research article on the generalization of the timoshenko...

Research ArticleOn the Generalization of the Timoshenko BeamModel Based on the Micropolar Linear Theory Static Case

Andrea Nobili

Dipartimento di Ingegneria ldquoEnzo Ferrarirdquo via Vignolese 905 41125 Modena Italy

Correspondence should be addressed to Andrea Nobili andreanobiliunimoreit

Received 3 December 2014 Revised 25 February 2015 Accepted 26 February 2015

Academic Editor Efstratios Tzirtzilakis

Copyright copy 2015 Andrea NobiliThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Three generalizations of the Timoshenko beammodel according to the linear theory ofmicropolar elasticity or its special cases thatis the couple stress theory or the modified couple stress theory recently developed in the literature are investigated and comparedThe analysis is carried out in a variational setting making use of Hamiltonrsquos principle It is shown that both the Timoshenko and the(possibly modified) couple stress models are based on a microstructural kinematics which is governed by kinosthenic (ignorable)terms in the Lagrangian Despite their difference all models bring in a beam-plane theory only one microstructural materialparameter Besides the micropolar model formally reduces to the couple stress model upon introducing the proper constraint onthe microstructure kinematics although the material parameter is generally different Line loading on the microstructure resultsin a nonconservative force potential Finally the Hamiltonian form of the micropolar beam model is derived and the canonicalequations are presented along with their general solution The latter exhibits a general oscillatory pattern for the microstructurerotation and stress whose behavior matches the numerical findings

1 Introduction

One truly remarkable feature of the theory of elasticity is thepossibility of dealing with the vast microscopic complexityand diversity of real materials in a unified and ldquoaveragedrdquofashion Such feature is best illustrated by the broad classof linear isotropic materials which can be described from amechanical standpoint by means of just two material param-eters Of course this far standing attitude cannot be expectedto work just as well when very small devices whose size is inthe order of the microstructure length scale are consideredTo overcome such short-coming and to effectively model thecurrent trend of micro- and nanoelectromechanical systemsMEMS and NEMS several nonclassical continuum theorieshave been developed Among these we will mention themicropolar continuum theory [1 2] hereinafter referred toas the Eringen-Nowacki (E-N) micropolar theory [3] whichadds at each material point a microstructure description interms of one rotation vector In general the adoption of anonclassical continuum theory takes a heavy toll in that eitherextra boundary conditions are demanded or a considerable

number of material parameters need to be somehow deter-mined For the case of the E-N micropolar theory threemicrostructural material parameters are required Conse-quently one very desirable feature of a workable nonclassicalcontinuum theory is the ability to capture the mechanicalbearing of the microstructure at the macroscale at the leastpossible cost in terms of experimental effort for the deter-mination of the material parameters The desire to strike thisdifficult balance has spawn a number of specialized versionsof the micropolar theory such as the Koiter-Mindlin (K-M)couple stress theory [4 5] and recently the reduced couplestress theory [6] wherein respectively two and one materialparameters are introduced When such diverse models areconfronted with approximate theories such as the beamrsquosgenerally unexpected results are in order [7] The modifiedcouple stress theory has been applied to develop an Euler-Bernoulli model in [8] However attention in the literature ismostly focused on Timoshenko-like models in light of theiradvantages Indeed the Timoshenko model incorporates theshear contribution to the deformation which ismost relevantin the thick-beam high-frequency regime proper of MEMS

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 914357 8 pageshttpdxdoiorg1011552015914357

2 Mathematical Problems in Engineering

qm

z

x

Ψ

(a)

T

M

Vw1

(b)

Figure 1 Kinematics of a Timoshenko beam (a) and positivity conventions (b)

Besides the Timoshenko model is variationally more aptto a FE approximation (no locking effect) Plates and rodmodels are mathematically justified through a proper scalingin a small parameter from the three-dimensional theoryof linear micropolar elasticity in [9 10] A Timoshenkobeam model based on the reduced couple stress theory ispresented in [11] where static bending and free vibration ofa simply supported beam are studied In [12] a First OrderShear Deformation Beam Theory (FSDBT) is developed formicropolar elastic beams and analytical results for the staticbending of a cantilever beam as well as for the dispersionrelation for longitudinal and flexural waves are given Couplestress Timoshenko beams have been investigated in [13]where a closed-form solution valid in the static case andfor homogeneous materials is obtained and then applied toanalyze a cantilever beam In a such a variety the questionof whether such models reconcile and under what conditionsseldom appears and when it does the answer rests in the neg-ativeThis paper attempts to address precisely this matter andto gather all models under a single variational frameworkThe issue of what assumptions really matter for the reductionin the number ofmaterial parameters is also investigatedThepaper is thus organized Section 2 revises some feature of theTimoshenko beam model Section 3 introduces the couplestress Timoshenko model and Section 4 introduces thebroader linear micropolar model The Hamiltonian form ofthe latter and its general solution are presented in Section 41Finally conclusions are drawn in Section 5

2 Variational Features ofthe Timoshenko Beam

Let us consider a prismatic body (beam) of cross-sectionalarea119860 and length 119871 whose central axis (the sectionsrsquo centroidline) rests along the 119909-axis in the reference configurationFigure 1 Traditionally a Timoshenko beam [14 Section 217]is characterized by the following Lagrangian density

L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

minus 119902119910 minus 119898120595 (1)

where119864119868 is the beam flexural rigidity119866 is the shearmodulus119910 is the transverse displacement 120595 is the cross-sectionrotation (positive clockwise) and 119902 and 119898 are the transverse

and the torque line load density respectively (This notationis Timoshenkorsquos and we abide by it for historical reasonsThenotation 119911 for the transverse displacement would better fitwith the reference system of Figure 1 which is commonlyadopted in the recent literature However confusion mayarise with the 119911 coordinate along the cross-section) Sincethis is a beam theory all field quantities depend on the singleindependent variable 119909 that is 119864119868 = 119864119868(119909) 119902 = 119902(119909)and so forth Here we use a subscript number to denotedifferentiation with respect to 119909 that is we use 119910

1in the

place of 119889119910119889119909 Besides we will restrict ourselves to the statictheory For the sake of notational compactness the shearfactor 119896

119904 usually associated with the Timoshenko model is

here omitted although it can be easily introduced whenevera factor 119860119866 appears The action integral is

A = int119871

0

L119889119909 (2)

and according to Hamiltonrsquos principle seeking for deforma-tion makes the action integral stationary Since here the axialcoordinate 119909 takes up the role which is usually played by time119905 in discrete mechanics we will refer to 119910

1as the velocity and

to 1199102as the accelerationThe Euler-Lagrange (E-L) equations

for the Lagrangian (1) are

(1198641198681205951)1015840

+ 119860119866 (1199101minus 120595) + 119898 = 0

[119860119866 (1199101minus 120595)]1015840

+ 119902 = 0

(3)

where prime denotes differentiation with respect to 119909The Lagrangian (1) is noteworthy because it contains

the gyroscopic term 1205951199101 that is a term which is linear in

the velocity It is well known that a gyroscopic term arisesfrom the elimination of kinosthenic (or ignorable) variables[15 Chapter V Section 4] Indeed we easily show that theLagrangian (1) is obtained from the enlarged Lagrangian

L =1

21198641198681205952

1+1

21198601198661199082

1minus 119902119910 minus 119898120595 minus 120583 (119910

1minus 1199081minus 120595) (4)

