating crackrdquo Journal of Elasticity vol 57 no 2 pp 85ndash103 1999
[37] A M Tarantino ldquoCrack propagation in finite elastodynamicsrdquoMathematics andMechanics of Solids vol 10 no 6 pp 577ndash6012005
[38] A M Tarantino ldquoHomogeneous equilibrium configurations ofa hyperelastic compressible cube under equitriaxial dead-loadtractionsrdquo Journal of Elasticity vol 92 no 3 pp 227ndash254 2008
[39] 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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2 Modelling and Simulation in Engineering
(3) 119870 is the influence function(4) 120591 = 1198900119886119897 a dimensionless nonlocal parameter defined
in terms of the material constant 1198900 and of theinternal and external characteristic lengths 119886 and 119897respectively
In agreement with the Eringen proposal in choosing thefollowing influence function 1minus1198882nabla2 the nonlocal elastic law(2) rewrites as
(1 minus (1198900119886)2 nabla2) t119894119895 = 120590119894119895 (3)
where nabla2 denotes the Laplace operator The differential formadopted for bending of nanobeams analogous to (3) isprovided by
120590 minus 1198882 11988921205901198891199092 = 120590 (4)
where 120590 is the nonlocal normal stress and 120590 is the macro-scopic normal stress on cross sections Note that the stress 120590is expressed in terms of elastic axial strains by
120590 = 119864120576 (5)
with 119864 Young modulus
3 Bending of Nonlocal Nanobeams
Let us consider a bent nanobeam of length 119871 with Youngmodulus 119864 functionally graded in the cross section Ω anduniform along the beam axis 119909 The cross-sectional elasticcentre and the principal axes of elastic inertia associated withthe scalar field 119864 are respectively denoted by G and by thepair (119910 119911)
The nanobeam is assumed to be subjected in the plane(119909 119910) to the following loading conditions119902119905 distributed load per unit length in the interval[0 119871]F119905 concentrated forces at the end cross sections0 119871M concentrated couples at the end cross sections0 119871
The bending stiffness is defined by
119896119887 fl intΩ1198641199102119889119860 (6)
Differential and boundary conditions of equilibrium areexpressed by
11988921198721198871198891199092 = 119902119905 in [0 119871] 119889119872119887119889119909 = plusmnF119905 at 0 119871 119872119887 = ∓M at 0 119871
(7)
where119872119887 is the bending moment
The bending curvature corresponding to the transversedisplacement V is given by
120594 (119909) = 1198892V1198891199092 (119909) (8)
The differential equation of nonlocal elastic equilibrium ofa nanobeam under flexure is formulated as follows Let uspreliminarily multiply (4) by the coordinate 119910 along thebending axis and integrate on the cross sectionΩ
intΩ120590119910119889119860 minus (1198900119886)2 int
Ω
11988921205901198891199092119910119889119860 = int
Ω119864120576119910 119889119860 (9)
with the axial dilation provided by the known formula 120576(119909) =minus120594(119909)119910Enforcing (8) and (7)1 and imposing the static equiva-
lence condition
119872119887 = minusintΩ120590119910119889119860 (10)
we obtain the relation
119872119887 minus (1198900119886)2 119902119905 = 119896119887 1198892V
1198891199092 (11)
This equation can be interpreted as decomposition formulaof the bending curvature 120594119887 = 1198892V1198891199092 into elastic 120594EL andinelastic 120594IN parts
120594119887 = 120594EL + 120594IN (12)
with
120594EL = 119872119887119896119887
120594IN = minus(1198900119886)2
119896119887 119902119905(13)
Accordingly the scale effect exhibited by bending momentsand displacements of a FG nonlocal nanobeam can beevaluated by solving a corresponding linearly elastic beamsubjected to the bending curvature distortion 120594IN (13)2
4 Examples
