research article stability analysis of nonlocal elastic...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 341232 12 pageshttpdxdoiorg1011552013341232

Research ArticleStability Analysis of Nonlocal Elastic Columns withInitial Imperfection

S P Xu12 M R Xu3 and C M Wang2

1 College of Engineering Ocean University of China Qingdao 266100 China2Department of Civil and Environmental Engineering National University of Singapore Singapore 1175763 School of Mathematics University of Jinan Jinan 250022 China

Correspondence should be addressed to S P Xu xusipengouceducn

Received 13 September 2012 Accepted 20 January 2013

Academic Editor Jyh Horng Chou

Copyright copy 2013 S P Xu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Investigated herein is the postbuckling behavior of an initially imperfect nonlocal elastic column which is simply supported atone end and subjected to an axial force at the other movable end The governing nonlinear differential equation of the axiallyloaded nonlocal elastic column experiencing large deflection is first establishedwithin the framework of Eringenrsquos nonlocal elasticitytheory in order to embrace the size effect Its semianalytical solutions by the virtue of homotopy perturbation method as well asthe successive approximation algorithm are determined in an explicit form through which the postbuckling equilibrium loads interms of the end rotation angle and the deformed configuration of the column at this end rotation are predicted By comparingthe degenerated results with the exact solutions available in the literature the validity and accuracy of the proposed methods arenumerically substantiated The size effect as well as the initial imperfection on the buckled configuration and the postbucklingequilibrium path is also thoroughly discussed through parametric studies

1 Introduction

The discovery of carbon nanotubes ushers a new era in thenano world [1] The enhanced features of nanomaterial inmechanical electrical optical and chemical properties overtraditional ones open up many application opportunities incutting-edge fields that range from sensing and communi-cations to energy harvesting However when the size of thestructure scales down to nanodomains an issue of consider-able importance namely the size effect arises and becomesprominent [2] The size effect may greatly alter the macro-scopic properties and moreover it calls the applicability ofclassical continuum models into question Various modifiedcontinuum theories (such as couple stress theory straingradient elasticity theory and modified couple stress theory)have thus been proposed to account for the size effect inthe micronanoscale structures Among them the nonlocalelasticity theory pioneered by Eringen [3] has been widelyaccepted and attracted an ever growing attention in recentyears This is because Eringenrsquos nonlocal continuum-based

models are mathematically easy to tackle and physically rea-sonable from the atomistic viewpoint of lattice dynamics andmolecular dynamics simulations [4] This theory abandonsthe classical assumption of locality and admits that the stressstate depends not only on the strain at that point but on thestrains of every point in the body In this way informationconcerning about the long-range forces between atoms isincorporated into the theory and consequently the internalsize scale is represented in the constitutive equations simplyas a material parameter

So far there have been considerable studies on size effectsin problems of bending vibration buckling and wave prop-agation of nanostructures (see [5] and references therein)based on the nonlocal elasticity theory Among them buck-ling is one of the important topics because nanoelements aremore susceptible to buckling instability when subjected tocompressive loads due to their high aspect ratio Moreoverthe onset of buckling will significantly compromise thestructural integrity of nanostructures [6] However it shouldbe mentioned that most studies assumed small deformations

2 Mathematical Problems in Engineering

for the nanoelements so that the curvature may be approxi-mated by the second derivative of the transverse deflectionNevertheless these nanoelements usually possess excellentflexibility just as experimental observations on carbon nan-otubes have indicated and they can undertake significantdeformations without experiencing apparent plasticity [7ndash9]Thus the general linearized theories fail to depict this afore-mentioned behavioral feature in a satisfactorymanner [10] Inorder to predict the postbuckling behavior of nanoelementsboth large deflection and size effect must be consideredHence a geometrically exact theory is indispensable forpostbuckling behavior of nanoelements

In the light of themathematical consistency in theway themechanical model is formulated hereinafter we will restrictour attention to the elastica theory [11 12] which entailsusing the exact expression of curvature The first study ofthe deformation of an elastica under loading was probablydue to James Bernoulli although it was Euler who establishedthe basic theory [11 13] By taking advantage of Eulerrsquoscelebrated work considerable efforts have been devoted tothe theory with applications extended from classical fieldsto statistical mechanics (see [14] and the references therein)biomechanics [15] and ocean engineering [16 17]Within thescope ofmost of these studies linear constitutive relations areutilized Encouraged by its effectiveness in dealing with thesegeometrical nonlinear problems the focus is also transferredto other kinds of materials and structures For exampleHaslach Jr [18] investigated the postbuckling behavior ofa column made of a material having a cubic constitutiveequation while Al-Sadder and Shatarat [19] examined thelarge deflection of a prismatic composite cantilever beamcomprising two different nonlinear elastic materials Withinthe framework of the elastica theory the nonlinear bucklingresponse of asymmetric laminated structures composed ofan arbitrary number of distinct material layers was treatedby Vinogradov and Derrick [20] Kang and Li [21] consid-ered the mechanical behaviors of a cantilevered functionallygraded material (FGM) beam that obeys the Ludwick-typelaw An interesting ldquopartially composite elasticardquo problemwasstudied by Challamel [22] where each subcolumn ismodeledwith the Euler-Bernoulli beam theory and connected to eachother via a linear constitutive law for the interlayer slip Inaddition Frostig [23] revealed the elastica behavior of a sand-wich panel with a softcompliant core that analysis includesboth the shear deformation and moderate strain The largedeflection of the fluid-saturated poroelastic beams which arepermeable in the axial direction and impermeable in thetransverse directions was examined by Li et al [24] Likewisean elastica-type dynamic stability of viscoelastic columnswith its behavior given in terms of the Boltzmann superpo-sition principle was discussed by Suire and Cederbaum [25]Moreover the boundary conditions were also extended fromthe commonly used conditions namely simply supportedand clamped-free conditions to clamped-hinged condition[26] frictional end [27] and slippery support [28] conditions

As for the governing equations of these elastica prob-lems the elliptical integral functions are often employedto express their exact solutions For example by utilizingthe elliptical integral transformation Kimball and Tsai [29]

presented the nonlinear deflection solution for a cantileverbeam subjected to an arbitrary combination of end forcesand moments Moreover the solution for double curvaturebending of variable-arc-length elasticas was proposed byChucheepsakul et al [30] in terms of elliptical integralsHowever from an engineering point of view these exactsolutions are not quite intuitive of any physical insight andone often needs to resort to numerical methods Besides theexact treatment remains intractable for most cases because ofthe intrinsic nonlinearity that resides in the elastica theoryTherefore another branch of treatment from the numericalpoint of view is developed simultaneously in conjunctionwith the availability of high-speed computers One popularapproach is the finite-element method for example bywhich along with Newton-Rhapson iteration technique thestatic analysis of risers under the influence of internal flowand hydrostatic pressure was performed by Chen et al[16] Another frequently used approach is the shootingmethod It converts the nonlinear boundary-value probleminto an initial-value problem which is then determinedwith or without iterations [31 32] Other newly developednumerical methods include the Lie-group shooting method[33] method of genetic algorithm [34 35] and differentialquadrature method [36] In view that an analytical solutionis always convenient for parametric studies and capturesclearly the physics of the problem it appears more appealingthan the numerical oneThus extensive research efforts werealso made in this third line of development which madeuse of all kinds of asymptotic methods Among them theperturbation technique [37ndash41] is popular in the broad area ofbuckling problems of structures For instance a higher-orderperturbation technique was employed by Lacarbonara [39] todetermine the postbuckling solutions for nonprismatic non-linearly elastic rods In surveying the postbuckling regimefor the five classical Euler buckling cases of an extensibleelastica the method of multiple (spatial) scales perturbationwas proposed by Mazzilli [40] The analysis of postbucklingas well as nonlinear bending and nonlinear vibration fora simply supported Euler-Bernoulli beam resting on a two-parameter elastic foundation was accomplished by Shen [41]with the help of a novel two-step perturbation techniqueAnother recently developedmethod the homotopy analyticalmethod was used by Wang et al [42] to solve a cantileverbeam problem in order to obtain an explicit solution of thedisplacement and rotation at the free end Later this problemwas revisited by Tolou and Herder by means of the Adomiandecompositionmethod [43]The remarkable accuracy of thismethod was also demonstrated in dealing with the tip loadedcantilever elastica and plastica problems [44] Besides withthe help of a series of admissible functional transformationsAndriotaki et al [45] furnished an implicit analytical solutionfor the elastica problem of a cantilever bar due to its ownweight

Although there are extensive studies related to the elasticaproblems just as clearly indicated earlier the scrutinies oflarge deflections of flexible nanocolumns with size effectincluded are less frequent Until recently efforts toward anelastica type buckling analysis of nanocolumn were con-ducted By employing the shooting method Wang et al

Mathematical Problems in Engineering 3

[46] examined the postbuckling problem of cantileverednanorodstubes in the absence of shear deformation Ratherthan relying on the numerical simulation Xu [47] inde-pendently investigated this problem theoretically with thehelp of the homotopy perturbation method For stubbynanocolumns Xu et al [48] exploited the Timoshenkohypothesis as a basis for the nonlinear postbuckling descrip-tion and in such a way the role of shear deformation in thepostbuckling behavior was revealed By using Pontryaginrsquosmaximum principle the optimal shape of a nonlocal elasticrod clamped at both ends was determined by Atanackovic etal [49] In these cited articles the exact curvature expressionwas used and the nonlocal elasticity theory was chosen toaccount for the size effect

Though meaningful results with allowance for the sizeeffect have been observed it appears that the postbuckling ofnanocolumns still merits some further investigation In factthe nanocolumns often suffer from geometric imperfectionsdue to variousmanufacturing and environmental factors andthe appearance of these initial imperfections in axially loadedstructures significantly affects their response in the prebuck-ling and postbuckling stages [50] On the other hand initialimperfectionsmay be introducedpurposely to take advantageof its beneficial features For example the postbucklingbehavior of nanocolumnwith an initial imperfection to createa very shallow arch can be used as a particular strategyto electronic devices [51] The imperfection amplitude offrames [52] or composite beams (see [53] and referencestherein) can also be properly manipulated for a desiredresponse such as maximizing the buckling load Motivatedby those observations the buckling behavior especially in thepostbuckling stage of an initially imperfect nonlocal elasticcolumn will be explored in this study The primary objectivewill focus on obtaining explicit semianalytical solutions viasimple but vigorous approximate techniques Moreover it isof interest to evaluate the effect of initial imperfection as wellas the size effect on the postbuckling behavior of nonlocalelastic columns

The remainder of this paper is organized as follows Theproblem formulation and exact governing equation are fur-nished in Section 2 Section 3 presents the novel employmentof two semianalyticalmethods namely the homotopy pertur-bationmethod (HPM) and successive approximate algorithm(SAA) for tackling the challenging geometrically nonlinearproblems and obtaining approximate solutions In Section 4the main numerical results by the proposed methods arepresented and discussed The concluding remarks are givenin Section 5

2 Theoretical Formulation

Consider the stability problem of an elastic nonlocal elasticcolumn with a slight geometrical curvature as an imperfec-tion This inextensible simply supported column of uniformcross-section 119860 and length 119871 is subjected to a conservativeforce 119875 at its right movable end as shown in Figure 1(a) TheCartesian (119909 119911) coordinate system is chosen in such amannerthat the abscissa axis coincides with the line connecting the

two hinged ends and the coordinate origin is located at itsleft end Let 119904 be the arc length reckoned along the column119908the deflection in 119911 direction and 120579

119879(119904) its angle of inclination

from the 119909 axis which is composed of two parts namely therotation 120579 of the cross-section induced by the pure bendingand rotation 120579

0due to the initial imperfection

Here the nonlocal elastic column is treated as an elasticaand the thickness-to-length ratio is assumed to be very smallsuch that the effect of transverse shear deformation may beneglected By assuming that the Euler-Bernoulli hypothesisholds irrespective of small or large deformation the strain ata distance 120578 from the neutral axis is given by

120576 = minus120578119889120579

119889119904 (1)

where 119889120579119889119904 is the exact bending curvatureFor the allowance of size effect Eringenrsquos nonlocal elas-

ticity theory is adopted Consequently by neglecting thenonlocal behavior in the thickness direction the constitutiverelation for a uniaxial stress state is written in the form[3 49 54]

120590 minus (1198900119886)2 1198892120590

1198891199042= 119864120576 (2)

where 120590 is the normal stress 119864 is the Youngrsquos modulusand 1198900119886 is the parameter that allows for the size effect 119886 is

an internal characteristic length (eg length of CndashC bondlattice spacing and granular distance) while 119890

0is a constant

appropriate to each material whose magnitude is usuallyidentified either by matching the dispersion curves of planewaves with those of atomic lattice dynamics or by calibratingit against molecular dynamic simulation results Generally aconservative estimate of the small-scale parameter can be setas 1198900119886 lt 20 nm for a single-wall carbon nanotube [55]In view of the definition of the resultant bendingmoment

119872 = int119860

120578120590119889119860 (3)

the exact moment-curvature relation can be written as

119872minus (1198900119886)2 1198892119872

1198891199042= minus119864119868

119889120579

119889119904 (4)

where119864119868 is the flexural rigidity and 119868 = int1198601205782119889119860 is the second

moment of inertia of the cross-sectionFrom static consideration on an arc-element 119889119904 (see

Figure 1(b)) we have

119889119872

119889119904= 119881 cos 120579

119879minus 119867 sin 120579

119879 (5)

where the horizontal and vertical internal forces 119867(119904) and119881(119904) for the problem considered are given by

119867(119904) = minus119875 119881 (119904) = 0 (6)