Here 1199081is the slope that arises from shear deformation

(see Figure 1(b)) on account of the shearing force 119881 =

1198601198661199081 On the other hand the total slope 119910

1is partly due to

Mathematical Problems in Engineering 3

the rigid rotation of the cross-section120595 and partly due to theshear deformation 119908

1 The last term of (4) stems from such

kinematical connection and its dealing through the Lagrangemultiplier method Nonetheless the variational principles (1)and (4) are not entirely equivalent Indeed the E-L equationsof (4) are

(1198641198681205951)1015840

minus 120583 + 119898 = 0 (5a)

1205831015840minus 119902 = 0 (5b)

(1198601198661199081+ 120583)1015840

= 0 (5c)

together with the constraint

1199101= 1199081+ 120595 (6)

Comparing such equations with the rotational and verticalequilibrium equations respectively

1198721015840minus 119879 + 119898 = 0

1198791015840minus 119902 = 0

(7)

shows that 120583 = 119879 is the shearing force acting on a cross-section and participates with the bending moment 119872 =

1198641198681205951to rotational equilibrium It is emphasized that only

the shearing force 119881 causes shear deformation 120574 = 119881(119860119866)while the shearing force 119879 is the traditional shearing forceassociated with a Euler-Bernoulli model that is it causes nodirect deformation So it can be seen that this model is moregeneral than the traditional Timoshenko beam for in it ashearing force 119881 exists which causes shear deformation andyet it is associated with no cross-section rotation that is thebending moment required to warrant rotational equilibriumappears at no energy cost In such a model the pure sheardeformation with no cross-section rotation shown at thebottom right corner of Figure 1 may exist on its own rightOnly through the boundary conditions (BCs) the twomodelsmay be entirely reconciled Indeed as it is the case withkinosthenic variables (5c) may be immediately integratedgiving

1198601198661199081+ 120583 = 119881 + 119879 = 119881 (8)

where 119881 is a constant and it equals the boundary shearingforce conjugated with 120575119908 If 119881 = 0 then 119879 = minus119881 andthe shearing force 119879 determines both shear deformation andcross-section rotation at the same time that is a pure sheardeformation is impossible This is the Timoshenko modelwhere only two natural BCs appear namely

(1198641198681205951) 120575120595 + 119860119866 (119910

1minus 120595) 120575119910 (9)

However if we were to consider 119881 different form zero wewould allow for a distinction between the shearing forces119879 and 119881 Indeed in this general case we would have threenatural boundary conditions namely

(1198641198681205951) 120575120595 minus 120583120575119910 + 119881120575119908 (10)

This difference in the BCs is a major issue when dealing withfree boundary problems [16 17]

3 The Couple Stress Timoshenko Beam

Many generalizations of the Timoshenko beam have beenrecently proposed in the literature which are based on themicropolar linear continuum theory of Eringen-Nowacki (E-N) [1] or its special cases the couple stress theory of Koiter-Mindlin [5] and the reduced couple stress theory [6] in anattempt to incorporate a scale effect in the theory In theE-N model a microstructure exists whose single degree offreedom with respect to the macrostructure is the rotation(a similar situation occurs in electro- or magneto-elasticmaterials where a vectorial microstructure is considered[18ndash20]) described by the microstructure pseudorotationvector 120579 (pseudo because it reverses sign upon passing from aright-handed to a left-handed reference system) Micro- andmacrostructure share the same macromotion described bythe displacement field u In the special case of the Koiter-Mindlin (K-M) couple stress theory the microstructurerotation is no longer an independent degree of freedom butrather it is bound to the macrostructure deformation that isit is a latent microstructure Consequently [5 Eq (217)]

120579 =1

2curlu (11)

where componentwise (curlu)119894= 120598119894119895119896120597119906119895120597119909119896and 120598

119894119895119896is

the permutation symbol or alternator Since we are hereconsidering a plane beam theory every motion takes placein the (119909 119911)-plane and we let

u = [119906119909 0 119906119911]

120579 = [0 120593 0]

(12)

being

119906119909= 119906119909(119909 119911)

119906119911= 119910 (119909)

120593 = 120593 (119909)

(13)

Note that here 120593 is positive when being counterclockwise(following the right hand screw rule about the 119910-axis)as opposed to 120595 which is positive when being clockwise(Figure 1) In a Timoshenko-like theory the cross-sectionoriginally at coordinate 119909 remains plane after deformationwhence

119906119909= 1199060minus 119911120595 (14)

which is sometimes referred to as a Timoshenkorsquos FirstOrder Shear Deformation Beam Theory (FSDBT) [12 21]More general theories can be constructed by considering thegeneral power series expansion along 119911 [21]

119906119909(119909 119911) =

119873

sum

119894=0

119911119894120595(119894)(119909) (15)

of which (14) corresponds to the case 119873 = 1 For simplicitywe will neglect the axial extension that is 119906

0= 0 given that


it turns out to be decoupled from the transverse deformationAgain this is a beam theory and all field variables120595 and 119910 arefunctions of 119909 Letting the strain energy for the beam occupythe three-dimensional prismatic regionΩ [5 Eq (225)]

119880 =1

2intΩ

(120590 sdot 120598 +mlowast sdot 120581) 119889V (16)

where 120590 is the Cauchy stress tensor 120598 = Sym[grad u] isthe linear strain tensor mlowast is the couple stress tensor and120581 is the microstrain tensor (also termed torsion-flexure orwryness) (A concise account of the E-N model and othermicropolar theories can be found in the recent monograph[3] As far as the notation goes the asymmetric strain tensor120574 and the microstrain tensor 120581 are the transpose of our e and120581 respectively given that div and grad are also defined in aldquotransposedrdquo way) Consider

120581 = grad 120579 (17)

The symmetric part of the wryness tensor is the curvaturetensor

120594 = Sym [grad 120579] = 12(grad 120579 + grad119879120579) (18)

Here the superscript 119879 denotes transposition and compo-nentwise (grad 120579)

119894119895= 120597120579119894120597119909119895 Besides a dot denotes scalar

product namely 120590 sdot120598 = 120590119894119895120598119894119895The usual constitutive equation

for linear elastic isotropic homogeneous material is assumedas

120590 = 120582 (tr 120598) 1 + 2119866120598 (19)

where 1 is the identity tensor and 120582 119866 are Lame constants Inlight of (11)ndash(14) it is (cf [11 Eq (10)])

120598119909119909= minus119911120595

1

120598119909119911= 120598119911119909=1

2(1199101minus 120595)

(20)

and 120598119910119910= 120598119911119911= 120598119909119910= 120598119910119911= 0 Again (11)ndash(14) give

the connection between the microstructure rotation vector119910-component and the cross-section macrorotation [11 Eq(11)]

120593 = minus1

2(120595 + 119910

1) (21)

Consequently the only nonzero component of the wrynesstensor is

120581119910119909= 1205931 (22)

while the only nonzero components of the curvature tensorare

120594119909119910= 120594119910119909=1

21205931 (23)

Letting the couple stress tensor mlowast be conjugated to thewryness tensor 120581 through the constitutive assumption (2) of[13]

mlowast = 120572 (div 120579) 1 + 120574lowast grad 120579 + 120573lowastgrad119879120579 (24)

wherein 120572 120573lowast and 120574lowast are three material constants Herediv 120579 = tr(grad 120579) or componentwise div 120579 = 120597120579