The solutionmethodology of the nonlocal elastic equilibriumproblem of a nanobeam enlightened in the previous sectionis here adopted in order to assess small-scale effects innanocantilever and clamped-simply supported nanobeamsunder a uniformly distributed load 119902119905 The nonlocality effecton the transverse displacement is thus due to the uniformbending curvature distortion formulated in (13)2 Graphicalevidences of the elastic displacements are provided in Figures1 and 2 in terms of the following dimensionless parameters120585 = 119909119871 and Vlowast(120585) = (100119896119887)(1199021199051198714)V(120585) for selected values ofthe nonlocal parameter 120591 fl 1198900119886119871 Details of the calculationsand some comments are reported below
41 Cantilever Nanobeam The bending moment is given by
119872119887 (119909) = 1199021199052 1199092 minus 119902119905119871119909 +11990211990511987122 (14)
Modelling and Simulation in Engineering 3
2
4
6
8
10
12
02 04 06 08 10
120591 = 0
120591
lowast
120585
Figure 1 Dimensionless transverse displacements Vlowast for thenanocantilever versus 120585 for 120591 isin 0 005 01 015 02 03 05
02
04
06
08
10
12
14
02 04 06 08 10
120591 = 0
120591
120585
lowast
Figure 2 Dimensionless transverse displacement Vlowast for theclamped-simply supported nanobeam versus 120585 for 120591 isin 0 005 01015 02 03 05
The lhs of (11) is hence known so that the differentialcondition of nonlocal elastic equilibrium to be integratedwrites explicitly as
119896119887 1198892V
1198891199092 (119909) = 119872119887 (119909) minus 12059121198712119902119905 (15)
The general integral of (15) takes the form
V (119909) = 119886119909 + 119887 + V (119909) (16)
where
V (119909) = minus 11990211990524119896119887 1199094 + 1199021199051198716119896119887 119909
3 minus 11990211990511987124 1199092 minus 120591211987121199021199052119896119887 1199092 (17)
is a particular solution of (15) The evaluation of the inte-gration constants 119886 and 119887 is carried out by prescribing theboundary conditions
V (0) = 0119889V119889119909 (0) = 0
(18)
The transversal deflection follows by a direct computation
V (119909) = 11990211990524119896119887 1199094 minus 1199021199051198716119896119887 119909
3 + 11990211990511987124119896119887 1199092 minus 120591211990211990511987122119896119887 1199092 (19)
The maximum displacement is given by
V119898 = V (119871) = (18 minus12059122 )
1199021199051198714119896119887 (20)
Nanocantilever becomes stiffer with increasing the nonlocalparameter 120591 Indeed according to the analysis proposed inSection 3 the sign of the prescribed distortion120594IN describingthe nonlocality effect is opposite to the one of the elasticcurvature 120594EL In particular the displacement of the free endvanishes if 120591 = 05 (see Figure 1) according to (20)
It is worth noting that with the structure being staticallydeterminate the bending moment (14) is not affected by thescale phenomenon
42 Clamped-Simply Supported Nanobeam Equilibratedbending moments are provided by the relation
119872119887 (119909) = 1198861 (119909 minus 119871) + 11990211990511987122 minus 1199021199052 1199092 (21)
with 1198861 isin R being an arbitrary constant The differentialcondition of nonlocal elastic equilibrium (11) to be integratedtakes thus the form
minus119896119887 1198892V
1198891199092 (119909) = 1198861 (119909 minus 119871) +11990211990511987122 minus 1199021199052 1199092 + 12059121198712119902119905 (22)
Equation (22) is solved by imposing the following kinematicboundary conditions
V (0) = 0119889V119889119909 (0) = 0V (119871) = 0
(23)
A direct computation gives the transversal displacement field
V (119909) = 11990211990524119896119887 1199094 minus 511990211990511987148119896119887 119909
3 + 119902119905119871216119896119887 1199092 + 12059121199021199051198714119896119887 119909
2 (119871 minus 119909) (24)
and the bending moment
119872119887 (119909) = 1199021199052 1199092 minus1199021199051198712 (54 + 31205912)119909 +
11990211990511987122 (31205912 + 14) (25)
The maximum displacement is given by
V119898 = V (119909) (26)
with 119909 = 3119871((32)(524+12059122)minusradic(916)1205914 + 11768 + (1396)1205912)solution of the equation (the known local maximum point119909 ≃ 0578119871 is recovered by setting 120591 = 0) (119889V119889119909)(119909) = 0