119871

1199080

119908

119875

119906(119871)

119909

119911

(a)

119909119881

119867

119872

119911

1205790

120579

119889119904

120579T119872+ d119872119881+ d119881

119867+ d119867

(b)

Figure 1 Initially imperfect nonlocal elastic column subjected to a conservative force (a) geometry and coordinate system and (b) aninfinitesimal element

In view of (5) and (6) and 120579 = 120579119879minus 1205790 then after differ-

entiating (4) once with respect to the arc length 119904 the generalgoverning equation may be written as

1198641198681198892120579119879

1198891199042+ 119875 sin 120579

119879

= 119875(1198900119886)2 cos 120579

119879(1198892120579119879

1198891199042minus tan 120579

119879(119889120579119879

119889119904)

2

)

+ 11986411986811988921205790

1198891199042

(7)

To facilitate the manipulation of the previous system thefollowing dimensionless parameters are introduced

120585 =119904

119871 120583 =

1198900119886

119871

120582 = radic1198751198712

119864119868 120582 = 120583120582

(8)

which imply that (7) can be rewritten as

12057910158401015840

119879+ 1205822 sin 120579

119879= 1205822 cos 120579

119879(12057910158401015840

119879minus tan 120579

11987912057910158402

119879) + 12057910158401015840

0 (9)

where the prime denotes differentiation with respect tospatial coordinate 120585

For the column with an initial imperfection its initialdeflection can be assumed to be [56]

1199080 (119904) = 1198860 sin120587120585 (10)

where 1198860is the columnrsquos midspan initial rise Then from the

geometrical relationships

119889119909 = 119889119904 cos 120579119879 119889119908 = 119889119904 sin 120579

119879 (11)

we have

1205790 (119904) = sinminus1 [

1198891199080 (119904)

119889119904] (12)

and correspondingly

12057910158401015840

0=

12057201205873(1205722

01205872minus 1) cos120587120585

(1 minus 1205722

01205872cos2120587120585)32

(13)

where 1205720= 1198860119871

The boundary condition for the postbuckling problem athand is [57]

119908 (0) = 119908 (119871) = 0 (14)

which can also be converted into one involving the firstderivative of slope 120579

119879 Bearing in mind that the internal

moment at any point 119904 is 119872 = 119875119908 then from (4) one canderive

119875119908 + [119864119868 minus 119875(1198900119886)2 cos 120579

119879]119889120579119879

119889119904= 119864119868

1198891205790

119889119904 (15)

through which and along with (12) and (14) the boundarycondition may be given by

119889120579119879

119889119904= 0 at 119904 = 0 119871 (16)

or

119864119868 minus 119875(1198900119886)2 cos 120579

119879= 0 at 119904 = 0 119871 (17)

The use of (17) yields a trivial solution Therefore theboundary condition given by (16) will be used to providea nontrivial solution Its nondimensionalized version is asfollows

1205791015840

119879(0) = 120579

1015840

119879(1) = 0 120579

119879 (0) = 120572119879 (18)

where 120572119879is the rotation at the end of the column

The governing differential equations (9) and (13) in con-junction with the corresponding boundary conditions (18)constitute a nonlinear two-point boundary value problemwhich is different from the classical elastica theory due to thetwo terms on the right-hand side of (9) arising respectivelyfrom the size effect and initial imperfection Owning to thestrong nonlinearity an exact or closed-form solution for thisclass of problem is unknown at present Thus the solutionof this problem can only be accomplished approximatelyor integrated numerically In what follows the homotopyperturbation method and successive approximate algorithmare adopted to seek the approximate solutions


3 Approximate Solution forthe Differential System

Before treating the elastica problem the zero-through third-order Taylor series expansions of trigonometric functions areemployed as follows

sin 120579119879= 120579119879minus1

61205793

119879

cos 120579119879= 1 minus

1

21205792

119879

tan 120579119879= 120579119879+1

31205793

119879

(19)

so as to reduce the complexity of the boundary-value problemdefined by (9) (13) and (18) and to capture the intrinsicgeometrical nonlinearity at the same timeThis simplificationwould introduce some small errors as will be verified laterby comparison with classical elastic solution Besides byassuming 120572

0le 1500 [56] (13) can be simplified as

12057910158401015840

0= minus12057201205873 cos120587120585 (20)

By substituting the expansions (19) and (20) into (9) andretaining terms up to O(1205793

119879) one obtains

L (120579119879) = N (120579

119879) + 119891 (120585) 120585 isin [0 1] (21)

where L is the linear operator which implies that L(120579119879) =

12057910158401015840

119879+ 1205742120579119879 while the nonlinear operator defines thatN(120579

119879) =

1205742[1205793

1198796 minus 120583

2120579119879(12057911987912057910158401015840

1198792 + 120579

10158402

119879)] and the analytical function

119891(120585) = minus1205873(1 + 120583

21205742)1205720cos120587120585 Besides the parameter 1205742 is

defined as 1205742 = 1205822(1 minus 1205822)

31 Homotopy Perturbation Method Among various tech-niques in dealing with the nonlinear differential equation(21) the homotopy perturbation method (HPM) developedby He [58 59] is one of the most effective methods Thismethod does not depend upon the assumption of smallparameters The main characteristic behind this approach isthat by embedding an auxiliary parameter 119901 HPM trans-forms a general nonlinear problem into an infinite number oflinear problems easy to solve Its effectiveness and accuracyhave been demonstrated in the analysis of various problems[60 61]

To investigate the solution of (21) using the homotopytechnique in topology we first construct a homotopy with anembedding parameter 119901 isin [0 1] in the form

H (Θ 119901)

= (1 minus 119901) [L (Θ) minus L (Θlowast

0)] + 119901 [L (Θ) minusN (Θ) minus 119891 (120585)]

= 0

(22)

where Θlowast0is an initial approximation here for the sake of

simplicity we take it as zero The differential equation (22)satisfies the boundary conditions

Θ1015840(0) = Θ

1015840(1) = 0 Θ (0) = 120572119879 (23)

It is obvious that by varying the embedding parameter 119901from zero to unity (22) approaches to the original (21) from asimple equation delineated by L(Θ) minus L(Θlowast

0) = 0 According

to the HPM we assume that the solution of (22) can beexpanded in a power series of the embedding parameter 119901as

Θ (120585) = Θ0 (120585) +

infin

sum

119895=1

119901119895Θ119895 (120585) (24)

Furthermore the dimensionless parameter 1205742 is also expand-ed in a series of 119901 namely

1205742= 1205742

0+

infin

sum

119895=1

1199011198951205742

119895 (25)

After the substitution of the series of (24) and (25) into(22) and (23) and splitting with respect to 119901 the followinghierarchies of linear boundary value problems are obtained

O (1199010) Θ

10158401015840

0+ 1205742

0Θ0= 0

Θ1015840

0(0) = Θ

1015840

0(0) = 0 Θ

0 (0) = 120572119879

(26a)

O (1199011) Θ

10158401015840

1+ 1205742

0Θ1= 1198911(Θ0 1205742

1 1205742

0)

Θ1015840

1(0) = Θ

1015840

1(1) = 0 Θ

1 (0) = 0

(26b)

O (1199012) Θ

10158401015840

2+ 1205742

0Θ2= 1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

Θ1015840

2(0) = Θ

1015840

2(1) = 0 Θ

2 (0) = 0

(26c)

where 119891119894(119894 = 1 2) are defined in Appendix A

The zero-order approximationΘ0(120585) is straightforwardly

determined by solving the homogeneous (26a) as

Θ0 (120585) = 120572119879 cos120587120585 (27)

which corresponds to the most important first linearizedbuckling mode and implies that 120574

0= 120587 Introducing the

last equation along with some trigonometric identities into(26b) results in

Θ10158401015840

1+ 1205742

0Θ1

= [minus1205742

1120572119879+1

81205872(1 + 120587

21205832) 1205723

119879minus 1205873(1 + 120587

21205832) 1205720] cos120587120585

+1

241205872(1 + 9120587

21205832) 1205723

119879cos 3120587120585

(28)

By solving (28) we have the following expression for Θ1

Θ1=1

192(1 + 9120587

21205832) 1205723

119879(cos120587120585 minus cos 3120587120585) (29)

which is accompanied by the following expression for 12057421

1205742

1=1

8(1 + 120587

21205832) [12058721205722

119879minus 812058731205720120572minus1

119879] (30)


Condition (30) is actually equivalent to the removal of asecular term as in the classical perturbation theory [40]It should be noted that the rotation of the nonlocal elasticcolumn in discussion can be expressed by the following basefunctions [30]

cos 1198991205740120585 119899 = 1 2 3 (31)

Therefore similar to the first-order approximation onecan set the coefficient of cos 120574

0120585 in the 119895th-order differential

equations (26a) (26b) and (26c) to zero This provides uswith the algebraic equations for the higher-order correc-tions of the load parameter namely 1205742

119895 By repeating the

procedures outlined earlier we can find sufficient accurateapproximations

If we stop at the second-order approximate solution then120579119879and 1205742 can be given by setting 119901 = 1 as

120579119879 (120585) = Θ (120585)

1003816100381610038161003816119901=1= 120572119879cos120587120585

+ Ξ1 (cos120587120585 minus cos 3120587120585) + Ξ2 (cos120587120585 minus cos 5120587120585)

(32)

1205742=

2

sum

119894=0

1205742

119894≜ 1198921(120572119879 1205720 120583) (33)

where Ξ119894(119894 = 1 2) and 119892

1(120572119879 1205720 120583) are defined in

Appendix B

32 Successive Approximate Algorithm The postbucklingbehavior of the nonlocal elastic column is now investigatedfrom an alternativemethodology developed by Kounadis andhis colleagues [62ndash64] Here we shall refer to this alternativemethod as successive approximate algorithm (SAA) Theconvergence uniqueness and upper bound error estimatesof solutions derived from SAA were thoroughly establishedin [65]

According to SAA the reduced homogeneous lineardifferential equation of (21) namely

L (120579119879) = 0 (34)

in conjunction with the boundary conditions (18) is firstsolved Obviously the solution 120579

1198790has the similar form as

that of (26a) Then introducing this solution into the right-hand side of (21) yields an inhomogeneous linear differentialequation

L (120579119879 1205742) = N (120579

1198790 1205742

0) + 119891 (120585 120574

2) 120585 isin [0 1] (35)

which associated with the conditions (18) is served as thefirst approximation Then a straightforward manipulationyields

1205791198791 (120585) = Λ 1 (120572119879 120574

2 1205742

0) cos120587120585 + Λ

2(120572119879 1205742 1205742

0) cos 3120587120585

(36)

where Λ119894(1205742 1205742

0 120572119879) (119894 = 1 2) are defined in Appendix C

Note that 1205791198791

is valid provided that 120574 = 1198991205740(119899 = 1 3 ) By

inserting (36) into the right-hand side of (21) one obtains

L (120579119879 1205742) = N (120579

1198791 1205742 1205742

0) + 119891 (120585 120574

2) 120585 isin [0 1] (37)

Along with the boundary condition (18) the secondapproximate solution can be determined By repeating thisprocedure more accurate results can be reached But ingeneral the first or second approximate is usually sufficientfor establishing a large part of the postbuckling path [63]In view of this and considering that higher approximationsrequire considerable computational efforts we choose thefirst approximate solution (36) as the final result which byapplying the condition 120579

1198791(0) = 120572

119879 furnishes us the relation

between 120572119879and 1205742 in the following form

1198871205744minus 1198881205742+ 119889 = 0 (38)

Here the coefficients 119887 119888 and 119889 are presented in Appendix CThe lowest root of last equation is given by

1205742=119888 minus radic1198871198882 minus 4119887119889

2119887≜ 1198922(120572119879 1205720 120583) (39)

Equation (39) provides uswith the functional relationshipof 1205742 versus 120572

119879for various initial imperfections and small-

scale parametersEquations (33) and (39) define the postbuckling equi-

librium path For a given rotation 120572119879at its left end the

equilibrium load 119875 is determined by

119875

119875119864

=119892119894(120572119879 1205720 120583)

1205872 [1 + 1205832119892119894(120572119879 1205720 120583)]

(119894 = 1 or 2) (40)

in which 119875119864= 119864119868120587

21198712 is the buckling load for the same

structure via local elasticity For the perfect nonlocal elasticcolumn let 120572

119879approach zero then one ends up with the

critical load 119875cr for the onset of buckling as follows

119875cr = 119875119864(1 + 12058721205832)minus1

(41)

which is an analytical solution without any approximationand it is identical with that given in [66] Obviously thecritical load 119875cr is a decreasing function with respect toincreasing small-scale parameter 120583

Once the functional relationship between 120579119879and 120585 is

known the expected 119909 and 119911 coordinates of any point alongthe deflected neutral axis of the column can be determinedby

119909

119871= int

120585

0

cos 120579119879 (120585) 119889120585

119911

119871= int

120585

0

sin 120579119879 (120585) 119889120585

(42)

4 Numerical Results and Discussion

In order to ascertain the accuracy and the range of appli-cability of the theoretical results developed previously aparallel model for perfect column via local elasticity theorydegenerated by setting 120572

0= 0 and 120583 = 0 is first evaluated

numerically against the exact elastica solution available inthe literature The comparisons of these results are presented


Table 1 Comparison of analytical approximations with the exact one for buckling loads

120572119879

119875exact119875119864(a)

119875HPM1119875119864

(b)119875HPM2

119875119864

(c)119875SAA119875119864

(d)