119894120597119909119894

and summation over repeated indexes is implied In light ofthe constraint (11) it is div 120579 = 0 We then see that in thecouple stress theory the material constants pertaining to themicrostructure are reduced to two Furthermore by the sameconstraint the spherical part of 120581 is zero (ie its first invariantvanishes tr 120581 = 0) whence the spherical part of the couplestress tensor rests undetermined [5] Equations (22) and (24)give

119898lowast

119909119910= 120573lowast1205931 (25)

In the case of the modified couple stress theory only thesymmetric part ofmlowast is considered thus further reducing thenumber of material constants to just one [11] that is

m = Sym [mlowast] = 2120573120594 (26)

where 2120573 = 120574lowast + 120573lowast Equations (23) and (26) yield

119898119909119910= 2120573120594

119909119910= 1205731205931 (27)

and regardless of whether the couple stress or the modifiedcouple stress theory is adopted only one constitutive parame-ter appears for the 119909119910-couple stress component respectively120573lowast or 120573 When the constitutive assumption (26) is compared

with (8) of [11] namely

m = 21198972119866120594 (28)

it is clearly seen that the single material parameter 120573 orig-inating from the couple stress theory is taken proportionalto the shear module 119866 through the material characteristiclength 119897 Besides it is noted that the bending moment 119898

119910119909

appears owing to the microstructure which requires a spe-cific constraint to dispensewith the out-of-plane deformationthis would cause

It is now possible to write the Lagrangian density whoseE-L equations are (17) and (18) of [13] namely

L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

8120573119860 (120595

1+ 1199102)2

minus 119902119910 minus 119898120595

(29)

This Lagrangian density is obtained integrating (16) alongthe cross-section lying in the (119910 119911) plane having introduced(19)ndash(26) together with the restricted kinematics (11) (12)[11] The result accounts for the first three terms at RHS of(29) whereas the last two terms convey the potential of theexternal loads It is observed that a quadratic term in theacceleration 119910

2appears in the Lagrangian (29) that is we

have an acceleration dependent Lagrangian Furthermorean extra gyroscopic-like term exists which is linear in theacceleration namely 120595

11199102 The E-L equations are

[1198641198681205951+1

4120573119860 (120595

1+ 1199102)]

1015840

+ 119860119866 (1199101minus 120595) + 119898 = 0 (30a)

[1

4120573119860 (120595

1+ 1199102)]

1015840

minus 119860119866 (1199101minus 120595)

1015840

minus 119902 = 0 (30b)


=

T

Mm

c

dx

dxMm

Tm

(a)

(b)

Figure 2 Actions (a) on the macro- and (b) on the microstructure

which are the self-adjoint form (ie the generalization toinhomogeneous materials) of the static version of (17) and(18) in [13] (It needs to be considered that 119908 and 120596 there arethe counterparts of our minus119910 and minus119902 given that the 119911-axis isoriented upwards that is 119908 997891rarr minus119910 and 120596 997891rarr minus119902 Besides120595 997891rarr minus120595) Equations (30a) and (30b) may be rewritten in theform of an equilibrium equation set (cf (7))

(119872 minus119872119898)1015840

+ 119881 + 119898 = 0

(119879119898minus 119881)1015840

minus 119902 = 0

(31)

wherein 119872119898= minus(14)120573119860(120595

1+ 1199102) is the bending moment

brought in by the microstructure (cf (23) of [11]) whichconveys the shearing force 119879

119898= minus119872

1015840

119898 It is observed that the

microstructure bending moment 119872119898 and shearing force

119879119898 are positive according to the convention of Figure 2Wewill now showhow fromamechanical standpoint the

couple stress theory appears in a new enlarged Lagrangianwherein the microstructure rotation is given the status ofdependent variable The Lagrangian (29) may be rewritten as

L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

21205731198601205932

1minus 119902119910 minus 119898120595

minus 119888120593 + 120582 (2120593 + 120595 + 1199101)

(32)

having introduced the Lagrangian multiplier 120582 and thedistributed torque 119888 acting upon the microstructure (positivewhen counterclockwise cf Figure 2) The third term in (32)may be reconciled with the couple stress theory in light of(27) The Lagrangian (32) gives the following E-L equations

(1198641198681205951)1015840

+ 119860119866 (1199101minus 120595) minus 120582 + 119898 = 0 (33a)

minus [119860119866 (1199101minus 120595) + 120582]

1015840

minus 119902 = 0 (33b)

(1

21205731198601205931)

1015840

minus 120582 +1

2119888 = 0 (33c)

together with the constraint (21) Recalling the general formof the equilibrium equation (7) (33c) may be interpreted asgiving the rotational equilibrium of themicrostructure whichis acted upon by the bending moment119872

119898= (12)120573119860120593

1and

the distributed torque 1198882 minus 120582 while the shearing force 119879119898=

0 Equation (33a) enforces the rotational equilibrium of themacrostructure

1198721015840+ 119881 + 119898 minus 120582 = 0 (34)

which shows that the distributed torque 120582 is exchangedwith the microstructure according to the action-reactionprinciple Finally (33b) is the vertical equilibrium for themacrostructure which is acted upon by the shearing force119881 =119860119866(119910

1minus 120595) and the shearing force 119879 = minus120582 This mechanical

interpretation of the equations where the distributed torque120582 appears also as a shearing force derives from the formof thelast term in the Lagrangian (32) wherein 120582 is conjugated tothe rotation 120593 like 119888 and to the shear-like term 119910

1+120595 like119881

(It is observed that the Lagrangemultiplier 120582may be given yetanothermechanical interpretation the distributed torque 120582 ismechanically equivalent to a line load distribution 119902

119898= 1205821015840 in

amanner similar to the dealing with that boundary conditionfor the twisting moment in the Kirchhoff-Love plate theoryHowever unlike the twisting moment in the Kirchhoff-Loveplate theory here 120582 appears in the equations in two forms ieboth as the line torque 120582 and as the line load 1205821015840)

We now consider the special case 119888 = 0 and modify theLagrangian (32) so that 120593

1is kinosthenic

L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

21205731198601205932

1minus 119902119910 minus 119898120595

+ 120583 (21205931+ 1205951+ 1199102)

(35)

Note that this step reintroduces an acceleration term and that4120583 is now conjugated with the curvature 120593

12 just like the

microstructure bending moment 119872119898 As a result this new

Lagrangian affords a simple mechanical interpretation Thenew E-L equations are

(1198641198681205951+ 120583)1015840

+ 119860119866 (1199101minus 120595) + 119898 = 0 (36a)

12058310158401015840minus [119860119866 (119910

1minus 120595)]1015840

minus 119902 = 0 (36b)

(1

21205731198601205931+ 120583)

1015840

= 0 (36c)

plus the derivative of the constraint (21) which determines 120593as a function of 119910

1+ 120595 up to a constant (This indeterminacy

is the reason why we need consider the case 119888 = 0) In thisnew variational setting 120583 is the exchange bending momentbetween the macro- and the microstructure the latter nowreceiving no exchange distributed torque The exchangebending moment participates in the vertical equilibriumequation (36b) in the formof a shearing force119879 = 1205831015840 Since120593