In agreement with the equivalence method exposedin Section 3 (24) and (25) provide the deflection and
4 Modelling and Simulation in Engineering
120591 = 0
120591
Mlowast
120585
02 04 06 08 10
minus10
minus20
minus30
minus40
minus50
Figure 3 Dimensionless bending moment 119872lowast versus dimension-less abscissa 120585 of a clamped-simply supported nanobeam
bending moment of a corresponding local nanobeam underthe transversal load distribution 119902119905 and the distortion 120594INequivalent to the nonlocality effect
A plot of the dimensionless bending moment 119872lowast =minus(1001199021199051198712)119872 versus the dimensionless parameter 120585 = 119909119871is given in Figure 3 for increasing values of the nonlocalparameter 1205915 Conclusion
TheEringen nonlocal law has been used in order to assess sizeeffects in nanobeams formulated according to the Bernoulli-Euler kinematics The treatment extends to functionallygraded materials the analysis carried out in [24] under thespecial assumption of elastically homogeneous nanobeamsTransverse deflections of cantilever and clamped-simply sup-ported nanobeams have been established for different valuesof the nonlocal parameter Such analytical solutions could beconveniently adopted by other scholars as simple referenceexamples for numerical evaluations in nonlocal compositemechanics
Competing Interests
The authors declare that they have no competing interests
References
[1] R de Borst andH-BMuehlhaus ldquoGradient-dependent plastic-ity formulation and algorithmic aspectsrdquo International Journalfor NumericalMethods in Engineering vol 35 no 3 pp 521ndash5391992
[2] E C Aifantis ldquoGradient deformation models at nano microand macro scalesrdquo Journal of Engineering Materials and Tech-nology vol 121 pp 189ndash202 1999
[3] 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
[4] R Rafiee and R M Moghadam ldquoOn the modeling of carbonnanotubes a critical reviewrdquo Composites Part B Engineeringvol 56 pp 435ndash449 2014
[5] J N Reddy ldquoNonlocal theories for bending buckling and vibra-tion of beamsrdquo International Journal of Engineering Science vol45 no 2ndash8 pp 288ndash307 2007
[6] 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
[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] 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
[9] R Barretta L Feo R Luciano and F Marotti de SciarraldquoApplication of an enhanced version of the Eringen differentialmodel to nanotechnologyrdquo Composites Part B Engineering vol96 pp 274ndash280 2016
[10] 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
[11] R Luciano and J R Willis ldquoNon-local constitutive response ofa random laminate subjected to configuration-dependent bodyforcerdquo Journal of the Mechanics and Physics of Solids vol 49 no2 pp 431ndash444 2001
[12] 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
[13] 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
[14] 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
[15] 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
[16] 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
[17] R Barretta ldquoAnalogies between Kirchhoff plates and Saint-Venant beams under flexurerdquo Acta Mechanica vol 225 no 7pp 2075ndash2083 2014
[18] 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
[19] 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
[20] FMarotti de Sciarra andM Salerno ldquoOn thermodynamic func-tions in thermoelasticity without energy dissipationrdquo EuropeanJournal of MechanicsmdashASolids vol 46 pp 84ndash95 2014
[21] F Marotti de Sciarra ldquoFinite element modelling of nonlocalbeamsrdquo Physica E vol 59 pp 144ndash149 2014
Modelling and Simulation in Engineering 5
[22] F Marotti de Sciarra and R Barretta ldquoA new nonlocal bendingmodel for Euler-Bernoulli nanobeamsrdquo Mechanics ResearchCommunications vol 62 no 1 pp 25ndash30 2014
[23] R Barretta R Luciano and F Marotti de Sciarra ldquoA fullygradient model for Euler-Bernoulli nanobeamsrdquo MathematicalProblems in Engineering vol 2015 Article ID 495095 8 pages2015