20∘ 10154 10152 (00197) 10155 (00098) 10155 (00098)40∘ 10637 10609 (02632) 10648 (01034) 10647 (00940)60∘ 11517 11371 (12677) 11567 (04341) 11576 (05123)80∘ 12939 12437 (38797) 13056 (09042) 13164 (17389)100∘ 15184 13808 (90622) 15318 (08825) 15890 (46496)120∘ 18848 15483 (178534) 18615 (12362) 20821 (104680)Number inside the bracket ( ) is the relative error computed(a)Results by the exact theory [12](b)Results by the HPM truncated to the first order(c)Results by the HPM truncated to the second order(d)Results by the SAA

Table 2 Comparison of analytical approximations with the exact one for midspan deflections

120572119879

119908exact119871(a)

119908HPM1119871

(b)119908HPM2

119871(c)

119908SAA119871(d)

20∘ 01097 01097 (00000) 01097 (00000) 01097 (00000)40∘ 02111 02111 (00000) 02111 (00000) 02112 (00474)60∘ 02966 02965 (00337) 02967 (00337) 02968 (00674)80∘ 03597 03592 (01390) 03601 (01112) 03606 (02502)100∘ 03958 03945 (03285) 03965 (01769) 03981 (05811)120∘ 04016 03993 (05727) 04019 (00747) 04051 (08715)140∘ 03752 03731 (05597) 03735 (04531) 03641 (29584)Number inside the bracket ( ) is the relative error computed(a)Results by the exact theory [12](b)Results by the HPM truncated to the first order(c)Results by the HPM truncated to the second order(d)Results by the SAA

in the tabular form Tables 1 and 2 collect the values of thebuckling loads and midspan deflections for a sequence ofvalues of end rotation where the relative errors are definedby

relative error

=

10038161003816100381610038161003816100381610038161003816

value obtained by the analytical approximationcorresponding value by exact solution

minus 1

10038161003816100381610038161003816100381610038161003816

times 100

(43)

As it can be seen from the presented results the bucklingloads obtained from HPM and SAA agree well with theexact elliptical integral solution when the end rotation is lessthan 40 degrees and the solution by the HPM truncatedto the second order provides reliable results even for theend rotation up to 120 degrees while for solutions providedby SAA more iterations are needed to get accurate resultsHowever for the midspan deflections the effectiveness of theaforementioned results getting both fromHPM and SAA canbe easily observed evenwhen the end rotation amounts to 140degrees

In view of the foregoing discussions the postbucklingbehavior of the nonlocal elastic column will be identifiedfrom the results by HPM truncated to the second order In

fact the almost identical results can be observed from usingthe SAA Numerical results for perfect nonlocal elastic col-umn are first presented in both tabular and graphical formsfor various small-scale parameter120583The results show that at aspecified end rotation the size effect becomes more obviousas the postbuckling deformation increases (see Table 3 andFigure 2) To investigate the postbuckling behaviors thestability of the column is also observed via the load-rotationcurves Figure 3 describes the size effect on the postbucklingpath It shows that the pitchfork bifurcation composed oftwo symmetrical stable branches and an unstable equilibriumbranch occurs at the critical load119875cr whatever the small-scaleparameter values Nevertheless the small-scale parameterdoes have an appreciable effect of reducing the buckling loadAs one can see from Figure 3 the deformation tends to belarger when compared to its local counterparts for the samemagnitude of postbuckling load

To illustrate the influence of the initial imperfectionseveral cases with or without the size effect are discussedThebifurcation response of the imperfect column is comparedin Figures 4 and 5 with that of its perfect local counterpartAs it can be seen from Figure 4 the introduction of theimperfection breaks the internal symmetry of the problemcompared with Figure 3 Buckling occurs through a saddle-node bifurcation which makes the critical load of thecolumn quantitatively less apparent since the critical state


Table 3 Midspan deflection of perfect nonlocal elastic column for various values of small-scale parameter

120572119879

120583 = 0 120583 = 005 120583 = 010 120583 = 015 120583 = 020

20∘ 01097 01097 01098 01099 0110040∘ 02111 02113 02118 02126 0213760∘ 02967 02972 02987 03012 0304580∘ 03601 03612 03642 03692 03759100∘ 03965 03980 04026 04098 04190120∘ 04019 04035 04078 04136 04196

0 02 04 06 08 10

01

02

03

04

05

119911119871

119909119871

Black lines 120583 = 000

Green lines 120583 = 010

Red lines 120583 = 020

120572T = 20∘

120572T = 60∘

120572T = 100∘

120572T = 120∘

Figure 2 Equilibrium configurations of a perfect nonlocal elasticcolumn for various end rotations and small-scale parameters

minus120 minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100 12006070809

11112131415

119875119875119864

Perfect column without size effectPerfect column with 120583 = 005Perfect column with 120583 = 010Perfect column with 120583 = 015Perfect column with 120583 = 020

120572T (∘)

Figure 3 Influence of size effect on the postbuckling equilibriumpaths of perfect nonlocal elastic columns

is represented by the point of zero slopes on each curveAlthough for this column the critical states other thanthe one for the perfect column cannot be reached underload control it is interesting to note that the critical statesof the imperfect column occur at loads higher than thecritical load for its perfect counterpart and the larger theimperfection the greater the critical load From Figure 4 itcan also be seen that the postbuckling behavior and growth

minus80 minus60 minus40 minus20 0 20 40 60 80

15

125

1

075

05

025

0

Stable StableUnstable

119875119875119864

Perfect columnImperfect column with 1205720 = 0002Imperfect column with 1205720 = 0004

120572T (∘)

Figure 4 Influence of initial imperfection on the postbucklingequilibrium paths with size effect precluded


15

125

1

075

05

025

0

119875119875119864

Perfect column (black lines)Imperfect column with 1205720 = 0002 (red lines)Imperfect column with 1205720 = 0004 (blue lines)

Nanocolumn without size effectNanocolumn with 120583 = 005Nanocolumn with 120583 = 010Nanocolumn with 120583 = 020

120572T (∘)

Figure 5 Influence of size effect on the postbuckling equilibriumpaths of imperfect nonlocal elastic column

of the end rotation are altered even for a seemingly smallimperfection particularly in the neighborhood of the criticalload of the perfect system within where any slight increase ofthe amplitude of the imperfection would bring about greaterdeformation for the same load Even so all postbucklingpaths of the imperfect system will eventually converge tothe symmetrical postbuckling path of its perfect counterpart


Unlike for the local elastic column the postbuckling path fornonlocal elastic column additionally depends on the small-scale parameter but the general trend of which is rathersimilar qualitatively as its local counterpart (see Figure 5)

5 Concluding Remarks

In this study a semianalytical treatment for calculating thelarge elastic deformation of an initially imperfect nonlocalelastic column is presented Herein the column is consideredto be a prismatic and inextensible one whose constitutiveequation corresponds to a differential type of Eringenrsquosnonlocal elasticity theory Moreover the Euler-Bernoulliassumption is adopted The described problem results in acomplicated two-point boundary value problemwith a strongnonlinearity and size effect incorporatedThis problem com-pletely precludes the use of elliptical integrals as a viablemethod of solution The load-rotation relation in an explicitform as well as the deformed curve is obtained by thehomotopy perturbation method and the successive approx-imate algorithm with a few iterations Presently computedvalues of the postbuckling deformation and correspondingload are found to agree very well with those elastic resultsavailable in the literature Parameter study reveals that thesize effect when the size of the column is scaled down tothe nanodomains and the initial imperfection can influencethe postbuckling behavior of a nanocolumn considerably Ingeneral an increase in the small-scale parameter gives rise toan increase in postbuckling deformation and a decrease in thebuckling load Also the greater the deformation becomes themore prominent the size effect is demonstrated Besides theappearance of the imperfection breaks the postbuckling pathfrom the form of an internal symmetrical pitchfork bifurca-tion into one of a saddle-node bifurcation The postbucklingpaths are affected primarily in the near-buckling regimeeventually all of themwill converge to its perfect counterpartThese findings will contribute to our better understanding ofthe special behavior of nanostructures

From the effectiveness and accuracy of the proposedmethods we can also conclude that the presented methodscan be potentially extended to a broad range of columnproblems under large deformations such as the postbucklingproblems of shallow arches subjected to lateral loads prob-lems for columns with initial imperfection having the shapeof the second or higher buckling modes [31] and problemsfor columns with the inextensibility assumption relaxed to anextensible one

Appendix

A Functions Used in Equations (26b) and (26c)

1198911(Θ0 1205742

1 1205742

0)

= minus1205742

1Θ0+1

61205742

0Θ3

0minus 12058321205742

0Θ0(1

2Θ0Θ10158401015840

0+ Θ10158402

0)

minus 1205873(1 + 120583

21205742

0) 1205720cos120587120585

1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

= minus1205742

2Θ0minus 1205742

1Θ1+1

61205742

1Θ3

0+1

21205742

0Θ2

0Θ2

1

minus1

21205832[(1205742

1Θ10158401015840

0+ 1205742

0Θ10158401015840

1)Θ2

0+ 21205742

0Θ0Θ1Θ10158401015840

0]

minus 1205832[(1205742

1Θ0+ 1205742

0Θ1)Θ10158402

0+ 21205742

0Θ0Θ1015840

0Θ1015840

1]

minus 120587312058321205742

11205720cos120587120585

(A1)

B Parameters Used in Equations (32)and (33)

Ξ1=

1

1536(1 + 9120587

21205832)

times [1205725

119879+ 81205723

119879minus 9120587 (1 + 120587

21205832) 12057201205722

119879]

Ξ2= minus

1

36864(1 + 9120587

21205832) (1 + 25120587

21205832) 1205725

119879

(B1)

1198921(120572119879 1205720 120583)

= 12058721 +

1

1536(1 + 120587

21205832)

times [ (25 + 3312058721205832) 1205724

119879

+ 192 (1 minus 120587312058321205720120572minus1

119879) (1205722

119879minus 8120587120572

0120572minus1

119879)

minus8120587 (23 + 1512058721205832) 1205720120572119879]

(B2)

C Coefficients Appeared in Equations (36)and (38)

Λ1(120572119879 1205742 1205742

0)

=

12057421205723

119879(1 + 120583

21205742

0) minus 8120587

31205720(1 + 120583

21205742)

8 (1205742 minus 1205742

0)

Λ2(120572119879 1205742 1205742

0) =

12057421205723

119879(1 + 9120583

21205742

0)

24 (1205742 minus 91205742

0)

(C1)

119887 = minus (1 + 312058721205832) 1205723

119879+ 6120572119879+ 6120587312058321205720

119888 = 1205872[minus (7 + 9120587

21205832) 1205723

119879+ 60120572

119879minus 6120587 (1 minus 9120587

21205832) 1205720]

119889 = 541205874(120572119879minus 1205871205720)

(C2)


Nomenclature

119860 Cross-sectional area119886 Internal characteristic length1198860 Midspan initial rise

119864 Youngrsquos modulus1198900 Constant appropriate to each material

119891 Analytical function119867 Horizontal internal force119868 Second moment of inertia of the cross-section119871 Length of the column119872 Moment resultant119875 Conservative force119901 Embedding auxiliary parameter119875119864 Euler load for local elastic columns

119875cr Critical load for nonlocal elastic columns119904 Arc length along the column119881 Vertical internal force119908 Deflection in 119911 direction1199080 Initial deflection in 119911 direction

119909 119911 In-plane coordinatesH HomotopyL Linear operatorN Nonlinear operator1205720 Dimensionless midspan initial rise

120572119879 End rotation of the column

120576 Normal strain120578 Distance from the neutral axis120582 Dimensionless force parameter (radic1198751198712119864119868)120582 Dimensionless parameter (120583120582)120574 Dimensionless parameter (120582radic1 minus 1205822)120583 Dimensionless small scale parameter120579119879 Rotation of the cross-section

120579 Rotation induced by pure bending1205790 Rotation due to the initial imperfection

120590 Normal stress120585 Dimensionless arc lengthΘ Unknown homotopy parameterΘlowast

0 Initial approximation of the rotation

Acknowledgments

Financial supports from National Natural Science Foun-dation of China (no 11002135) and the China ScholarshipCouncil are gratefully acknowledged

References

[1] S Iijima ldquoHelicalmicrotubules of graphitic carbonrdquoNature vol354 no 6348 pp 56ndash58 1991

[2] T Murmu and S Adhikari ldquoNonlocal elasticity based vibrationof initially pre-stressed coupled nanobeam systemsrdquo EuropeanJournal of Mechanics A vol 34 pp 52ndash62 2012

[3] A C Eringen Nonlocal Continuum Field Theories SpringerNew York NY USA 2002

[4] Y Chen J D Lee and A Eskandarian ldquoAtomistic viewpointof the applicability of microcontinuum theoriesrdquo InternationalJournal of Solids and Structures vol 41 no 8 pp 2085ndash20972004

[5] 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

[6] C M Wang Y Y Zhang Y Xiang and J N Reddy ldquoRecentstudies on buckling of carbon nanotubesrdquo Applied MechanicsReviews vol 63 no 3 Article ID 030804 18 pages 2010

[7] E W Wong P E Sheehan and C M Lieber ldquoNanobeammechanics elasticity strength and toughness of nanorods andnanotubesrdquo Science vol 277 no 5334 pp 1971ndash1975 1997

[8] M R Falvo G J Clary R M Taylor et al ldquoBending andbuckling of carbon nanotubes under large strainrdquo Nature vol389 no 6651 pp 582ndash584 1997

[9] B I Yakobson C J Brabec and J Bernholc ldquoNanomechanicsof carbon tubes instabilities beyond linear responserdquo PhysicalReview Letters vol 76 no 14 pp 2511ndash2514 1996