1

is nowkinosthenic it can be eliminated giving the Lagrangian(29) in a similar manner as with the Timoshenko modelIndeed integrating (36c) gives

119872119898+ 120583 = 119872

119898 (37)

where 119872119898is the BC imposing the microstructure bending

moment When119872119898= 0 we have 120583 = minus119872

119898= (14)120573119860(120595

1+

1199102) and (36a) and (36b) turn into (30a) and (30b) respec-

tively

4 The E-N Micropolar Timoshenko Beam

When the E-N micropolar linear theory is employedthe microstructure rotation 120579 is no longer bound to themacrostructure rotation and in particular the constraint (21)


drops out As a consequence the constitutive equation (24)may be applied in full form

mlowast = 120572 (div 120579) 1 + 120574lowast120581 + 120573lowast120581119879 (38)

which shows that three constitutive parameters come in fromthe microstructure Furthermore the micropolar theoryentails that the stress tensor 120590lowast is no longer symmetric andits skew-symmetric part Skw[120590lowast] is energy conjugated withthemacrostructure rotation that is it does provide an energycontribution Letting the linear micropolar strain tensor

e = gradu +Ω

Ω119894119895= 120598119894119895119896120579119896

(39)

where Ω is the skew-symmetric tensor associated with therotation vector 120579 that is the latter is the axial vector of theformer The constitutive equation now needs to specify thefull stress tensor namely

120590lowast= 120582 (tr e) 1 + 2119866Sym [e] + 120578Skw [e] (40)

Obviously 120582 and 119866 are the usual Lame constants whichspecify the symmetric part of the stress tensor while 120578determines the skew symmetric partThe Lagrangian densitycorresponding to the E-L equations (35)ndash(37) of [12] is

L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

2120573lowast1198601205932

1

minus1

4120578119860 (119910

1+ 120595)2

minus 119902119910 minus 119898120595 minus 119888120593

minus 119860120578120593 (120593 + 1199101+ 120595)

(41)

This Lagrangian has been obtained integrating along thebeam cross-section the strain energy

119880lowast=1

2intΩ

(120590lowastsdot e +mlowast sdot 120581) 119889V (42)

and accounting for the external load potential Introducingthe constraint (21) and assuming the special case 119888 = 0 thefourth term and the last term cancel each other andwe are leftwith the Lagrangian density (35) Indeed this result may be atthe root of the equivalence observed in [8] for the deflectionexpression of a cantilever beam within the modified couplestress and the micropolar theory However attention shouldbe drawn to the fact that the material constant 120573lowast appearsinstead of 120573 = (120573lowast + 120574lowast)2 Besides the last of (12) and thebeam theory assumptions (13) give

div 120579 = 0 (43)

whence the material parameter 120572 never appears Indeed inthis plane deformation only the microstructural constitutiveparameter 120573lowast plays a role In the general case 119888 = 0 theexternal load potential appears as

1

2119888 (1199101+ 120595) (44)

This loading potential provides us with an example of amonogenic yet nonconservative loading [15 Chapter I Sec-tion 7] just like the case of a follower force It is then seen thatthe E-N micropolar beam model and the couple stress beammodel formally reconcile although they actually coincideonly in the very special case when 119888 equiv 0 and 120574lowast equiv 120573lowastFurthermore the case 120578 = 0 is especially interesting becausethen the E-N model and the couple stress model formallycoincide apart from the last term of (35)The latter disappearswhen neither exchange distributed torque 120583 acts nor theconstraint (21) holds Obviously in the case 120578 equiv 0 the macro-and the microstructure equilibrium equations decouple andthe former reduce to the Timoshenko model Clearly thefurther assumption 119860119866 rarr infin lends the Euler-Bernoullimodel

41 Hamiltonian Form and General Solution For the E-Nmodel we now derive the Hamiltonian form and the corre-sponding canonical equations Following [16] let the gener-alized momenta

1199011= 119860119866 (119910

1minus 120595) minus

1

2120578119860 (119910

1+ 120595) minus 119860120578120593

1199012= 119864119868120595

1

1199013= 120573lowast1198601205931

(45)

be the shearing force themacrostructure andmicrostructurebending moment respectively The Hamiltonian function is

H =1199012

1

119860 (2119866 minus 120578)+1199012

2

2119864119868+1199012

3

2119860120573lowast

+2120578120593 + (2119866 + 120578)120595

2119866 minus 1205781199011+2119860120578119866 (120595 + 120593)

2

2119866 minus 120578+ 119902119910

+ 119898120595 + 119888120593

(46)

and hence the canonical equations read (a first order systemis given in (39)ndash(41) of [12])

1199101015840minus 2

1199011

119860 (2119866 minus 120578)minus2120578120593 + (2119866 + 120578)120595

2119866 minus 120578= 0 (47a)

1205951015840minus1199012

119864119868= 0 (47b)

1205931015840minus1199013

119860120573lowast= 0 (47c)

1199011015840

1+ 119902 = 0 (47d)

1199011015840

2+2119866 + 120578

2119866 minus 1205781199011+ 4120578119860119866

2119866 minus 120578(120595 + 120593) + 119898 = 0 (47e)

1199011015840

3+2120578

2119866 minus 1205781199011+ 4120578119860119866

2119866 minus 120578(120595 + 120593) + 119888 = 0 (47f)


In this system 119910 never appears whence (47a) may be solvedindependently Indeed (47a) and (47d) give 119910 and 119901

1by a

quadrature respectively In particular

1199011(119909) = 119875

1(119909) = 119901

1+ int

119909

0

119902 (120591) 119889120591 (48)

and 1199011is given by the BC 119901

1(0) = 119901

1 Furthermore taking

the difference of (47e) with (47f) it is

(1199013minus 1199012)1015840

= 119898 minus 119888 +2119866

2119866 minus 1205781198751 (49)

where the RHS is known Let

119875 (119909) = int

119909

0

(119898 minus 119888 +2119866

2119866 minus 1205781198751)119889119909 + 119888

0 (50)

where 1198880= 1199013(0) minus 119901

2(0) Then the remaining equations

(47b) (47c) and (47e) read

1205951015840minus1199012

119864119868= 0 (51a)

1205931015840minus1199012

119860120573lowast=119875 (119909)

119860120573lowast (51b)

1199011015840

2+ 4120578119860119866

2119866 minus 120578(120595 + 120593) = minus119898 minus

2119866 + 120578

2119866 minus 1205781198751 (51c)

Such linear system can be easily reduced to a single secondorder linear inhomogeneous ODE in 119901

2 namely

11990110158401015840

2+ 412058521199012= minus119898

1015840minus2119866 + 120578

2119866 minus 120578119902

minus 4120578119866

(2119866 minus 120578) 120573lowast(119898 minus 119888 +

2119866

2119866 minus 1205781198751)

(52)

where it is let that

120585 = radic120578119866 (119864119868 + 119860120573

lowast)

119864119868120573lowast (2119866 minus 120578) (53)

The coefficient (53) corresponds to (44) of [12] although afactor radic2 seems to be missing Besides the limit of 120585 as120578 rarr infin appears in the first of (24) of [13] which deals withthe couple stress situation Indeed it is observed that (23)of [13] which governs the behavior of the shearing force isequivalent to (52) although aminus sign in front of 120585 appearsSuch minus sign determines an exponential-type solutionwhich is clearly unphysical in the case of an infinite beamThe ODE (52) is the celebrated linear pendulum equationwhich gives rise to oscillatory solutions The finding of anoscillatory behavior for the couple stress and microrotationin the cantilever beam numerically investigated in is thenmotivated [12] In the special case 120578 = 0 (52) reduces tothe shearing force equilibrium for a Timoshenko beam Once(52) is solved120595 and 120593may be found by a quadrature through(51a) and (51b) respectively