[24] R Barretta and F Marotti de Sciarra ldquoAnalogies betweennonlocal and local BernoullindashEuler nanobeamsrdquo Archive ofApplied Mechanics vol 85 no 1 pp 89ndash99 2015
[25] R Barretta L Feo R Luciano F Marotti de Sciarra and RPenna ldquoFunctionally graded Timoshenko nanobeams a novelnonlocal gradient formulationrdquoComposites Part B Engineeringvol 100 pp 208ndash219 2016
[26] 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
[27] 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
[28] V A Salomoni C Majorana G M Giannuzzi and A MiliozzildquoThermal-fluid flow within innovative heat storage concretesystems for solar power plantsrdquo International Journal of Numer-ical Methods for Heat amp Fluid Flow vol 18 pp 969ndash999 2008
[29] 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
[30] 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
[31] G Dinelli G Belz C E Majorana and B A SchreflerldquoExperimental investigation on the use of fly ash for lightweightprecast structural elementsrdquo Materials and Structures vol 29no 194 pp 632ndash638 1996
[32] 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
[33] A M Tarantino ldquoNonlinear fracture mechanics for an elasticBell materialrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 50 part 3 pp 435ndash456 1997
[34] 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
[35] AMTarantino ldquoOn extreme thinning at the notch-tip of a neo-Hookean sheetrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 51 part 2 pp 179ndash190 1998
[36] A M Tarantino ldquoOn the finite motions generated by a modeI propagating crackrdquo Journal of Elasticity vol 57 no 2 pp 85ndash103 1999
[37] A M Tarantino ldquoCrack propagation in finite elastodynamicsrdquoMathematics andMechanics of Solids vol 10 no 6 pp 577ndash6012005
[38] A M Tarantino ldquoHomogeneous equilibrium configurations ofa hyperelastic compressible cube under equitriaxial dead-loadtractionsrdquo Journal of Elasticity vol 92 no 3 pp 227ndash254 2008
[39] 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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Modelling and Simulation in Engineering 3
2
4
6
8
10
12
02 04 06 08 10
120591 = 0
120591
lowast
120585
Figure 1 Dimensionless transverse displacements Vlowast for thenanocantilever versus 120585 for 120591 isin 0 005 01 015 02 03 05
02
04
06
08
10
12
14
02 04 06 08 10
120591 = 0
120591
120585
lowast
Figure 2 Dimensionless transverse displacement Vlowast for theclamped-simply supported nanobeam versus 120585 for 120591 isin 0 005 01015 02 03 05
The lhs of (11) is hence known so that the differentialcondition of nonlocal elastic equilibrium to be integratedwrites explicitly as
119896119887 1198892V
1198891199092 (119909) = 119872119887 (119909) minus 12059121198712119902119905 (15)
The general integral of (15) takes the form
V (119909) = 119886119909 + 119887 + V (119909) (16)
where
V (119909) = minus 11990211990524119896119887 1199094 + 1199021199051198716119896119887 119909
3 minus 11990211990511987124 1199092 minus 120591211987121199021199052119896119887 1199092 (17)
is a particular solution of (15) The evaluation of the inte-gration constants 119886 and 119887 is carried out by prescribing theboundary conditions
V (0) = 0119889V119889119909 (0) = 0
(18)
The transversal deflection follows by a direct computation
V (119909) = 11990211990524119896119887 1199094 minus 1199021199051198716119896119887 119909
3 + 11990211990511987124119896119887 1199092 minus 120591211990211990511987122119896119887 1199092 (19)
The maximum displacement is given by
V119898 = V (119871) = (18 minus12059122 )
1199021199051198714119896119887 (20)
Nanocantilever becomes stiffer with increasing the nonlocalparameter 120591 Indeed according to the analysis proposed inSection 3 the sign of the prescribed distortion120594IN describingthe nonlocality effect is opposite to the one of the elasticcurvature 120594EL In particular the displacement of the free endvanishes if 120591 = 05 (see Figure 1) according to (20)