[10] G Alici ldquoAn effectivemodelling approach to estimate nonlinearbending behaviour of cantilever type conducting polymeractuatorsrdquo Sensors and Actuators B vol 141 no 1 pp 284ndash2922009

[11] A E H Love A Treatise on the Mathematical Theory of Elastic-ity Dover New York NY USA 4th edition 1944

[12] S P Timoshenko Theory of Elastic Stability EngineeringSocieties Monographs McGraw-Hill New York NY USA 2ndedition 1961

[13] V G A Goss ldquoThe history of the planar elastica insights intomechanics and scientific methodrdquo Science and Education vol18 no 8 pp 1057ndash1082 2009

[14] S Matsutani ldquoEulerrsquos elastica and beyondrdquo Journal of Geometryand Symmetry in Physics vol 17 pp 45ndash86 2010

[15] J C Lotz OM OrsquoReilly andDM Peters ldquoSome comments onthe absence of buckling of the ligamentous human spine in thesagittal planerdquoMechanics Research Communications vol 40 pp11ndash15 2012

[16] H F Chen S P Xu and H Y Guo ldquoNonlinear analysis offlexible and steel catenary risers with internal flow and seabedinteraction effectsrdquo Journal of Marine Science and Applicationvol 10 no 2 pp 156ndash162 2011

[17] H F Chen S P Xu and H Y Guo ldquoParametric study of globalresponse behavior of deepwater free standing hybrid risersrdquoJournal of Ship Mechanics vol 15 pp 996ndash1004 2011

[18] H W Haslach Jr ldquoPost-buckling behavior of columns withnon-linear constitutive equationsrdquo International Journal of Non-Linear Mechanics vol 20 no 1 pp 53ndash67 1985

[19] S Al-Sadder and N Shatarat ldquoA proposed technique for largedeflection analysis of cantilever beams composed of two non-linear elastic materials subjected to an inclined tip concentratedforcerdquoAdvances in Structural Engineering vol 10 no 3 pp 319ndash335 2007

[20] A M Vinogradov and W R Derrick ldquoStructure-material rela-tions in the buckling problem of asymmetric compositecolumnsrdquo International Journal of Non-Linear Mechanics vol35 no 1 pp 167ndash175 2000

[21] Y A Kang and X F Li ldquoBending of functionally gradedcantilever beam with power-law non-linearity subjected to anend forcerdquo International Journal of Non-Linear Mechanics vol44 no 6 pp 696ndash703 2009

[22] N Challamel ldquoOn geometrically exact post-buckling of com-posite columns with interlayer slipmdashthe partially compositeelasticardquo International Journal of Non-Linear Mechanics vol 47no 3 pp 7ndash17 2012


[23] Y Frostig ldquoElastica of sandwich panels with a transverselyflexible coremdasha high-order theory approachrdquo InternationalJournal of Solids and Structures vol 46 no 10 pp 2043ndash20592009

[24] L P Li K Schulgasser and G Cederbaum ldquoLarge deflectionanalysis of poroelastic beamsrdquo International Journal of Non-Linear Mechanics vol 33 no 1 pp 1ndash14 1998

[25] G Suire and G Cederbaum ldquoElastica type dynamic stabilityanalysis of viscoelastic columnsrdquo Archive of Applied Mechanicsvol 64 no 5 pp 307ndash316 1994

[26] Y Mikata ldquoComplete solution of elastica for a clamped-hinged beam and its applications to a carbon nanotuberdquo ActaMechanica vol 190 no 1ndash4 pp 133ndash150 2007

[27] X Q He C MWang and K Y Lam ldquoAnalytical bending solu-tions of elastica with one end held while the other end portionslides on a friction supportrdquo Archive of Applied Mechanics vol67 no 8 pp 543ndash554 1997

[28] J S Chen H C Li and W C Ro ldquoSlip-through of a heavyelastica on point supportsrdquo International Journal of Solids andStructures vol 47 no 2 pp 261ndash268 2010

[29] C Kimball and LW Tsai ldquoModeling of flexural beams subject-ed to arbitrary end loadsrdquo Journal ofMechanical Design vol 124no 2 pp 223ndash235 2002

[30] S Chucheepsakul C M Wang and X Q He ldquoDouble curva-ture bending of variable-arc-length elasticasrdquo Journal of AppliedMechanics vol 66 no 1 pp 87ndash94 1999

[31] R H Plaut D A Dillard and L N Virgin ldquoPostbuckling ofelastic columns with second-mode imperfectionrdquo Journal ofEngineering Mechanics vol 132 no 8 pp 898ndash901 2006

[32] B S Shvartsman ldquoDirect method for analysis of flexiblecantilever beam subjected to two follower forcesrdquo InternationalJournal of Non-Linear Mechanics vol 44 no 2 pp 249ndash2522009

[33] C S Liu ldquoA Lie-group shooting method for post bucklingcalculations of elasticardquo Computer Modeling in Engineering andSciences vol 30 no 1 pp 1ndash16 2008

[34] R Kumar L S Ramachandra and D Roy ldquoTechniques basedon genetic algorithms for large deflection analysis of beamsrdquoSadhana vol 29 no 6 pp 589ndash604 2004

[35] T Y Wang C G Koh and C Y Liaw ldquoPost-buckling analysisof planar elastica using a hybrid numerical strategyrdquo Computersand Structures vol 88 no 11-12 pp 785ndash795 2010

[36] O Sepahi M R Forouzan and P Malekzadeh ldquoDifferentialquadrature application in post-buckling analysis of a hinged-fixed elastica under terminal forces and self-weightrdquo Journal ofMechanical Science and Technology vol 24 no 1 pp 331ndash3362010

[37] D D Berkey and M I Freedman ldquoA perturbation methodapplied to the buckling of a compressed elasticardquo Journal ofComputational and Applied Mathematics vol 4 no 3 pp 213ndash221 1978

[38] C Y Wang ldquoAsymptotic formula for the flexible barrdquo Mecha-nism and Machine Theory vol 34 no 4 pp 645ndash655 1999

[39] W Lacarbonara ldquoBuckling and post-buckling of non-uniformnon-linearly elastic rodsrdquo International Journal of MechanicalSciences vol 50 no 8 pp 1316ndash1325 2008

[40] C E N Mazzilli ldquoBuckling and post-buckling of extensiblerods revisited a multiple-scale solutionrdquo International Journalof Non-Linear Mechanics vol 44 no 2 pp 200ndash208 2009

[41] H S Shen ldquoA novel technique for nonlinear analysis of beamson two-parameter elastic foundationsrdquo International Journal of

Structural Stability and Dynamics vol 11 no 6 pp 999ndash10142011

[42] JWang J K Chen and S Liao ldquoAn explicit solution of the largedeformation of a cantilever beam under point load at the freetiprdquo Journal of Computational and Applied Mathematics vol212 no 2 pp 320ndash330 2008

[43] N Tolou and J L Herder ldquoA seminalytical approach to largedeflections in compliant beams under point loadrdquoMathematicalProblems in Engineering vol 2009 Article ID 910896 13 pages2009

[44] S Ghosh and D Roy ldquoNumeric-analytic form of the adomiandecompositionmethod for two-point boundary value problemsin nonlinear mechanicsrdquo Journal of Engineering Mechanics vol133 no 10 pp 1124ndash1133 2007

[45] P N Andriotaki I H Stampouloglou and E E TheotokoglouldquoNonlinear asymptotic analysis in elastica of straight bars-analytical parametric solutionsrdquo Archive of Applied Mechanicsvol 76 no 9-10 pp 525ndash536 2006

[46] C M Wang Y Xiang and S Kitipornchai ldquoPostbuckling ofnano rodstubes based on nonlocal beam theoryrdquo InternationalJournal of Applied Mechanics vol 1 no 2 pp 259ndash266 2009

[47] S P Xu ldquoElastica type buckling analysis of micro-nano-rodsusing nonlocal elasticity theoryrdquo in Proceedings of the 2nd AsianConference on Mechanics of Functional Materials and Structures(ACMFMSrsquo10) pp 219ndash222 Nanjing China 2010

[48] S P Xu C M Wang and M R Xu ldquoBuckling analysis ofshear deformable nanorods within the framework of nonlocalelasticity theoryrdquo Physica E vol 44 no 7-8 pp 1380ndash1385 2012

[49] T M Atanackovic B N Novakovic and Z Vrcelj ldquoApplicationof Pontryaginrsquos principle to bimodal optimization of nano rodsrdquoInternational Journal of Structural Stability and Dynamics vol12 no 3 Article ID 1250012 11 pages 2012

[50] T R Tauchert andW Y Lu ldquoLarge deformation and postbuck-ling behavior of an initially deformed rodrdquo International Journalof Non-Linear Mechanics vol 22 no 6 pp 511ndash520 1987

[51] M D Williams F V Keulen and M Sheplak ldquoModelingof initially curved beam structures for design of multistableMEMSrdquo Journal of Applied Mechanics vol 79 no 1 Article ID011006 11 pages 2012

[52] A N Kounadis and A F Economou ldquoThe effects of initialcurvature and other parameters on the nonlinear buckling ofsimple framesrdquo Journal of Structural Mechanics vol 12 no 1pp 27ndash42 1984

[53] S A Emam ldquoA static and dynamic analysis of the postbucklingof geometrically imperfect composite beamsrdquo Composite Struc-tures vol 90 no 2 pp 247ndash253 2009

[54] WH Duan and CMWang ldquoExact solutions for axisymmetricbending of micronanoscale circular plates based on nonlocalplate theoryrdquoNanotechnology vol 18 no 38 Article ID 3857042007

[55] Q Wang and C M Wang ldquoThe constitutive relation andsmall scale parameter of nonlocal continuum mechanics formodelling carbon nanotubesrdquo Nanotechnology vol 18 no 7Article ID 075702 2007

[56] J Mallis and A N Kounadis ldquoOn the accuracy of various largeaxial displacement formulae for crooked columnsrdquo Computa-tional Mechanics vol 4 no 1 pp 47ndash58 1988

[57] T M Atanackovic B N Novakovic and Z Vrcelj ldquoShape opti-mization against buckling of micro- and nano-rodsrdquo Archive ofApplied Mechanics vol 82 no 10-11 pp 1303ndash1311 2012


[58] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[59] J H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[60] M Mojahedi M M Zand and M T Ahmadian ldquoStatic pull-inanalysis of electrostatically actuated microbeams using homo-topy perturbation methodrdquo Applied Mathematical Modellingvol 34 no 4 pp 1032ndash1041 2010

[61] M R Xu S P Xu andH Y Guo ldquoDetermination of natural fre-quencies of fluid-conveying pipes using homotopy perturbationmethodrdquoComputers andMathematics withApplications vol 60no 3 pp 520ndash527 2010

[62] A N Kounadis and J GMallis ldquoElastica type buckling analysisof bars from non-linearly elastic materialrdquo International Journalof Non-Linear Mechanics vol 22 no 2 pp 99ndash107 1987

[63] G Kandakis and A N Kounadis ldquoOn the large postbucklingresponse of nonconservative continuous systemsrdquo Archive ofApplied Mechanics vol 62 no 4 pp 256ndash265 1992

[64] A N Kounadis J Mallis and A Sbarounis ldquoPostbucklinganalysis of columns resting on an elastic foundationrdquo Archiveof Applied Mechanics vol 75 no 6-7 pp 395ndash404 2006

[65] AN Kounadis ldquoAn efficient and simple approximate techniquefor solving nonlinear initial and boundary-value problemsrdquoComputational Mechanics vol 9 no 3 pp 221ndash231 1992

[66] N Challamel and C M Wang ldquoOn lateral-torsional bucklingof non-local beamsrdquo Advances in Applied Mathematics andMechanics vol 2 no 3 pp 389ndash398 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of


for the nanoelements so that the curvature may be approxi-mated by the second derivative of the transverse deflectionNevertheless these nanoelements usually possess excellentflexibility just as experimental observations on carbon nan-otubes have indicated and they can undertake significantdeformations without experiencing apparent plasticity [7ndash9]Thus the general linearized theories fail to depict this afore-mentioned behavioral feature in a satisfactorymanner [10] Inorder to predict the postbuckling behavior of nanoelementsboth large deflection and size effect must be consideredHence a geometrically exact theory is indispensable forpostbuckling behavior of nanoelements

In the light of themathematical consistency in theway themechanical model is formulated hereinafter we will restrictour attention to the elastica theory [11 12] which entailsusing the exact expression of curvature The first study ofthe deformation of an elastica under loading was probablydue to James Bernoulli although it was Euler who establishedthe basic theory [11 13] By taking advantage of Eulerrsquoscelebrated work considerable efforts have been devoted tothe theory with applications extended from classical fieldsto statistical mechanics (see [14] and the references therein)biomechanics [15] and ocean engineering [16 17]Within thescope ofmost of these studies linear constitutive relations areutilized Encouraged by its effectiveness in dealing with thesegeometrical nonlinear problems the focus is also transferredto other kinds of materials and structures For exampleHaslach Jr [18] investigated the postbuckling behavior ofa column made of a material having a cubic constitutiveequation while Al-Sadder and Shatarat [19] examined thelarge deflection of a prismatic composite cantilever beamcomprising two different nonlinear elastic materials Withinthe framework of the elastica theory the nonlinear bucklingresponse of asymmetric laminated structures composed ofan arbitrary number of distinct material layers was treatedby Vinogradov and Derrick [20] Kang and Li [21] consid-ered the mechanical behaviors of a cantilevered functionallygraded material (FGM) beam that obeys the Ludwick-typelaw An interesting ldquopartially composite elasticardquo problemwasstudied by Challamel [22] where each subcolumn ismodeledwith the Euler-Bernoulli beam theory and connected to eachother via a linear constitutive law for the interlayer slip Inaddition Frostig [23] revealed the elastica behavior of a sand-wich panel with a softcompliant core that analysis includesboth the shear deformation and moderate strain The largedeflection of the fluid-saturated poroelastic beams which arepermeable in the axial direction and impermeable in thetransverse directions was examined by Li et al [24] Likewisean elastica-type dynamic stability of viscoelastic columnswith its behavior given in terms of the Boltzmann superpo-sition principle was discussed by Suire and Cederbaum [25]Moreover the boundary conditions were also extended fromthe commonly used conditions namely simply supportedand clamped-free conditions to clamped-hinged condition[26] frictional end [27] and slippery support [28] conditions