5 Conclusions

In this paper three Timoshenko-like beam models availablein the literature have been considered the first model isbased on the micropolar linear continuum theory of Eringenand Nowacki (E-N) the second is based on the couple stresstheory of Koiter and Mindlin (K-M) and the third modelis based on the modified couple stress theory proposedin [6] A comparison of such models has been developedtogether with a confrontation with the original Timoshenkomodel The analysis is carried out in a variational settingmaking use of Hamiltonrsquos principle It has been shown thatboth the Timoshenko and the couple stress model are basedon a microstructural kinematics which disappears owingto it being governed by kinosthenic (ignorable) terms inthe Lagrangian Both the micropolar and the couple stressmodels within a restricted beam-plane kinematics requirea single microstructural material parameter regardless ofwhether the modified couple stress theory is adopted Fur-thermore it has been shown that the micropolar modelformally reduces to the couple stressmodel upon introducingthe proper constraint on the microstructure kinematicsalthough the material parameter is generally different (cf[13]) Line loading on the microstructure results in a non-conservative force potential Finally the Hamiltonian form ofthe micropolar beam model has been derived together withthe corresponding canonical equations The general solutionof the micropolar model has been presented which shows ageneral oscillatory behavior for the microstructure rotationand couple stress Such behavior is found in the numericalstudies available in the literature

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Financial support from the Fondazione Cassa di RisparmiodiModena Pratica Sime no 20130662 is gratefully acknowl-edged

References

[1] A C Eringen ldquoTheory of micropolar elasticityrdquo Tech RepPrinceton University 1967

[2] AC Eringen ldquoPart I Polar field theoriesrdquo inContinuumPhysicsVolume 4 Polar and Nonlocal FieldTheories pp 1ndash73 AcademicPress New York NY USA 1976

[3] J Dyszlewicz Micropolar Theory of Elasticity vol 15 of Lec-ture Notes in Applied and Computational Mechanics SpringerBerlin Germany 2004

[4] R D Mindlin and H F Tiersten ldquoEffects of couple-stresses inlinear elasticityrdquo Archive for Rational Mechanics and Analysisvol 11 no 1 pp 415ndash448 1962

[5] W T Koiter ldquoCouple stresses in the theory of elasticity partsI and IIrdquo Poceedings Koninklijke Nederlandse Akademie vanWetenschappen Series B vol 67 no 1 pp 17ndash44 1964


[6] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[7] N Challamel and CMWang ldquoThe small length scale effect fora non-local cantilever beam a paradox solvedrdquoNanotechnologyvol 19 no 34 Article ID 345703 2008

[8] S K Park and X-L Gao ldquoBernoulli-Euler beam model basedon a modified couple stress theoryrdquo Journal of Micromechanicsand Microengineering vol 16 no 11 pp 2355ndash2359 2006

[9] I Aganovic J Tambaca and Z Tutek ldquoDerivation and jus-tification of the models of rods and plates from linearizedthree-dimensional micropolar elasticityrdquo Journal of ElasticityThe Physical and Mathematical Science of Solids vol 84 no 2pp 131ndash152 2006

[10] G Riey and G Tomassetti ldquoMicropolar linearly elastic rodsrdquoCommunications in Applied Analysis vol 13 no 4 pp 647ndash6572009

[11] H M Ma X-L Gao and J N Reddy ldquoA microstructure-dependent Timoshenko beam model based on a modifiedcouple stress theoryrdquo Journal of the Mechanics and Physics ofSolids vol 56 no 12 pp 3379ndash3391 2008

[12] S Ramezani R Naghdabadi and S Sohrabpour ldquoAnalysis ofmicropolar elastic beamsrdquo European Journal of MechanicsASolids vol 28 no 2 pp 202ndash208 2009

[13] M Asghari M H Kahrobaiyan M Rahaeifard and M TAhmadian ldquoInvestigation of the size effects in Timoshenkobeams based on the couple stress theoryrdquo Archive of AppliedMechanics vol 81 no 7 pp 863ndash874 2011

[14] S P Timoshenko and J M Gere Theory of Elastic StabilityDover Publications 2nd edition 1961

[15] C Lanczos Variational Principles of Mechanics Dover 1986[16] ANobili ldquoVariationalapproachtobeams resting on two-param-

eter tensionless elastic foundationsrdquo Transactions ASMEmdashJournal of Applied Mechanics vol 79 no 2 Article ID 0210102012

[17] ANobili ldquoSuperposition principle for the tensionless contact ofa beam resting on a winkler or a pasternak foundationrdquo Journalof Engineering Mechanics vol 139 no 10 pp 1470ndash1478 2013

[18] A Nobili and A M Tarantino ldquoMagnetostriction of a hardferromagnetic and elastic thin-film structurerdquoMathematics andMechanics of Solids vol 13 no 2 pp 95ndash123 2008

[19] A Nobili and L Lanzoni ldquoElectromechanical instability in lay-ered materialsrdquo Mechanics of Materials vol 42 no 5 pp 581ndash591 2010

[20] A Nobili and A M Tarantino ldquoPseudo-spectral methods inone-dimensional magnetostrictionrdquo Meccanica vol 50 no 1pp 99ndash108 2015

[21] CMWang JN Reddy andKH Lee ShearDeformable Beamsand Plates Relationships with Classical Solutions Elsevier 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


qm

z

x

Ψ

(a)

T

M

Vw1

(b)

Figure 1 Kinematics of a Timoshenko beam (a) and positivity conventions (b)

Besides the Timoshenko model is variationally more aptto a FE approximation (no locking effect) Plates and rodmodels are mathematically justified through a proper scalingin a small parameter from the three-dimensional theoryof linear micropolar elasticity in [9 10] A Timoshenkobeam model based on the reduced couple stress theory ispresented in [11] where static bending and free vibration ofa simply supported beam are studied In [12] a First OrderShear Deformation Beam Theory (FSDBT) is developed formicropolar elastic beams and analytical results for the staticbending of a cantilever beam as well as for the dispersionrelation for longitudinal and flexural waves are given Couplestress Timoshenko beams have been investigated in [13]where a closed-form solution valid in the static case andfor homogeneous materials is obtained and then applied toanalyze a cantilever beam In a such a variety the questionof whether such models reconcile and under what conditionsseldom appears and when it does the answer rests in the neg-ativeThis paper attempts to address precisely this matter andto gather all models under a single variational frameworkThe issue of what assumptions really matter for the reductionin the number ofmaterial parameters is also investigatedThepaper is thus organized Section 2 revises some feature of theTimoshenko beam model Section 3 introduces the couplestress Timoshenko model and Section 4 introduces thebroader linear micropolar model The Hamiltonian form ofthe latter and its general solution are presented in Section 41Finally conclusions are drawn in Section 5

2 Variational Features ofthe Timoshenko Beam

Let us consider a prismatic body (beam) of cross-sectionalarea119860 and length 119871 whose central axis (the sectionsrsquo centroidline) rests along the 119909-axis in the reference configurationFigure 1 Traditionally a Timoshenko beam [14 Section 217]is characterized by the following Lagrangian density