It is worth noting that with the structure being staticallydeterminate the bending moment (14) is not affected by thescale phenomenon
42 Clamped-Simply Supported Nanobeam Equilibratedbending moments are provided by the relation
119872119887 (119909) = 1198861 (119909 minus 119871) + 11990211990511987122 minus 1199021199052 1199092 (21)
with 1198861 isin R being an arbitrary constant The differentialcondition of nonlocal elastic equilibrium (11) to be integratedtakes thus the form
minus119896119887 1198892V
1198891199092 (119909) = 1198861 (119909 minus 119871) +11990211990511987122 minus 1199021199052 1199092 + 12059121198712119902119905 (22)
Equation (22) is solved by imposing the following kinematicboundary conditions
V (0) = 0119889V119889119909 (0) = 0V (119871) = 0
(23)
A direct computation gives the transversal displacement field
V (119909) = 11990211990524119896119887 1199094 minus 511990211990511987148119896119887 119909
3 + 119902119905119871216119896119887 1199092 + 12059121199021199051198714119896119887 119909
2 (119871 minus 119909) (24)
and the bending moment
119872119887 (119909) = 1199021199052 1199092 minus1199021199051198712 (54 + 31205912)119909 +
11990211990511987122 (31205912 + 14) (25)
The maximum displacement is given by
V119898 = V (119909) (26)
with 119909 = 3119871((32)(524+12059122)minusradic(916)1205914 + 11768 + (1396)1205912)solution of the equation (the known local maximum point119909 ≃ 0578119871 is recovered by setting 120591 = 0) (119889V119889119909)(119909) = 0
In agreement with the equivalence method exposedin Section 3 (24) and (25) provide the deflection and
4 Modelling and Simulation in Engineering
120591 = 0
120591
Mlowast
120585
02 04 06 08 10
minus10
minus20
minus30
minus40
minus50
Figure 3 Dimensionless bending moment 119872lowast versus dimension-less abscissa 120585 of a clamped-simply supported nanobeam
bending moment of a corresponding local nanobeam underthe transversal load distribution 119902119905 and the distortion 120594INequivalent to the nonlocality effect
A plot of the dimensionless bending moment 119872lowast =minus(1001199021199051198712)119872 versus the dimensionless parameter 120585 = 119909119871is given in Figure 3 for increasing values of the nonlocalparameter 1205915 Conclusion
TheEringen nonlocal law has been used in order to assess sizeeffects in nanobeams formulated according to the Bernoulli-Euler kinematics The treatment extends to functionallygraded materials the analysis carried out in [24] under thespecial assumption of elastically homogeneous nanobeamsTransverse deflections of cantilever and clamped-simply sup-ported nanobeams have been established for different valuesof the nonlocal parameter Such analytical solutions could beconveniently adopted by other scholars as simple referenceexamples for numerical evaluations in nonlocal compositemechanics
Competing Interests
The authors declare that they have no competing interests
References
[1] R de Borst andH-BMuehlhaus ldquoGradient-dependent plastic-ity formulation and algorithmic aspectsrdquo International Journalfor NumericalMethods in Engineering vol 35 no 3 pp 521ndash5391992
[2] E C Aifantis ldquoGradient deformation models at nano microand macro scalesrdquo Journal of Engineering Materials and Tech-nology vol 121 pp 189ndash202 1999
[3] 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
[4] R Rafiee and R M Moghadam ldquoOn the modeling of carbonnanotubes a critical reviewrdquo Composites Part B Engineeringvol 56 pp 435ndash449 2014
[5] J N Reddy ldquoNonlocal theories for bending buckling and vibra-tion of beamsrdquo International Journal of Engineering Science vol45 no 2ndash8 pp 288ndash307 2007
[6] 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
[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] 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
[9] R Barretta L Feo R Luciano and F Marotti de SciarraldquoApplication of an enhanced version of the Eringen differentialmodel to nanotechnologyrdquo Composites Part B Engineering vol96 pp 274ndash280 2016
[10] 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