As for the governing equations of these elastica prob-lems the elliptical integral functions are often employedto express their exact solutions For example by utilizingthe elliptical integral transformation Kimball and Tsai [29]

presented the nonlinear deflection solution for a cantileverbeam subjected to an arbitrary combination of end forcesand moments Moreover the solution for double curvaturebending of variable-arc-length elasticas was proposed byChucheepsakul et al [30] in terms of elliptical integralsHowever from an engineering point of view these exactsolutions are not quite intuitive of any physical insight andone often needs to resort to numerical methods Besides theexact treatment remains intractable for most cases because ofthe intrinsic nonlinearity that resides in the elastica theoryTherefore another branch of treatment from the numericalpoint of view is developed simultaneously in conjunctionwith the availability of high-speed computers One popularapproach is the finite-element method for example bywhich along with Newton-Rhapson iteration technique thestatic analysis of risers under the influence of internal flowand hydrostatic pressure was performed by Chen et al[16] Another frequently used approach is the shootingmethod It converts the nonlinear boundary-value probleminto an initial-value problem which is then determinedwith or without iterations [31 32] Other newly developednumerical methods include the Lie-group shooting method[33] method of genetic algorithm [34 35] and differentialquadrature method [36] In view that an analytical solutionis always convenient for parametric studies and capturesclearly the physics of the problem it appears more appealingthan the numerical oneThus extensive research efforts werealso made in this third line of development which madeuse of all kinds of asymptotic methods Among them theperturbation technique [37ndash41] is popular in the broad area ofbuckling problems of structures For instance a higher-orderperturbation technique was employed by Lacarbonara [39] todetermine the postbuckling solutions for nonprismatic non-linearly elastic rods In surveying the postbuckling regimefor the five classical Euler buckling cases of an extensibleelastica the method of multiple (spatial) scales perturbationwas proposed by Mazzilli [40] The analysis of postbucklingas well as nonlinear bending and nonlinear vibration fora simply supported Euler-Bernoulli beam resting on a two-parameter elastic foundation was accomplished by Shen [41]with the help of a novel two-step perturbation techniqueAnother recently developedmethod the homotopy analyticalmethod was used by Wang et al [42] to solve a cantileverbeam problem in order to obtain an explicit solution of thedisplacement and rotation at the free end Later this problemwas revisited by Tolou and Herder by means of the Adomiandecompositionmethod [43]The remarkable accuracy of thismethod was also demonstrated in dealing with the tip loadedcantilever elastica and plastica problems [44] Besides withthe help of a series of admissible functional transformationsAndriotaki et al [45] furnished an implicit analytical solutionfor the elastica problem of a cantilever bar due to its ownweight

Although there are extensive studies related to the elasticaproblems just as clearly indicated earlier the scrutinies oflarge deflections of flexible nanocolumns with size effectincluded are less frequent Until recently efforts toward anelastica type buckling analysis of nanocolumn were con-ducted By employing the shooting method Wang et al












120576 = minus120578119889120579

119889119904 (1)



120590 minus (1198900119886)2 1198892120590

1198891199042= 119864120576 (2)



0is a constant


119872 = int119860

120578120590119889119860 (3)


119872minus (1198900119886)2 1198892119872

1198891199042= minus119864119868

119889120579

119889119904 (4)




119889119872

119889119904= 119881 cos 120579

119879minus 119867 sin 120579

119879 (5)


119867(119904) = minus119875 119881 (119904) = 0 (6)


119871

1199080

119908

119875

119906(119871)

119909

119911

(a)

119909119881

119867

119872

119911

1205790

120579

119889119904

120579T119872+ d119872119881+ d119881

119867+ d119867

(b)




1198641198681198892120579119879

1198891199042+ 119875 sin 120579

119879

= 119875(1198900119886)2 cos 120579

119879(1198892120579119879

1198891199042minus tan 120579

119879(119889120579119879

119889119904)

2

)

+ 11986411986811988921205790

1198891199042

(7)


120585 =119904

119871 120583 =

1198900119886

119871

120582 = radic1198751198712

119864119868 120582 = 120583120582

(8)


12057910158401015840

119879+ 1205822 sin 120579

119879= 1205822 cos 120579

119879(12057910158401015840

119879minus tan 120579

11987912057910158402

119879) + 12057910158401015840

0 (9)



1199080 (119904) = 1198860 sin120587120585 (10)



119889119909 = 119889119904 cos 120579119879 119889119908 = 119889119904 sin 120579

119879 (11)

we have

1205790 (119904) = sinminus1 [

1198891199080 (119904)

119889119904] (12)

and correspondingly

12057910158401015840

0=

12057201205873(1205722

01205872minus 1) cos120587120585

(1 minus 1205722

01205872cos2120587120585)32

(13)

where 1205720= 1198860119871


119908 (0) = 119908 (119871) = 0 (14)




119875119908 + [119864119868 minus 119875(1198900119886)2 cos 120579

119879]119889120579119879

119889119904= 119864119868

1198891205790

119889119904 (15)


119889120579119879

119889119904= 0 at 119904 = 0 119871 (16)

or

119864119868 minus 119875(1198900119886)2 cos 120579

119879= 0 at 119904 = 0 119871 (17)


1205791015840

119879(0) = 120579

1015840

119879(1) = 0 120579

119879 (0) = 120572119879 (18)






sin 120579119879= 120579119879minus1

61205793

119879

cos 120579119879= 1 minus

1

21205792

119879

tan 120579119879= 120579119879+1

31205793

119879

(19)



12057910158401015840

0= minus12057201205873 cos120587120585 (20)


119879) one obtains

L (120579119879) = N (120579

119879) + 119891 (120585) 120585 isin [0 1] (21)


12057910158401015840


119879) =

1205742[1205793

1198796 minus 120583

2120579119879(12057911987912057910158401015840

1198792 + 120579

10158402


119891(120585) = minus1205873(1 + 120583


defined as 1205742 = 1205822(1 minus 1205822)



H (Θ 119901)


0)] + 119901 [L (Θ) minusN (Θ) minus 119891 (120585)]

= 0

(22)



Θ1015840(0) = Θ

1015840(1) = 0 Θ (0) = 120572119879 (23)


0) = 0 According


Θ (120585) = Θ0 (120585) +

infin

sum

119895=1

119901119895Θ119895 (120585) (24)


1205742= 1205742

0+

infin

sum

119895=1

1199011198951205742

119895 (25)


O (1199010) Θ

10158401015840

0+ 1205742

0Θ0= 0

Θ1015840

0(0) = Θ

1015840

0(0) = 0 Θ

0 (0) = 120572119879

(26a)

O (1199011) Θ

10158401015840

1+ 1205742

0Θ1= 1198911(Θ0 1205742

1 1205742

0)

Θ1015840

1(0) = Θ

1015840

1(1) = 0 Θ

1 (0) = 0

(26b)

O (1199012) Θ

10158401015840

2+ 1205742

0Θ2= 1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

Θ1015840

2(0) = Θ

1015840

2(1) = 0 Θ

2 (0) = 0

(26c)




Θ0 (120585) = 120572119879 cos120587120585 (27)




Θ10158401015840

1+ 1205742

0Θ1

= [minus1205742

1120572119879+1

81205872(1 + 120587

21205832) 1205723

119879minus 1205873(1 + 120587

21205832) 1205720] cos120587120585

+1

241205872(1 + 9120587

21205832) 1205723

119879cos 3120587120585

(28)


Θ1=1

192(1 + 9120587

21205832) 1205723

119879(cos120587120585 minus cos 3120587120585) (29)


1205742

1=1

8(1 + 120587

21205832) [12058721205722

119879minus 812058731205720120572minus1

119879] (30)



cos 1198991205740120585 119899 = 1 2 3 (31)







120579119879 (120585) = Θ (120585)

1003816100381610038161003816119901=1= 120572119879cos120587120585


(32)

1205742=

2

sum

119894=0

1205742

119894≜ 1198921(120572119879 1205720 120583) (33)

where Ξ119894(119894 = 1 2) and 119892

1(120572119879 1205720 120583) are defined in

Appendix B



L (120579119879) = 0 (34)




L (120579119879 1205742) = N (120579

1198790 1205742

0) + 119891 (120585 120574

2) 120585 isin [0 1] (35)


1205791198791 (120585) = Λ 1 (120572119879 120574

2 1205742

0) cos120587120585 + Λ

2(120572119879 1205742 1205742

0) cos 3120587120585

(36)

where Λ119894(1205742 1205742


Note that 1205791198791



L (120579119879 1205742) = N (120579

1198791 1205742 1205742

0) + 119891 (120585 120574

2) 120585 isin [0 1] (37)


1198791(0) = 120572



1198871205744minus 1198881205742+ 119889 = 0 (38)



2119887≜ 1198922(120572119879 1205720 120583) (39)






119875

119875119864

=119892119894(120572119879 1205720 120583)

1205872 [1 + 1205832119892119894(120572119879 1205720 120583)]

(119894 = 1 or 2) (40)

in which 119875119864= 119864119868120587





119875cr = 119875119864(1 + 12058721205832)minus1

(41)




119909

119871= int

120585

0

cos 120579119879 (120585) 119889120585

119911

119871= int

120585

0

sin 120579119879 (120585) 119889120585

(42)







120572119879

119875exact119875119864(a)

119875HPM1119875119864

(b)119875HPM2

119875119864

(c)119875SAA119875119864

(d)



120572119879

119908exact119871(a)

119908HPM1119871

(b)119908HPM2

119871(c)

119908SAA119871(d)



relative error

=

10038161003816100381610038161003816100381610038161003816


minus 1

10038161003816100381610038161003816100381610038161003816

times 100

(43)







120572119879

120583 = 0 120583 = 005 120583 = 010 120583 = 015 120583 = 020

20∘ 01097 01097 01098 01099 0110040∘ 02111 02113 02118 02126 0213760∘ 02967 02972 02987 03012 0304580∘ 03601 03612 03642 03692 03759100∘ 03965 03980 04026 04098 04190120∘ 04019 04035 04078 04136 04196

0 02 04 06 08 10

01

02

03

04

05

119911119871

119909119871



Red lines 120583 = 020

120572T = 20∘

120572T = 60∘

120572T = 100∘

120572T = 120∘



11112131415

119875119875119864


120572T (∘)




15

125

1

075

05

025

0


119875119875119864


120572T (∘)



15

125

1

075

05

025

0

119875119875119864



120572T (∘)








Appendix


1198911(Θ0 1205742

1 1205742

0)

= minus1205742

1Θ0+1

61205742

0Θ3

0minus 12058321205742

0Θ0(1

2Θ0Θ10158401015840

0+ Θ10158402

0)

minus 1205873(1 + 120583

21205742

0) 1205720cos120587120585

1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

= minus1205742

2Θ0minus 1205742

1Θ1+1

61205742

1Θ3

0+1

21205742

0Θ2

0Θ2

1

minus1

21205832[(1205742

1Θ10158401015840

0+ 1205742

0Θ10158401015840

1)Θ2

0+ 21205742

0Θ0Θ1Θ10158401015840

0]

minus 1205832[(1205742

1Θ0+ 1205742

0Θ1)Θ10158402

0+ 21205742

0Θ0Θ1015840

0Θ1015840

1]

minus 120587312058321205742

11205720cos120587120585

(A1)


Ξ1=

1

1536(1 + 9120587

21205832)

times [1205725

119879+ 81205723

119879minus 9120587 (1 + 120587

21205832) 12057201205722

119879]

Ξ2= minus

1

36864(1 + 9120587

21205832) (1 + 25120587

21205832) 1205725

119879

(B1)

1198921(120572119879 1205720 120583)

= 12058721 +

1

1536(1 + 120587

21205832)

times [ (25 + 3312058721205832) 1205724

119879

+ 192 (1 minus 120587312058321205720120572minus1

119879) (1205722

119879minus 8120587120572

0120572minus1

119879)

minus8120587 (23 + 1512058721205832) 1205720120572119879]

(B2)


Λ1(120572119879 1205742 1205742

0)

=

12057421205723

119879(1 + 120583

21205742

0) minus 8120587

31205720(1 + 120583

21205742)

8 (1205742 minus 1205742

0)

Λ2(120572119879 1205742 1205742

0) =

12057421205723

119879(1 + 9120583

21205742

0)

24 (1205742 minus 91205742

0)

(C1)

119887 = minus (1 + 312058721205832) 1205723

119879+ 6120572119879+ 6120587312058321205720

119888 = 1205872[minus (7 + 9120587

21205832) 1205723

119879+ 60120572

119879minus 6120587 (1 minus 9120587

21205832) 1205720]

119889 = 541205874(120572119879minus 1205871205720)

(C2)


Nomenclature











Acknowledgments


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















120576 = minus120578119889120579

119889119904 (1)