L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

minus 119902119910 minus 119898120595 (1)

where119864119868 is the beam flexural rigidity119866 is the shearmodulus119910 is the transverse displacement 120595 is the cross-sectionrotation (positive clockwise) and 119902 and 119898 are the transverse

and the torque line load density respectively (This notationis Timoshenkorsquos and we abide by it for historical reasonsThenotation 119911 for the transverse displacement would better fitwith the reference system of Figure 1 which is commonlyadopted in the recent literature However confusion mayarise with the 119911 coordinate along the cross-section) Sincethis is a beam theory all field quantities depend on the singleindependent variable 119909 that is 119864119868 = 119864119868(119909) 119902 = 119902(119909)and so forth Here we use a subscript number to denotedifferentiation with respect to 119909 that is we use 119910

1in the

place of 119889119910119889119909 Besides we will restrict ourselves to the statictheory For the sake of notational compactness the shearfactor 119896

119904 usually associated with the Timoshenko model is

here omitted although it can be easily introduced whenevera factor 119860119866 appears The action integral is

A = int119871

0

L119889119909 (2)

and according to Hamiltonrsquos principle seeking for deforma-tion makes the action integral stationary Since here the axialcoordinate 119909 takes up the role which is usually played by time119905 in discrete mechanics we will refer to 119910

1as the velocity and

to 1199102as the accelerationThe Euler-Lagrange (E-L) equations

for the Lagrangian (1) are

(1198641198681205951)1015840

+ 119860119866 (1199101minus 120595) + 119898 = 0

[119860119866 (1199101minus 120595)]1015840

+ 119902 = 0

(3)

where prime denotes differentiation with respect to 119909The Lagrangian (1) is noteworthy because it contains

the gyroscopic term 1205951199101 that is a term which is linear in

the velocity It is well known that a gyroscopic term arisesfrom the elimination of kinosthenic (or ignorable) variables[15 Chapter V Section 4] Indeed we easily show that theLagrangian (1) is obtained from the enlarged Lagrangian

L =1

21198641198681205952

1+1

21198601198661199082

1minus 119902119910 minus 119898120595 minus 120583 (119910

1minus 1199081minus 120595) (4)

Here 1199081is the slope that arises from shear deformation

(see Figure 1(b)) on account of the shearing force 119881 =

1198601198661199081 On the other hand the total slope 119910

1is partly due to





(1198641198681205951)1015840

minus 120583 + 119898 = 0 (5a)

1205831015840minus 119902 = 0 (5b)

(1198601198661199081+ 120583)1015840

= 0 (5c)


1199101= 1199081+ 120595 (6)


1198721015840minus 119879 + 119898 = 0

1198791015840minus 119902 = 0

(7)




1198601198661199081+ 120583 = 119881 + 119879 = 119881 (8)


(1198641198681205951) 120575120595 + 119860119866 (119910

1minus 120595) 120575119910 (9)


(1198641198681205951) 120575120595 minus 120583120575119910 + 119881120575119908 (10)




120579 =1

2curlu (11)


119894119895119896is


u = [119906119909 0 119906119911]

120579 = [0 120593 0]

(12)

being

119906119909= 119906119909(119909 119911)

119906119911= 119910 (119909)

120593 = 120593 (119909)

(13)


119906119909= 1199060minus 119911120595 (14)


119906119909(119909 119911) =

119873

sum

119894=0

119911119894120595(119894)(119909) (15)


0= 0 given that



119880 =1

2intΩ



120581 = grad 120579 (17)







120590 = 120582 (tr 120598) 1 + 2119866120598 (19)


120598119909119909= minus119911120595

1

120598119909119911= 120598119911119909=1

2(1199101minus 120595)

(20)



120593 = minus1

2(120595 + 119910

1) (21)


120581119910119909= 1205931 (22)


120594119909119910= 120594119910119909=1

21205931 (23)




119894120597119909119894


119898lowast

119909119910= 120573lowast1205931 (25)


m = Sym [mlowast] = 2120573120594 (26)


119898119909119910= 2120573120594

119909119910= 1205731205931 (27)



m = 21198972119866120594 (28)


119910119909



L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

8120573119860 (120595

1+ 1199102)2

minus 119902119910 minus 119898120595

(29)





[1198641198681205951+1

4120573119860 (120595

1+ 1199102)]

1015840

+ 119860119866 (1199101minus 120595) + 119898 = 0 (30a)

[1

4120573119860 (120595

1+ 1199102)]

1015840

minus 119860119866 (1199101minus 120595)

1015840

minus 119902 = 0 (30b)


=

T

Mm

c

dx

dxMm

Tm

(a)

(b)



(119872 minus119872119898)1015840

+ 119881 + 119898 = 0

(119879119898minus 119881)1015840

minus 119902 = 0

(31)

wherein 119872119898= minus(14)120573119860(120595



119898= minus119872

1015840





L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

21205731198601205932

1minus 119902119910 minus 119898120595

minus 119888120593 + 120582 (2120593 + 120595 + 1199101)

(32)


(1198641198681205951)1015840

+ 119860119866 (1199101minus 120595) minus 120582 + 119898 = 0 (33a)

minus [119860119866 (1199101minus 120595) + 120582]

1015840

minus 119902 = 0 (33b)

(1

21205731198601205931)

1015840

minus 120582 +1

2119888 = 0 (33c)


119898= (12)120573119860120593

1and



1198721015840+ 119881 + 119898 minus 120582 = 0 (34)




1+120595 like119881


119898= 1205821015840 in



1is kinosthenic

L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

21205731198601205932

1minus 119902119910 minus 119898120595

+ 120583 (21205931+ 1205951+ 1199102)

(35)


12 just like the



(1198641198681205951+ 120583)1015840

+ 119860119866 (1199101minus 120595) + 119898 = 0 (36a)

12058310158401015840minus [119860119866 (119910

1minus 120595)]1015840

minus 119902 = 0 (36b)

(1

21205731198601205931+ 120583)

1015840

= 0 (36c)




1


119872119898+ 120583 = 119872

119898 (37)



119898= (14)120573119860(120595

1+


tively







e = gradu +Ω

Ω119894119895= 120598119894119895119896120579119896

(39)




L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

2120573lowast1198601205932

1

minus1

4120578119860 (119910

1+ 120595)2


minus 119860120578120593 (120593 + 1199101+ 120595)

(41)


119880lowast=1

2intΩ



div 120579 = 0 (43)


1

2119888 (1199101+ 120595) (44)



1199011= 119860119866 (119910


1

2120578119860 (119910

1+ 120595) minus 119860120578120593

1199012= 119864119868120595

1

1199013= 120573lowast1198601205931

(45)


H =1199012

1

119860 (2119866 minus 120578)+1199012

2

2119864119868+1199012

3

2119860120573lowast

+2120578120593 + (2119866 + 120578)120595

2119866 minus 1205781199011+2119860120578119866 (120595 + 120593)

2

2119866 minus 120578+ 119902119910

+ 119898120595 + 119888120593

(46)


1199101015840minus 2

1199011

119860 (2119866 minus 120578)minus2120578120593 + (2119866 + 120578)120595

2119866 minus 120578= 0 (47a)