[11] R Luciano and J R Willis ldquoNon-local constitutive response ofa random laminate subjected to configuration-dependent bodyforcerdquo Journal of the Mechanics and Physics of Solids vol 49 no2 pp 431ndash444 2001
[12] 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
[13] 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
[14] 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
[15] 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
[16] 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
[17] R Barretta ldquoAnalogies between Kirchhoff plates and Saint-Venant beams under flexurerdquo Acta Mechanica vol 225 no 7pp 2075ndash2083 2014
[18] 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
[19] 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
[20] FMarotti de Sciarra andM Salerno ldquoOn thermodynamic func-tions in thermoelasticity without energy dissipationrdquo EuropeanJournal of MechanicsmdashASolids vol 46 pp 84ndash95 2014
[21] F Marotti de Sciarra ldquoFinite element modelling of nonlocalbeamsrdquo Physica E vol 59 pp 144ndash149 2014
Modelling and Simulation in Engineering 5
[22] F Marotti de Sciarra and R Barretta ldquoA new nonlocal bendingmodel for Euler-Bernoulli nanobeamsrdquo Mechanics ResearchCommunications vol 62 no 1 pp 25ndash30 2014
[23] R Barretta R Luciano and F Marotti de Sciarra ldquoA fullygradient model for Euler-Bernoulli nanobeamsrdquo MathematicalProblems in Engineering vol 2015 Article ID 495095 8 pages2015
[24] R Barretta and F Marotti de Sciarra ldquoAnalogies betweennonlocal and local BernoullindashEuler nanobeamsrdquo Archive ofApplied Mechanics vol 85 no 1 pp 89ndash99 2015
[25] R Barretta L Feo R Luciano F Marotti de Sciarra and RPenna ldquoFunctionally graded Timoshenko nanobeams a novelnonlocal gradient formulationrdquoComposites Part B Engineeringvol 100 pp 208ndash219 2016
[26] 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
[27] 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
[28] V A Salomoni C Majorana G M Giannuzzi and A MiliozzildquoThermal-fluid flow within innovative heat storage concretesystems for solar power plantsrdquo International Journal of Numer-ical Methods for Heat amp Fluid Flow vol 18 pp 969ndash999 2008
[29] 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
[30] 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
[31] G Dinelli G Belz C E Majorana and B A SchreflerldquoExperimental investigation on the use of fly ash for lightweightprecast structural elementsrdquo Materials and Structures vol 29no 194 pp 632ndash638 1996
[32] 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
[33] A M Tarantino ldquoNonlinear fracture mechanics for an elasticBell materialrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 50 part 3 pp 435ndash456 1997
[34] 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
[35] AMTarantino ldquoOn extreme thinning at the notch-tip of a neo-Hookean sheetrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 51 part 2 pp 179ndash190 1998
[36] A M Tarantino ldquoOn the finite motions generated by a modeI propagating crackrdquo Journal of Elasticity vol 57 no 2 pp 85ndash103 1999
[37] A M Tarantino ldquoCrack propagation in finite elastodynamicsrdquoMathematics andMechanics of Solids vol 10 no 6 pp 577ndash6012005
[38] A M Tarantino ldquoHomogeneous equilibrium configurations ofa hyperelastic compressible cube under equitriaxial dead-loadtractionsrdquo Journal of Elasticity vol 92 no 3 pp 227ndash254 2008
[39] 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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 Modelling and Simulation in Engineering
120591 = 0
120591
Mlowast
120585
02 04 06 08 10
minus10
minus20
minus30
minus40
minus50
Figure 3 Dimensionless bending moment 119872lowast versus dimension-less abscissa 120585 of a clamped-simply supported nanobeam
bending moment of a corresponding local nanobeam underthe transversal load distribution 119902119905 and the distortion 120594INequivalent to the nonlocality effect