120590 minus (1198900119886)2 1198892120590

1198891199042= 119864120576 (2)



0is a constant


119872 = int119860

120578120590119889119860 (3)


119872minus (1198900119886)2 1198892119872

1198891199042= minus119864119868

119889120579

119889119904 (4)




119889119872

119889119904= 119881 cos 120579

119879minus 119867 sin 120579

119879 (5)


119867(119904) = minus119875 119881 (119904) = 0 (6)


119871

1199080

119908

119875

119906(119871)

119909

119911

(a)

119909119881

119867

119872

119911

1205790

120579

119889119904

120579T119872+ d119872119881+ d119881

119867+ d119867

(b)




1198641198681198892120579119879

1198891199042+ 119875 sin 120579

119879

= 119875(1198900119886)2 cos 120579

119879(1198892120579119879

1198891199042minus tan 120579

119879(119889120579119879

119889119904)

2

)

+ 11986411986811988921205790

1198891199042

(7)


120585 =119904

119871 120583 =

1198900119886

119871

120582 = radic1198751198712

119864119868 120582 = 120583120582

(8)


12057910158401015840

119879+ 1205822 sin 120579

119879= 1205822 cos 120579

119879(12057910158401015840

119879minus tan 120579

11987912057910158402

119879) + 12057910158401015840

0 (9)



1199080 (119904) = 1198860 sin120587120585 (10)



119889119909 = 119889119904 cos 120579119879 119889119908 = 119889119904 sin 120579

119879 (11)

we have

1205790 (119904) = sinminus1 [

1198891199080 (119904)

119889119904] (12)

and correspondingly

12057910158401015840

0=

12057201205873(1205722

01205872minus 1) cos120587120585

(1 minus 1205722

01205872cos2120587120585)32

(13)

where 1205720= 1198860119871


119908 (0) = 119908 (119871) = 0 (14)




119875119908 + [119864119868 minus 119875(1198900119886)2 cos 120579

119879]119889120579119879

119889119904= 119864119868

1198891205790

119889119904 (15)


119889120579119879

119889119904= 0 at 119904 = 0 119871 (16)

or

119864119868 minus 119875(1198900119886)2 cos 120579

119879= 0 at 119904 = 0 119871 (17)


1205791015840

119879(0) = 120579

1015840

119879(1) = 0 120579

119879 (0) = 120572119879 (18)






sin 120579119879= 120579119879minus1

61205793

119879

cos 120579119879= 1 minus

1

21205792

119879

tan 120579119879= 120579119879+1

31205793

119879

(19)



12057910158401015840

0= minus12057201205873 cos120587120585 (20)


119879) one obtains

L (120579119879) = N (120579

119879) + 119891 (120585) 120585 isin [0 1] (21)


12057910158401015840


119879) =

1205742[1205793

1198796 minus 120583

2120579119879(12057911987912057910158401015840

1198792 + 120579

10158402


119891(120585) = minus1205873(1 + 120583


defined as 1205742 = 1205822(1 minus 1205822)



H (Θ 119901)


0)] + 119901 [L (Θ) minusN (Θ) minus 119891 (120585)]

= 0

(22)



Θ1015840(0) = Θ

1015840(1) = 0 Θ (0) = 120572119879 (23)


0) = 0 According


Θ (120585) = Θ0 (120585) +

infin

sum

119895=1

119901119895Θ119895 (120585) (24)


1205742= 1205742

0+

infin

sum

119895=1

1199011198951205742

119895 (25)


O (1199010) Θ

10158401015840

0+ 1205742

0Θ0= 0

Θ1015840

0(0) = Θ

1015840

0(0) = 0 Θ

0 (0) = 120572119879

(26a)

O (1199011) Θ

10158401015840

1+ 1205742

0Θ1= 1198911(Θ0 1205742

1 1205742

0)

Θ1015840

1(0) = Θ

1015840

1(1) = 0 Θ

1 (0) = 0

(26b)

O (1199012) Θ

10158401015840

2+ 1205742

0Θ2= 1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

Θ1015840

2(0) = Θ

1015840

2(1) = 0 Θ

2 (0) = 0

(26c)




Θ0 (120585) = 120572119879 cos120587120585 (27)




Θ10158401015840

1+ 1205742

0Θ1

= [minus1205742

1120572119879+1

81205872(1 + 120587

21205832) 1205723

119879minus 1205873(1 + 120587

21205832) 1205720] cos120587120585

+1

241205872(1 + 9120587

21205832) 1205723

119879cos 3120587120585

(28)


Θ1=1

192(1 + 9120587

21205832) 1205723

119879(cos120587120585 minus cos 3120587120585) (29)


1205742

1=1

8(1 + 120587

21205832) [12058721205722

119879minus 812058731205720120572minus1

119879] (30)



cos 1198991205740120585 119899 = 1 2 3 (31)







120579119879 (120585) = Θ (120585)

1003816100381610038161003816119901=1= 120572119879cos120587120585


(32)

1205742=

2

sum

119894=0

1205742

119894≜ 1198921(120572119879 1205720 120583) (33)

where Ξ119894(119894 = 1 2) and 119892

1(120572119879 1205720 120583) are defined in

Appendix B



L (120579119879) = 0 (34)




L (120579119879 1205742) = N (120579

1198790 1205742

0) + 119891 (120585 120574

2) 120585 isin [0 1] (35)


1205791198791 (120585) = Λ 1 (120572119879 120574

2 1205742

0) cos120587120585 + Λ

2(120572119879 1205742 1205742

0) cos 3120587120585

(36)

where Λ119894(1205742 1205742


Note that 1205791198791



L (120579119879 1205742) = N (120579

1198791 1205742 1205742

0) + 119891 (120585 120574

2) 120585 isin [0 1] (37)


1198791(0) = 120572



1198871205744minus 1198881205742+ 119889 = 0 (38)



2119887≜ 1198922(120572119879 1205720 120583) (39)






119875

119875119864

=119892119894(120572119879 1205720 120583)

1205872 [1 + 1205832119892119894(120572119879 1205720 120583)]

(119894 = 1 or 2) (40)

in which 119875119864= 119864119868120587





119875cr = 119875119864(1 + 12058721205832)minus1

(41)




119909

119871= int

120585

0

cos 120579119879 (120585) 119889120585

119911

119871= int

120585

0

sin 120579119879 (120585) 119889120585

(42)







120572119879

119875exact119875119864(a)

119875HPM1119875119864

(b)119875HPM2

119875119864

(c)119875SAA119875119864

(d)



120572119879

119908exact119871(a)

119908HPM1119871

(b)119908HPM2

119871(c)

119908SAA119871(d)



relative error

=

10038161003816100381610038161003816100381610038161003816


minus 1

10038161003816100381610038161003816100381610038161003816

times 100

(43)







120572119879

120583 = 0 120583 = 005 120583 = 010 120583 = 015 120583 = 020

20∘ 01097 01097 01098 01099 0110040∘ 02111 02113 02118 02126 0213760∘ 02967 02972 02987 03012 0304580∘ 03601 03612 03642 03692 03759100∘ 03965 03980 04026 04098 04190120∘ 04019 04035 04078 04136 04196

0 02 04 06 08 10

01

02

03

04

05

119911119871

119909119871



Red lines 120583 = 020

120572T = 20∘

120572T = 60∘

120572T = 100∘

120572T = 120∘



11112131415

119875119875119864


120572T (∘)




15

125

1

075

05

025

0


119875119875119864


120572T (∘)



15

125

1

075

05

025

0

119875119875119864



120572T (∘)








Appendix


1198911(Θ0 1205742

1 1205742

0)

= minus1205742

1Θ0+1

61205742

0Θ3

0minus 12058321205742

0Θ0(1

2Θ0Θ10158401015840

0+ Θ10158402

0)

minus 1205873(1 + 120583

21205742

0) 1205720cos120587120585

1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

= minus1205742

2Θ0minus 1205742

1Θ1+1

61205742

1Θ3

0+1

21205742

0Θ2

0Θ2

1

minus1

21205832[(1205742

1Θ10158401015840

0+ 1205742

0Θ10158401015840

1)Θ2

0+ 21205742

0Θ0Θ1Θ10158401015840

0]

minus 1205832[(1205742

1Θ0+ 1205742

0Θ1)Θ10158402

0+ 21205742

0Θ0Θ1015840

0Θ1015840

1]

minus 120587312058321205742

11205720cos120587120585

(A1)


Ξ1=

1

1536(1 + 9120587

21205832)

times [1205725

119879+ 81205723

119879minus 9120587 (1 + 120587

21205832) 12057201205722

119879]

Ξ2= minus

1

36864(1 + 9120587

21205832) (1 + 25120587

21205832) 1205725

119879

(B1)

1198921(120572119879 1205720 120583)

= 12058721 +

1

1536(1 + 120587

21205832)

times [ (25 + 3312058721205832) 1205724

119879

+ 192 (1 minus 120587312058321205720120572minus1

119879) (1205722

119879minus 8120587120572

0120572minus1

119879)

minus8120587 (23 + 1512058721205832) 1205720120572119879]

(B2)


Λ1(120572119879 1205742 1205742

0)

=

12057421205723

119879(1 + 120583

21205742

0) minus 8120587

31205720(1 + 120583

21205742)

8 (1205742 minus 1205742

0)

Λ2(120572119879 1205742 1205742

0) =

12057421205723

119879(1 + 9120583

21205742

0)

24 (1205742 minus 91205742

0)

(C1)

119887 = minus (1 + 312058721205832) 1205723

119879+ 6120572119879+ 6120587312058321205720

119888 = 1205872[minus (7 + 9120587

21205832) 1205723

119879+ 60120572

119879minus 6120587 (1 minus 9120587

21205832) 1205720]

119889 = 541205874(120572119879minus 1205871205720)

(C2)


Nomenclature











Acknowledgments


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119871

1199080

119908

119875

119906(119871)

119909

119911

(a)

119909119881

119867

119872

119911

1205790

120579

119889119904

120579T119872+ d119872119881+ d119881

119867+ d119867

(b)




1198641198681198892120579119879

1198891199042+ 119875 sin 120579

119879

= 119875(1198900119886)2 cos 120579

119879(1198892120579119879

1198891199042minus tan 120579

119879(119889120579119879

119889119904)

2

)

+ 11986411986811988921205790

1198891199042

(7)


120585 =119904

119871 120583 =

1198900119886

119871

120582 = radic1198751198712

119864119868 120582 = 120583120582

(8)


12057910158401015840

119879+ 1205822 sin 120579

119879= 1205822 cos 120579

119879(12057910158401015840

119879minus tan 120579

11987912057910158402

119879) + 12057910158401015840

0 (9)



1199080 (119904) = 1198860 sin120587120585 (10)



119889119909 = 119889119904 cos 120579119879 119889119908 = 119889119904 sin 120579

119879 (11)

we have

1205790 (119904) = sinminus1 [

1198891199080 (119904)

119889119904] (12)

and correspondingly

12057910158401015840

0=

12057201205873(1205722

01205872minus 1) cos120587120585

(1 minus 1205722

01205872cos2120587120585)32

(13)

where 1205720= 1198860119871


119908 (0) = 119908 (119871) = 0 (14)




119875119908 + [119864119868 minus 119875(1198900119886)2 cos 120579

119879]119889120579119879

119889119904= 119864119868

1198891205790

119889119904 (15)


119889120579119879

119889119904= 0 at 119904 = 0 119871 (16)

or

119864119868 minus 119875(1198900119886)2 cos 120579

119879= 0 at 119904 = 0 119871 (17)


1205791015840

119879(0) = 120579

1015840

119879(1) = 0 120579

119879 (0) = 120572119879 (18)






sin 120579119879= 120579119879minus1

61205793

119879

cos 120579119879= 1 minus

1

21205792

119879

tan 120579119879= 120579119879+1

31205793

119879

(19)



12057910158401015840

0= minus12057201205873 cos120587120585 (20)


119879) one obtains

L (120579119879) = N (120579

119879) + 119891 (120585) 120585 isin [0 1] (21)


12057910158401015840


119879) =

1205742[1205793

1198796 minus 120583

2120579119879(12057911987912057910158401015840

1198792 + 120579

10158402


119891(120585) = minus1205873(1 + 120583


defined as 1205742 = 1205822(1 minus 1205822)



H (Θ 119901)


0)] + 119901 [L (Θ) minusN (Θ) minus 119891 (120585)]

= 0

(22)



Θ1015840(0) = Θ

1015840(1) = 0 Θ (0) = 120572119879 (23)


0) = 0 According


Θ (120585) = Θ0 (120585) +

infin

sum

119895=1

119901119895Θ119895 (120585) (24)


1205742= 1205742

0+

infin

sum

119895=1

1199011198951205742

119895 (25)


O (1199010) Θ

10158401015840

0+ 1205742

0Θ0= 0

Θ1015840

0(0) = Θ

1015840

0(0) = 0 Θ

0 (0) = 120572119879

(26a)

O (1199011) Θ

10158401015840

1+ 1205742

0Θ1= 1198911(Θ0 1205742

1 1205742

0)

Θ1015840

1(0) = Θ

1015840

1(1) = 0 Θ

1 (0) = 0

(26b)

O (1199012) Θ

10158401015840

2+ 1205742

0Θ2= 1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

Θ1015840

2(0) = Θ

1015840

2(1) = 0 Θ

2 (0) = 0

(26c)




Θ0 (120585) = 120572119879 cos120587120585 (27)