1205951015840minus1199012

119864119868= 0 (47b)

1205931015840minus1199013

119860120573lowast= 0 (47c)

1199011015840

1+ 119902 = 0 (47d)

1199011015840

2+2119866 + 120578

2119866 minus 1205781199011+ 4120578119860119866

2119866 minus 120578(120595 + 120593) + 119898 = 0 (47e)

1199011015840

3+2120578

2119866 minus 1205781199011+ 4120578119860119866

2119866 minus 120578(120595 + 120593) + 119888 = 0 (47f)



1by a


1199011(119909) = 119875

1(119909) = 119901

1+ int

119909

0

119902 (120591) 119889120591 (48)


1(0) = 119901



(1199013minus 1199012)1015840

= 119898 minus 119888 +2119866

2119866 minus 1205781198751 (49)


119875 (119909) = int

119909

0

(119898 minus 119888 +2119866

2119866 minus 1205781198751)119889119909 + 119888

0 (50)

where 1198880= 1199013(0) minus 119901


(47b) (47c) and (47e) read

1205951015840minus1199012

119864119868= 0 (51a)

1205931015840minus1199012

119860120573lowast=119875 (119909)

119860120573lowast (51b)

1199011015840

2+ 4120578119860119866


2119866 + 120578

2119866 minus 1205781198751 (51c)


2 namely

11990110158401015840

2+ 412058521199012= minus119898

1015840minus2119866 + 120578

2119866 minus 120578119902

minus 4120578119866


2119866

2119866 minus 1205781198751)

(52)


120585 = radic120578119866 (119864119868 + 119860120573

lowast)

119864119868120573lowast (2119866 minus 120578) (53)


5 Conclusions




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













(1198641198681205951)1015840

minus 120583 + 119898 = 0 (5a)

1205831015840minus 119902 = 0 (5b)

(1198601198661199081+ 120583)1015840

= 0 (5c)


1199101= 1199081+ 120595 (6)


1198721015840minus 119879 + 119898 = 0

1198791015840minus 119902 = 0

(7)




1198601198661199081+ 120583 = 119881 + 119879 = 119881 (8)


(1198641198681205951) 120575120595 + 119860119866 (119910

1minus 120595) 120575119910 (9)


(1198641198681205951) 120575120595 minus 120583120575119910 + 119881120575119908 (10)




120579 =1

2curlu (11)


119894119895119896is


u = [119906119909 0 119906119911]

120579 = [0 120593 0]

(12)

being

119906119909= 119906119909(119909 119911)

119906119911= 119910 (119909)

120593 = 120593 (119909)

(13)


119906119909= 1199060minus 119911120595 (14)


119906119909(119909 119911) =

119873

sum

119894=0

119911119894120595(119894)(119909) (15)


0= 0 given that



119880 =1

2intΩ



120581 = grad 120579 (17)







120590 = 120582 (tr 120598) 1 + 2119866120598 (19)


120598119909119909= minus119911120595

1

120598119909119911= 120598119911119909=1

2(1199101minus 120595)

(20)



120593 = minus1

2(120595 + 119910

1) (21)


120581119910119909= 1205931 (22)


120594119909119910= 120594119910119909=1

21205931 (23)




119894120597119909119894


119898lowast

119909119910= 120573lowast1205931 (25)


m = Sym [mlowast] = 2120573120594 (26)


119898119909119910= 2120573120594

119909119910= 1205731205931 (27)



m = 21198972119866120594 (28)


119910119909



L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

8120573119860 (120595

1+ 1199102)2

minus 119902119910 minus 119898120595

(29)





[1198641198681205951+1

4120573119860 (120595

1+ 1199102)]

1015840

+ 119860119866 (1199101minus 120595) + 119898 = 0 (30a)

[1

4120573119860 (120595

1+ 1199102)]

1015840

minus 119860119866 (1199101minus 120595)

1015840

minus 119902 = 0 (30b)


=

T

Mm

c

dx

dxMm

Tm

(a)

(b)



(119872 minus119872119898)1015840

+ 119881 + 119898 = 0

(119879119898minus 119881)1015840

minus 119902 = 0

(31)

wherein 119872119898= minus(14)120573119860(120595



119898= minus119872

1015840





L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

21205731198601205932

1minus 119902119910 minus 119898120595

minus 119888120593 + 120582 (2120593 + 120595 + 1199101)

(32)


(1198641198681205951)1015840

+ 119860119866 (1199101minus 120595) minus 120582 + 119898 = 0 (33a)

minus [119860119866 (1199101minus 120595) + 120582]

1015840

minus 119902 = 0 (33b)

(1

21205731198601205931)

1015840

minus 120582 +1

2119888 = 0 (33c)


119898= (12)120573119860120593

1and



1198721015840+ 119881 + 119898 minus 120582 = 0 (34)




1+120595 like119881


119898= 1205821015840 in



1is kinosthenic

L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

21205731198601205932

1minus 119902119910 minus 119898120595

+ 120583 (21205931+ 1205951+ 1199102)

(35)


12 just like the



(1198641198681205951+ 120583)1015840

+ 119860119866 (1199101minus 120595) + 119898 = 0 (36a)

12058310158401015840minus [119860119866 (119910

1minus 120595)]1015840

minus 119902 = 0 (36b)

(1

21205731198601205931+ 120583)

1015840

= 0 (36c)




1


119872119898+ 120583 = 119872

119898 (37)



119898= (14)120573119860(120595

1+


tively







e = gradu +Ω

Ω119894119895= 120598119894119895119896120579119896

(39)




L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

2120573lowast1198601205932

1

minus1

4120578119860 (119910

1+ 120595)2


minus 119860120578120593 (120593 + 1199101+ 120595)

(41)


119880lowast=1

2intΩ



div 120579 = 0 (43)


1

2119888 (1199101+ 120595) (44)



1199011= 119860119866 (119910


1

2120578119860 (119910

1+ 120595) minus 119860120578120593

1199012= 119864119868120595

1

1199013= 120573lowast1198601205931

(45)


H =1199012

1

119860 (2119866 minus 120578)+1199012

2

2119864119868+1199012

3

2119860120573lowast

+2120578120593 + (2119866 + 120578)120595

2119866 minus 1205781199011+2119860120578119866 (120595 + 120593)

2

2119866 minus 120578+ 119902119910

+ 119898120595 + 119888120593

(46)


1199101015840minus 2

1199011

119860 (2119866 minus 120578)minus2120578120593 + (2119866 + 120578)120595

2119866 minus 120578= 0 (47a)

1205951015840minus1199012

119864119868= 0 (47b)

1205931015840minus1199013

119860120573lowast= 0 (47c)

1199011015840

1+ 119902 = 0 (47d)

1199011015840

2+2119866 + 120578

2119866 minus 1205781199011+ 4120578119860119866

2119866 minus 120578(120595 + 120593) + 119898 = 0 (47e)

1199011015840

3+2120578

2119866 minus 1205781199011+ 4120578119860119866

2119866 minus 120578(120595 + 120593) + 119888 = 0 (47f)



1by a


1199011(119909) = 119875

1(119909) = 119901

1+ int

119909

0

119902 (120591) 119889120591 (48)


1(0) = 119901



(1199013minus 1199012)1015840

= 119898 minus 119888 +2119866

2119866 minus 1205781198751 (49)