A plot of the dimensionless bending moment 119872lowast =minus(1001199021199051198712)119872 versus the dimensionless parameter 120585 = 119909119871is given in Figure 3 for increasing values of the nonlocalparameter 1205915 Conclusion
TheEringen nonlocal law has been used in order to assess sizeeffects in nanobeams formulated according to the Bernoulli-Euler kinematics The treatment extends to functionallygraded materials the analysis carried out in [24] under thespecial assumption of elastically homogeneous nanobeamsTransverse deflections of cantilever and clamped-simply sup-ported nanobeams have been established for different valuesof the nonlocal parameter Such analytical solutions could beconveniently adopted by other scholars as simple referenceexamples for numerical evaluations in nonlocal compositemechanics
Competing Interests
The authors declare that they have no competing interests
References
[1] R de Borst andH-BMuehlhaus ldquoGradient-dependent plastic-ity formulation and algorithmic aspectsrdquo International Journalfor NumericalMethods in Engineering vol 35 no 3 pp 521ndash5391992
[2] E C Aifantis ldquoGradient deformation models at nano microand macro scalesrdquo Journal of Engineering Materials and Tech-nology vol 121 pp 189ndash202 1999
[3] 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
[4] R Rafiee and R M Moghadam ldquoOn the modeling of carbonnanotubes a critical reviewrdquo Composites Part B Engineeringvol 56 pp 435ndash449 2014
[5] J N Reddy ldquoNonlocal theories for bending buckling and vibra-tion of beamsrdquo International Journal of Engineering Science vol45 no 2ndash8 pp 288ndash307 2007
[6] 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
[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] 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
[9] R Barretta L Feo R Luciano and F Marotti de SciarraldquoApplication of an enhanced version of the Eringen differentialmodel to nanotechnologyrdquo Composites Part B Engineering vol96 pp 274ndash280 2016
[10] 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
[11] R Luciano and J R Willis ldquoNon-local constitutive response ofa random laminate subjected to configuration-dependent bodyforcerdquo Journal of the Mechanics and Physics of Solids vol 49 no2 pp 431ndash444 2001
[12] 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
[13] 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
[14] 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
[15] 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
[16] 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
[17] R Barretta ldquoAnalogies between Kirchhoff plates and Saint-Venant beams under flexurerdquo Acta Mechanica vol 225 no 7pp 2075ndash2083 2014
[18] 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
[19] 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
[20] FMarotti de Sciarra andM Salerno ldquoOn thermodynamic func-tions in thermoelasticity without energy dissipationrdquo EuropeanJournal of MechanicsmdashASolids vol 46 pp 84ndash95 2014
[21] F Marotti de Sciarra ldquoFinite element modelling of nonlocalbeamsrdquo Physica E vol 59 pp 144ndash149 2014
Modelling and Simulation in Engineering 5
[22] F Marotti de Sciarra and R Barretta ldquoA new nonlocal bendingmodel for Euler-Bernoulli nanobeamsrdquo Mechanics ResearchCommunications vol 62 no 1 pp 25ndash30 2014
[23] R Barretta R Luciano and F Marotti de Sciarra ldquoA fullygradient model for Euler-Bernoulli nanobeamsrdquo MathematicalProblems in Engineering vol 2015 Article ID 495095 8 pages2015
[24] R Barretta and F Marotti de Sciarra ldquoAnalogies betweennonlocal and local BernoullindashEuler nanobeamsrdquo Archive ofApplied Mechanics vol 85 no 1 pp 89ndash99 2015
[25] R Barretta L Feo R Luciano F Marotti de Sciarra and RPenna ldquoFunctionally graded Timoshenko nanobeams a novelnonlocal gradient formulationrdquoComposites Part B Engineeringvol 100 pp 208ndash219 2016
[26] 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
[27] 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