Θ10158401015840

1+ 1205742

0Θ1

= [minus1205742

1120572119879+1

81205872(1 + 120587

21205832) 1205723

119879minus 1205873(1 + 120587

21205832) 1205720] cos120587120585

+1

241205872(1 + 9120587

21205832) 1205723

119879cos 3120587120585

(28)


Θ1=1

192(1 + 9120587

21205832) 1205723

119879(cos120587120585 minus cos 3120587120585) (29)


1205742

1=1

8(1 + 120587

21205832) [12058721205722

119879minus 812058731205720120572minus1

119879] (30)



cos 1198991205740120585 119899 = 1 2 3 (31)







120579119879 (120585) = Θ (120585)

1003816100381610038161003816119901=1= 120572119879cos120587120585


(32)

1205742=

2

sum

119894=0

1205742

119894≜ 1198921(120572119879 1205720 120583) (33)

where Ξ119894(119894 = 1 2) and 119892

1(120572119879 1205720 120583) are defined in

Appendix B



L (120579119879) = 0 (34)




L (120579119879 1205742) = N (120579

1198790 1205742

0) + 119891 (120585 120574

2) 120585 isin [0 1] (35)


1205791198791 (120585) = Λ 1 (120572119879 120574

2 1205742

0) cos120587120585 + Λ

2(120572119879 1205742 1205742

0) cos 3120587120585

(36)

where Λ119894(1205742 1205742


Note that 1205791198791



L (120579119879 1205742) = N (120579

1198791 1205742 1205742

0) + 119891 (120585 120574

2) 120585 isin [0 1] (37)


1198791(0) = 120572



1198871205744minus 1198881205742+ 119889 = 0 (38)



2119887≜ 1198922(120572119879 1205720 120583) (39)






119875

119875119864

=119892119894(120572119879 1205720 120583)

1205872 [1 + 1205832119892119894(120572119879 1205720 120583)]

(119894 = 1 or 2) (40)

in which 119875119864= 119864119868120587





119875cr = 119875119864(1 + 12058721205832)minus1

(41)




119909

119871= int

120585

0

cos 120579119879 (120585) 119889120585

119911

119871= int

120585

0

sin 120579119879 (120585) 119889120585

(42)







120572119879

119875exact119875119864(a)

119875HPM1119875119864

(b)119875HPM2

119875119864

(c)119875SAA119875119864

(d)



120572119879

119908exact119871(a)

119908HPM1119871

(b)119908HPM2

119871(c)

119908SAA119871(d)



relative error

=

10038161003816100381610038161003816100381610038161003816


minus 1

10038161003816100381610038161003816100381610038161003816

times 100

(43)







120572119879

120583 = 0 120583 = 005 120583 = 010 120583 = 015 120583 = 020

20∘ 01097 01097 01098 01099 0110040∘ 02111 02113 02118 02126 0213760∘ 02967 02972 02987 03012 0304580∘ 03601 03612 03642 03692 03759100∘ 03965 03980 04026 04098 04190120∘ 04019 04035 04078 04136 04196

0 02 04 06 08 10

01

02

03

04

05

119911119871

119909119871



Red lines 120583 = 020

120572T = 20∘

120572T = 60∘

120572T = 100∘

120572T = 120∘



11112131415

119875119875119864


120572T (∘)




15

125

1

075

05

025

0


119875119875119864


120572T (∘)



15

125

1

075

05

025

0

119875119875119864



120572T (∘)








Appendix


1198911(Θ0 1205742

1 1205742

0)

= minus1205742

1Θ0+1

61205742

0Θ3

0minus 12058321205742

0Θ0(1

2Θ0Θ10158401015840

0+ Θ10158402

0)

minus 1205873(1 + 120583

21205742

0) 1205720cos120587120585

1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

= minus1205742

2Θ0minus 1205742

1Θ1+1

61205742

1Θ3

0+1

21205742

0Θ2

0Θ2

1

minus1

21205832[(1205742

1Θ10158401015840

0+ 1205742

0Θ10158401015840

1)Θ2

0+ 21205742

0Θ0Θ1Θ10158401015840

0]

minus 1205832[(1205742

1Θ0+ 1205742

0Θ1)Θ10158402

0+ 21205742

0Θ0Θ1015840

0Θ1015840

1]

minus 120587312058321205742

11205720cos120587120585

(A1)


Ξ1=

1

1536(1 + 9120587

21205832)

times [1205725

119879+ 81205723

119879minus 9120587 (1 + 120587

21205832) 12057201205722

119879]

Ξ2= minus

1

36864(1 + 9120587

21205832) (1 + 25120587

21205832) 1205725

119879

(B1)

1198921(120572119879 1205720 120583)

= 12058721 +

1

1536(1 + 120587

21205832)

times [ (25 + 3312058721205832) 1205724

119879

+ 192 (1 minus 120587312058321205720120572minus1

119879) (1205722

119879minus 8120587120572

0120572minus1

119879)

minus8120587 (23 + 1512058721205832) 1205720120572119879]

(B2)


Λ1(120572119879 1205742 1205742

0)

=

12057421205723

119879(1 + 120583

21205742

0) minus 8120587

31205720(1 + 120583

21205742)

8 (1205742 minus 1205742

0)

Λ2(120572119879 1205742 1205742

0) =

12057421205723

119879(1 + 9120583

21205742

0)

24 (1205742 minus 91205742

0)

(C1)

119887 = minus (1 + 312058721205832) 1205723

119879+ 6120572119879+ 6120587312058321205720

119888 = 1205872[minus (7 + 9120587

21205832) 1205723

119879+ 60120572

119879minus 6120587 (1 minus 9120587

21205832) 1205720]

119889 = 541205874(120572119879minus 1205871205720)

(C2)


Nomenclature











Acknowledgments


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












sin 120579119879= 120579119879minus1

61205793

119879

cos 120579119879= 1 minus

1

21205792

119879

tan 120579119879= 120579119879+1

31205793

119879

(19)



12057910158401015840

0= minus12057201205873 cos120587120585 (20)


119879) one obtains

L (120579119879) = N (120579

119879) + 119891 (120585) 120585 isin [0 1] (21)


12057910158401015840


119879) =

1205742[1205793

1198796 minus 120583

2120579119879(12057911987912057910158401015840

1198792 + 120579

10158402


119891(120585) = minus1205873(1 + 120583


defined as 1205742 = 1205822(1 minus 1205822)



H (Θ 119901)


0)] + 119901 [L (Θ) minusN (Θ) minus 119891 (120585)]

= 0

(22)



Θ1015840(0) = Θ

1015840(1) = 0 Θ (0) = 120572119879 (23)


0) = 0 According


Θ (120585) = Θ0 (120585) +

infin

sum

119895=1

119901119895Θ119895 (120585) (24)


1205742= 1205742

0+

infin

sum

119895=1

1199011198951205742

119895 (25)


O (1199010) Θ

10158401015840

0+ 1205742

0Θ0= 0

Θ1015840

0(0) = Θ

1015840

0(0) = 0 Θ

0 (0) = 120572119879

(26a)

O (1199011) Θ

10158401015840

1+ 1205742

0Θ1= 1198911(Θ0 1205742

1 1205742

0)

Θ1015840

1(0) = Θ

1015840

1(1) = 0 Θ

1 (0) = 0

(26b)

O (1199012) Θ

10158401015840

2+ 1205742

0Θ2= 1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

Θ1015840

2(0) = Θ

1015840

2(1) = 0 Θ

2 (0) = 0

(26c)




Θ0 (120585) = 120572119879 cos120587120585 (27)




Θ10158401015840

1+ 1205742

0Θ1

= [minus1205742

1120572119879+1

81205872(1 + 120587

21205832) 1205723

119879minus 1205873(1 + 120587

21205832) 1205720] cos120587120585

+1

241205872(1 + 9120587

21205832) 1205723

119879cos 3120587120585

(28)


Θ1=1

192(1 + 9120587

21205832) 1205723

119879(cos120587120585 minus cos 3120587120585) (29)


1205742

1=1

8(1 + 120587

21205832) [12058721205722

119879minus 812058731205720120572minus1

119879] (30)



cos 1198991205740120585 119899 = 1 2 3 (31)







120579119879 (120585) = Θ (120585)

1003816100381610038161003816119901=1= 120572119879cos120587120585


(32)

1205742=

2

sum

119894=0

1205742

119894≜ 1198921(120572119879 1205720 120583) (33)

where Ξ119894(119894 = 1 2) and 119892

1(120572119879 1205720 120583) are defined in

Appendix B



L (120579119879) = 0 (34)




L (120579119879 1205742) = N (120579

1198790 1205742

0) + 119891 (120585 120574

2) 120585 isin [0 1] (35)


1205791198791 (120585) = Λ 1 (120572119879 120574

2 1205742

0) cos120587120585 + Λ

2(120572119879 1205742 1205742

0) cos 3120587120585

(36)

where Λ119894(1205742 1205742


Note that 1205791198791



L (120579119879 1205742) = N (120579

1198791 1205742 1205742

0) + 119891 (120585 120574

2) 120585 isin [0 1] (37)


1198791(0) = 120572



1198871205744minus 1198881205742+ 119889 = 0 (38)



2119887≜ 1198922(120572119879 1205720 120583) (39)






119875

119875119864

=119892119894(120572119879 1205720 120583)

1205872 [1 + 1205832119892119894(120572119879 1205720 120583)]

(119894 = 1 or 2) (40)

in which 119875119864= 119864119868120587





119875cr = 119875119864(1 + 12058721205832)minus1

(41)




119909

119871= int

120585

0

cos 120579119879 (120585) 119889120585

119911

119871= int

120585

0

sin 120579119879 (120585) 119889120585

(42)







120572119879

119875exact119875119864(a)

119875HPM1119875119864

(b)119875HPM2

119875119864

(c)119875SAA119875119864

(d)



120572119879

119908exact119871(a)

119908HPM1119871

(b)119908HPM2

119871(c)

119908SAA119871(d)



relative error

=

10038161003816100381610038161003816100381610038161003816


minus 1

10038161003816100381610038161003816100381610038161003816

times 100

(43)







120572119879

120583 = 0 120583 = 005 120583 = 010 120583 = 015 120583 = 020

20∘ 01097 01097 01098 01099 0110040∘ 02111 02113 02118 02126 0213760∘ 02967 02972 02987 03012 0304580∘ 03601 03612 03642 03692 03759100∘ 03965 03980 04026 04098 04190120∘ 04019 04035 04078 04136 04196

0 02 04 06 08 10

01

02

03

04

05

119911119871

119909119871



Red lines 120583 = 020

120572T = 20∘

120572T = 60∘

120572T = 100∘

120572T = 120∘



11112131415

119875119875119864


120572T (∘)




15

125

1

075

05

025

0


119875119875119864


120572T (∘)



15

125

1

075

05

025

0

119875119875119864



120572T (∘)








Appendix


1198911(Θ0 1205742

1 1205742

0)

= minus1205742

1Θ0+1

61205742

0Θ3

0minus 12058321205742

0Θ0(1

2Θ0Θ10158401015840

0+ Θ10158402

0)

minus 1205873(1 + 120583

21205742

0) 1205720cos120587120585

1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

= minus1205742

2Θ0minus 1205742

1Θ1+1

61205742

1Θ3

0+1

21205742

0Θ2

0Θ2

1

minus1

21205832[(1205742

1Θ10158401015840

0+ 1205742

0Θ10158401015840

1)Θ2

0+ 21205742

0Θ0Θ1Θ10158401015840

0]

minus 1205832[(1205742

1Θ0+ 1205742

0Θ1)Θ10158402

0+ 21205742

0Θ0Θ1015840

0Θ1015840

1]

minus 120587312058321205742

11205720cos120587120585

(A1)


Ξ1=

1

1536(1 + 9120587

21205832)

times [1205725

119879+ 81205723

119879minus 9120587 (1 + 120587

21205832) 12057201205722

119879]

Ξ2= minus

1

36864(1 + 9120587

21205832) (1 + 25120587

21205832) 1205725

119879

(B1)

1198921(120572119879 1205720 120583)

= 12058721 +

1

1536(1 + 120587

21205832)

times [ (25 + 3312058721205832) 1205724

119879

+ 192 (1 minus 120587312058321205720120572minus1

119879) (1205722

119879minus 8120587120572

0120572minus1

119879)

minus8120587 (23 + 1512058721205832) 1205720120572119879]

(B2)


Λ1(120572119879 1205742 1205742

0)

=

12057421205723

119879(1 + 120583

21205742

0) minus 8120587

31205720(1 + 120583

21205742)

8 (1205742 minus 1205742

0)

Λ2(120572119879 1205742 1205742

0) =

12057421205723

119879(1 + 9120583

21205742

0)

24 (1205742 minus 91205742

0)

(C1)

119887 = minus (1 + 312058721205832) 1205723

119879+ 6120572119879+ 6120587312058321205720

119888 = 1205872[minus (7 + 9120587

21205832) 1205723

119879+ 60120572

119879minus 6120587 (1 minus 9120587

21205832) 1205720]

119889 = 541205874(120572119879minus 1205871205720)

(C2)


Nomenclature











Acknowledgments


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











cos 1198991205740120585 119899 = 1 2 3 (31)







120579119879 (120585) = Θ (120585)

1003816100381610038161003816119901=1= 120572119879cos120587120585


(32)

1205742=

2

sum

119894=0

1205742

119894≜ 1198921(120572119879 1205720 120583) (33)

where Ξ119894(119894 = 1 2) and 119892

1(120572119879 1205720 120583) are defined in

Appendix B



L (120579119879) = 0 (34)




L (120579119879 1205742) = N (120579

1198790 1205742

0) + 119891 (120585 120574

2) 120585 isin [0 1] (35)