[28] V A Salomoni C Majorana G M Giannuzzi and A MiliozzildquoThermal-fluid flow within innovative heat storage concretesystems for solar power plantsrdquo International Journal of Numer-ical Methods for Heat amp Fluid Flow vol 18 pp 969ndash999 2008
[29] 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
[30] 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
[31] G Dinelli G Belz C E Majorana and B A SchreflerldquoExperimental investigation on the use of fly ash for lightweightprecast structural elementsrdquo Materials and Structures vol 29no 194 pp 632ndash638 1996
[32] 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
[33] A M Tarantino ldquoNonlinear fracture mechanics for an elasticBell materialrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 50 part 3 pp 435ndash456 1997
[34] 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
[35] AMTarantino ldquoOn extreme thinning at the notch-tip of a neo-Hookean sheetrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 51 part 2 pp 179ndash190 1998
[36] A M Tarantino ldquoOn the finite motions generated by a modeI propagating crackrdquo Journal of Elasticity vol 57 no 2 pp 85ndash103 1999
[37] A M Tarantino ldquoCrack propagation in finite elastodynamicsrdquoMathematics andMechanics of Solids vol 10 no 6 pp 577ndash6012005
[38] A M Tarantino ldquoHomogeneous equilibrium configurations ofa hyperelastic compressible cube under equitriaxial dead-loadtractionsrdquo Journal of Elasticity vol 92 no 3 pp 227ndash254 2008
[39] 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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Modelling and Simulation in Engineering 5
[22] F Marotti de Sciarra and R Barretta ldquoA new nonlocal bendingmodel for Euler-Bernoulli nanobeamsrdquo Mechanics ResearchCommunications vol 62 no 1 pp 25ndash30 2014
[23] R Barretta R Luciano and F Marotti de Sciarra ldquoA fullygradient model for Euler-Bernoulli nanobeamsrdquo MathematicalProblems in Engineering vol 2015 Article ID 495095 8 pages2015
[24] R Barretta and F Marotti de Sciarra ldquoAnalogies betweennonlocal and local BernoullindashEuler nanobeamsrdquo Archive ofApplied Mechanics vol 85 no 1 pp 89ndash99 2015
[25] R Barretta L Feo R Luciano F Marotti de Sciarra and RPenna ldquoFunctionally graded Timoshenko nanobeams a novelnonlocal gradient formulationrdquoComposites Part B Engineeringvol 100 pp 208ndash219 2016
[26] 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
[27] 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
[28] V A Salomoni C Majorana G M Giannuzzi and A MiliozzildquoThermal-fluid flow within innovative heat storage concretesystems for solar power plantsrdquo International Journal of Numer-ical Methods for Heat amp Fluid Flow vol 18 pp 969ndash999 2008
[29] 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
[30] 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
[31] G Dinelli G Belz C E Majorana and B A SchreflerldquoExperimental investigation on the use of fly ash for lightweightprecast structural elementsrdquo Materials and Structures vol 29no 194 pp 632ndash638 1996
[32] 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
[33] A M Tarantino ldquoNonlinear fracture mechanics for an elasticBell materialrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 50 part 3 pp 435ndash456 1997
[34] 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
[35] AMTarantino ldquoOn extreme thinning at the notch-tip of a neo-Hookean sheetrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 51 part 2 pp 179ndash190 1998
[36] A M Tarantino ldquoOn the finite motions generated by a modeI propagating crackrdquo Journal of Elasticity vol 57 no 2 pp 85ndash103 1999
[37] A M Tarantino ldquoCrack propagation in finite elastodynamicsrdquoMathematics andMechanics of Solids vol 10 no 6 pp 577ndash6012005
[38] A M Tarantino ldquoHomogeneous equilibrium configurations ofa hyperelastic compressible cube under equitriaxial dead-loadtractionsrdquo Journal of Elasticity vol 92 no 3 pp 227ndash254 2008
[39] 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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