1205791198791 (120585) = Λ 1 (120572119879 120574

2 1205742

0) cos120587120585 + Λ

2(120572119879 1205742 1205742

0) cos 3120587120585

(36)

where Λ119894(1205742 1205742


Note that 1205791198791



L (120579119879 1205742) = N (120579

1198791 1205742 1205742

0) + 119891 (120585 120574

2) 120585 isin [0 1] (37)


1198791(0) = 120572



1198871205744minus 1198881205742+ 119889 = 0 (38)



2119887≜ 1198922(120572119879 1205720 120583) (39)






119875

119875119864

=119892119894(120572119879 1205720 120583)

1205872 [1 + 1205832119892119894(120572119879 1205720 120583)]

(119894 = 1 or 2) (40)

in which 119875119864= 119864119868120587





119875cr = 119875119864(1 + 12058721205832)minus1

(41)




119909

119871= int

120585

0

cos 120579119879 (120585) 119889120585

119911

119871= int

120585

0

sin 120579119879 (120585) 119889120585

(42)







120572119879

119875exact119875119864(a)

119875HPM1119875119864

(b)119875HPM2

119875119864

(c)119875SAA119875119864

(d)



120572119879

119908exact119871(a)

119908HPM1119871

(b)119908HPM2

119871(c)

119908SAA119871(d)



relative error

=

10038161003816100381610038161003816100381610038161003816


minus 1

10038161003816100381610038161003816100381610038161003816

times 100

(43)







120572119879

120583 = 0 120583 = 005 120583 = 010 120583 = 015 120583 = 020

20∘ 01097 01097 01098 01099 0110040∘ 02111 02113 02118 02126 0213760∘ 02967 02972 02987 03012 0304580∘ 03601 03612 03642 03692 03759100∘ 03965 03980 04026 04098 04190120∘ 04019 04035 04078 04136 04196

0 02 04 06 08 10

01

02

03

04

05

119911119871

119909119871



Red lines 120583 = 020

120572T = 20∘

120572T = 60∘

120572T = 100∘

120572T = 120∘



11112131415

119875119875119864


120572T (∘)




15

125

1

075

05

025

0


119875119875119864


120572T (∘)



15

125

1

075

05

025

0

119875119875119864



120572T (∘)








Appendix


1198911(Θ0 1205742

1 1205742

0)

= minus1205742

1Θ0+1

61205742

0Θ3

0minus 12058321205742

0Θ0(1

2Θ0Θ10158401015840

0+ Θ10158402

0)

minus 1205873(1 + 120583

21205742

0) 1205720cos120587120585

1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

= minus1205742

2Θ0minus 1205742

1Θ1+1

61205742

1Θ3

0+1

21205742

0Θ2

0Θ2

1

minus1

21205832[(1205742

1Θ10158401015840

0+ 1205742

0Θ10158401015840

1)Θ2

0+ 21205742

0Θ0Θ1Θ10158401015840

0]

minus 1205832[(1205742

1Θ0+ 1205742

0Θ1)Θ10158402

0+ 21205742

0Θ0Θ1015840

0Θ1015840

1]

minus 120587312058321205742

11205720cos120587120585

(A1)


Ξ1=

1

1536(1 + 9120587

21205832)

times [1205725

119879+ 81205723

119879minus 9120587 (1 + 120587

21205832) 12057201205722

119879]

Ξ2= minus

1

36864(1 + 9120587

21205832) (1 + 25120587

21205832) 1205725

119879

(B1)

1198921(120572119879 1205720 120583)

= 12058721 +

1

1536(1 + 120587

21205832)

times [ (25 + 3312058721205832) 1205724

119879

+ 192 (1 minus 120587312058321205720120572minus1

119879) (1205722

119879minus 8120587120572

0120572minus1

119879)

minus8120587 (23 + 1512058721205832) 1205720120572119879]

(B2)


Λ1(120572119879 1205742 1205742

0)

=

12057421205723

119879(1 + 120583

21205742

0) minus 8120587

31205720(1 + 120583

21205742)

8 (1205742 minus 1205742

0)

Λ2(120572119879 1205742 1205742

0) =

12057421205723

119879(1 + 9120583

21205742

0)

24 (1205742 minus 91205742

0)

(C1)

119887 = minus (1 + 312058721205832) 1205723

119879+ 6120572119879+ 6120587312058321205720

119888 = 1205872[minus (7 + 9120587

21205832) 1205723

119879+ 60120572

119879minus 6120587 (1 minus 9120587

21205832) 1205720]

119889 = 541205874(120572119879minus 1205871205720)

(C2)


Nomenclature











Acknowledgments


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572119879

119875exact119875119864(a)

119875HPM1119875119864

(b)119875HPM2

119875119864

(c)119875SAA119875119864

(d)



120572119879

119908exact119871(a)

119908HPM1119871

(b)119908HPM2

119871(c)

119908SAA119871(d)



relative error

=

10038161003816100381610038161003816100381610038161003816


minus 1

10038161003816100381610038161003816100381610038161003816

times 100

(43)







120572119879

120583 = 0 120583 = 005 120583 = 010 120583 = 015 120583 = 020

20∘ 01097 01097 01098 01099 0110040∘ 02111 02113 02118 02126 0213760∘ 02967 02972 02987 03012 0304580∘ 03601 03612 03642 03692 03759100∘ 03965 03980 04026 04098 04190120∘ 04019 04035 04078 04136 04196

0 02 04 06 08 10

01

02

03

04

05

119911119871

119909119871



Red lines 120583 = 020

120572T = 20∘

120572T = 60∘

120572T = 100∘

120572T = 120∘



11112131415

119875119875119864


120572T (∘)




15

125

1

075

05

025

0


119875119875119864


120572T (∘)



15

125

1

075

05

025

0

119875119875119864



120572T (∘)








Appendix


1198911(Θ0 1205742

1 1205742

0)

= minus1205742

1Θ0+1

61205742

0Θ3

0minus 12058321205742

0Θ0(1

2Θ0Θ10158401015840

0+ Θ10158402

0)

minus 1205873(1 + 120583

21205742

0) 1205720cos120587120585

1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

= minus1205742

2Θ0minus 1205742

1Θ1+1

61205742

1Θ3

0+1

21205742

0Θ2

0Θ2

1

minus1

21205832[(1205742

1Θ10158401015840

0+ 1205742

0Θ10158401015840

1)Θ2

0+ 21205742

0Θ0Θ1Θ10158401015840

0]

minus 1205832[(1205742

1Θ0+ 1205742

0Θ1)Θ10158402

0+ 21205742

0Θ0Θ1015840

0Θ1015840

1]

minus 120587312058321205742

11205720cos120587120585

(A1)


Ξ1=

1

1536(1 + 9120587

21205832)

times [1205725

119879+ 81205723

119879minus 9120587 (1 + 120587

21205832) 12057201205722

119879]

Ξ2= minus

1

36864(1 + 9120587

21205832) (1 + 25120587

21205832) 1205725

119879

(B1)

1198921(120572119879 1205720 120583)

= 12058721 +

1

1536(1 + 120587

21205832)

times [ (25 + 3312058721205832) 1205724

119879

+ 192 (1 minus 120587312058321205720120572minus1

119879) (1205722

119879minus 8120587120572

0120572minus1

119879)

minus8120587 (23 + 1512058721205832) 1205720120572119879]

(B2)


Λ1(120572119879 1205742 1205742

0)

=

12057421205723

119879(1 + 120583

21205742

0) minus 8120587

31205720(1 + 120583

21205742)

8 (1205742 minus 1205742

0)

Λ2(120572119879 1205742 1205742

0) =

12057421205723

119879(1 + 9120583

21205742

0)

24 (1205742 minus 91205742

0)

(C1)

119887 = minus (1 + 312058721205832) 1205723

119879+ 6120572119879+ 6120587312058321205720

119888 = 1205872[minus (7 + 9120587

21205832) 1205723

119879+ 60120572

119879minus 6120587 (1 minus 9120587

21205832) 1205720]

119889 = 541205874(120572119879minus 1205871205720)

(C2)


Nomenclature











Acknowledgments


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572119879

120583 = 0 120583 = 005 120583 = 010 120583 = 015 120583 = 020

20∘ 01097 01097 01098 01099 0110040∘ 02111 02113 02118 02126 0213760∘ 02967 02972 02987 03012 0304580∘ 03601 03612 03642 03692 03759100∘ 03965 03980 04026 04098 04190120∘ 04019 04035 04078 04136 04196

0 02 04 06 08 10

01

02

03

04

05

119911119871

119909119871



Red lines 120583 = 020

120572T = 20∘

120572T = 60∘

120572T = 100∘

120572T = 120∘



11112131415

119875119875119864


120572T (∘)




15

125

1

075

05

025

0


119875119875119864


120572T (∘)



15

125

1

075

05

025

0

119875119875119864



120572T (∘)








Appendix


1198911(Θ0 1205742

1 1205742

0)

= minus1205742

1Θ0+1

61205742

0Θ3

0minus 12058321205742

0Θ0(1

2Θ0Θ10158401015840

0+ Θ10158402

0)

minus 1205873(1 + 120583

21205742

0) 1205720cos120587120585

1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

= minus1205742

2Θ0minus 1205742

1Θ1+1

61205742

1Θ3

0+1

21205742

0Θ2

0Θ2

1

minus1

21205832[(1205742

1Θ10158401015840

0+ 1205742

0Θ10158401015840

1)Θ2

0+ 21205742

0Θ0Θ1Θ10158401015840

0]

minus 1205832[(1205742

1Θ0+ 1205742

0Θ1)Θ10158402

0+ 21205742

0Θ0Θ1015840

0Θ1015840

1]

minus 120587312058321205742

11205720cos120587120585

(A1)


Ξ1=

1

1536(1 + 9120587

21205832)

times [1205725

119879+ 81205723

119879minus 9120587 (1 + 120587

21205832) 12057201205722

119879]

Ξ2= minus

1

36864(1 + 9120587

21205832) (1 + 25120587

21205832) 1205725

119879

(B1)

1198921(120572119879 1205720 120583)

= 12058721 +

1

1536(1 + 120587

21205832)

times [ (25 + 3312058721205832) 1205724

119879

+ 192 (1 minus 120587312058321205720120572minus1

119879) (1205722

119879minus 8120587120572

0120572minus1

119879)

minus8120587 (23 + 1512058721205832) 1205720120572119879]

(B2)


Λ1(120572119879 1205742 1205742

0)

=

12057421205723

119879(1 + 120583

21205742

0) minus 8120587

31205720(1 + 120583

21205742)

8 (1205742 minus 1205742

0)

Λ2(120572119879 1205742 1205742

0) =

12057421205723

119879(1 + 9120583

21205742

0)

24 (1205742 minus 91205742

0)

(C1)

119887 = minus (1 + 312058721205832) 1205723

119879+ 6120572119879+ 6120587312058321205720

119888 = 1205872[minus (7 + 9120587

21205832) 1205723

119879+ 60120572

119879minus 6120587 (1 minus 9120587

21205832) 1205720]

119889 = 541205874(120572119879minus 1205871205720)

(C2)


Nomenclature











Acknowledgments


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Appendix


1198911(Θ0 1205742

1 1205742

0)

= minus1205742

1Θ0+1

61205742

0Θ3

0minus 12058321205742

0Θ0(1

2Θ0Θ10158401015840

0+ Θ10158402

0)

minus 1205873(1 + 120583

21205742

0) 1205720cos120587120585

1198912(Θ1 Θ0 1205742

2 1205742

1 1205742

0)

= minus1205742

2Θ0minus 1205742

1Θ1+1

61205742

1Θ3

0+1

21205742

0Θ2

0Θ2

1

minus1

21205832[(1205742

1Θ10158401015840

0+ 1205742

0Θ10158401015840

1)Θ2

0+ 21205742

0Θ0Θ1Θ10158401015840

0]

minus 1205832[(1205742

1Θ0+ 1205742

0Θ1)Θ10158402

0+ 21205742

0Θ0Θ1015840

0Θ1015840

1]

minus 120587312058321205742

11205720cos120587120585

(A1)


Ξ1=

1

1536(1 + 9120587

21205832)

times [1205725

119879+ 81205723

119879minus 9120587 (1 + 120587

21205832) 12057201205722

119879]

Ξ2= minus

1

36864(1 + 9120587

21205832) (1 + 25120587

21205832) 1205725

119879

(B1)

1198921(120572119879 1205720 120583)

= 12058721 +

1

1536(1 + 120587

21205832)

times [ (25 + 3312058721205832) 1205724

119879

+ 192 (1 minus 120587312058321205720120572minus1

119879) (1205722

119879minus 8120587120572

0120572minus1

119879)

minus8120587 (23 + 1512058721205832) 1205720120572119879]

(B2)


Λ1(120572119879 1205742 1205742

0)

=

12057421205723

119879(1 + 120583

21205742

0) minus 8120587

31205720(1 + 120583

21205742)

8 (1205742 minus 1205742

0)

Λ2(120572119879 1205742 1205742

0) =

12057421205723

119879(1 + 9120583

21205742

0)

24 (1205742 minus 91205742

0)

(C1)

119887 = minus (1 + 312058721205832) 1205723

119879+ 6120572119879+ 6120587312058321205720

119888 = 1205872[minus (7 + 9120587

21205832) 1205723

119879+ 60120572

119879minus 6120587 (1 minus 9120587

21205832) 1205720]

119889 = 541205874(120572119879minus 1205871205720)

(C2)


Nomenclature











Acknowledgments


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Nomenclature











Acknowledgments


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article stability analysis of nonlocal elastic...

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