119875 (119909) = int

119909

0

(119898 minus 119888 +2119866

2119866 minus 1205781198751)119889119909 + 119888

0 (50)

where 1198880= 1199013(0) minus 119901


(47b) (47c) and (47e) read

1205951015840minus1199012

119864119868= 0 (51a)

1205931015840minus1199012

119860120573lowast=119875 (119909)

119860120573lowast (51b)

1199011015840

2+ 4120578119860119866


2119866 + 120578

2119866 minus 1205781198751 (51c)


2 namely

11990110158401015840

2+ 412058521199012= minus119898

1015840minus2119866 + 120578

2119866 minus 120578119902

minus 4120578119866


2119866

2119866 minus 1205781198751)

(52)


120585 = radic120578119866 (119864119868 + 119860120573

lowast)

119864119868120573lowast (2119866 minus 120578) (53)


5 Conclusions




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880 =1

2intΩ



120581 = grad 120579 (17)







120590 = 120582 (tr 120598) 1 + 2119866120598 (19)


120598119909119909= minus119911120595

1

120598119909119911= 120598119911119909=1

2(1199101minus 120595)

(20)



120593 = minus1

2(120595 + 119910

1) (21)


120581119910119909= 1205931 (22)


120594119909119910= 120594119910119909=1

21205931 (23)




119894120597119909119894


119898lowast

119909119910= 120573lowast1205931 (25)


m = Sym [mlowast] = 2120573120594 (26)


119898119909119910= 2120573120594

119909119910= 1205731205931 (27)



m = 21198972119866120594 (28)


119910119909



L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

8120573119860 (120595

1+ 1199102)2

minus 119902119910 minus 119898120595

(29)





[1198641198681205951+1

4120573119860 (120595

1+ 1199102)]

1015840

+ 119860119866 (1199101minus 120595) + 119898 = 0 (30a)

[1

4120573119860 (120595

1+ 1199102)]

1015840

minus 119860119866 (1199101minus 120595)

1015840

minus 119902 = 0 (30b)


=

T

Mm

c

dx

dxMm

Tm

(a)

(b)



(119872 minus119872119898)1015840

+ 119881 + 119898 = 0

(119879119898minus 119881)1015840

minus 119902 = 0

(31)

wherein 119872119898= minus(14)120573119860(120595



119898= minus119872

1015840





L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

21205731198601205932

1minus 119902119910 minus 119898120595

minus 119888120593 + 120582 (2120593 + 120595 + 1199101)

(32)


(1198641198681205951)1015840

+ 119860119866 (1199101minus 120595) minus 120582 + 119898 = 0 (33a)

minus [119860119866 (1199101minus 120595) + 120582]

1015840

minus 119902 = 0 (33b)

(1

21205731198601205931)

1015840

minus 120582 +1

2119888 = 0 (33c)


119898= (12)120573119860120593

1and



1198721015840+ 119881 + 119898 minus 120582 = 0 (34)




1+120595 like119881


119898= 1205821015840 in



1is kinosthenic

L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

21205731198601205932

1minus 119902119910 minus 119898120595

+ 120583 (21205931+ 1205951+ 1199102)

(35)


12 just like the



(1198641198681205951+ 120583)1015840

+ 119860119866 (1199101minus 120595) + 119898 = 0 (36a)

12058310158401015840minus [119860119866 (119910

1minus 120595)]1015840

minus 119902 = 0 (36b)

(1

21205731198601205931+ 120583)

1015840

= 0 (36c)




1


119872119898+ 120583 = 119872

119898 (37)



119898= (14)120573119860(120595

1+


tively







e = gradu +Ω

Ω119894119895= 120598119894119895119896120579119896

(39)




L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

2120573lowast1198601205932

1

minus1

4120578119860 (119910

1+ 120595)2


minus 119860120578120593 (120593 + 1199101+ 120595)

(41)


119880lowast=1

2intΩ



div 120579 = 0 (43)


1

2119888 (1199101+ 120595) (44)



1199011= 119860119866 (119910


1

2120578119860 (119910

1+ 120595) minus 119860120578120593

1199012= 119864119868120595

1

1199013= 120573lowast1198601205931

(45)


H =1199012

1

119860 (2119866 minus 120578)+1199012

2

2119864119868+1199012

3

2119860120573lowast

+2120578120593 + (2119866 + 120578)120595

2119866 minus 1205781199011+2119860120578119866 (120595 + 120593)

2

2119866 minus 120578+ 119902119910

+ 119898120595 + 119888120593

(46)


1199101015840minus 2

1199011

119860 (2119866 minus 120578)minus2120578120593 + (2119866 + 120578)120595

2119866 minus 120578= 0 (47a)

1205951015840minus1199012

119864119868= 0 (47b)

1205931015840minus1199013

119860120573lowast= 0 (47c)

1199011015840

1+ 119902 = 0 (47d)

1199011015840

2+2119866 + 120578

2119866 minus 1205781199011+ 4120578119860119866

2119866 minus 120578(120595 + 120593) + 119898 = 0 (47e)

1199011015840

3+2120578

2119866 minus 1205781199011+ 4120578119860119866

2119866 minus 120578(120595 + 120593) + 119888 = 0 (47f)



1by a


1199011(119909) = 119875

1(119909) = 119901

1+ int

119909

0

119902 (120591) 119889120591 (48)


1(0) = 119901



(1199013minus 1199012)1015840

= 119898 minus 119888 +2119866

2119866 minus 1205781198751 (49)


119875 (119909) = int

119909

0

(119898 minus 119888 +2119866

2119866 minus 1205781198751)119889119909 + 119888

0 (50)

where 1198880= 1199013(0) minus 119901


(47b) (47c) and (47e) read

1205951015840minus1199012

119864119868= 0 (51a)

1205931015840minus1199012

119860120573lowast=119875 (119909)

119860120573lowast (51b)

1199011015840

2+ 4120578119860119866


2119866 + 120578

2119866 minus 1205781198751 (51c)


2 namely

11990110158401015840

2+ 412058521199012= minus119898

1015840minus2119866 + 120578

2119866 minus 120578119902

minus 4120578119866


2119866

2119866 minus 1205781198751)

(52)


120585 = radic120578119866 (119864119868 + 119860120573

lowast)

119864119868120573lowast (2119866 minus 120578) (53)


5 Conclusions




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










=

T

Mm

c

dx

dxMm

Tm

(a)

(b)



(119872 minus119872119898)1015840

+ 119881 + 119898 = 0

(119879119898minus 119881)1015840

minus 119902 = 0

(31)

wherein 119872119898= minus(14)120573119860(120595



119898= minus119872

1015840





L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

21205731198601205932

1minus 119902119910 minus 119898120595

minus 119888120593 + 120582 (2120593 + 120595 + 1199101)

(32)


(1198641198681205951)1015840

+ 119860119866 (1199101minus 120595) minus 120582 + 119898 = 0 (33a)

minus [119860119866 (1199101minus 120595) + 120582]

1015840

minus 119902 = 0 (33b)

(1

21205731198601205931)

1015840

minus 120582 +1

2119888 = 0 (33c)


119898= (12)120573119860120593

1and



1198721015840+ 119881 + 119898 minus 120582 = 0 (34)




1+120595 like119881


119898= 1205821015840 in



1is kinosthenic

L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

21205731198601205932

1minus 119902119910 minus 119898120595

+ 120583 (21205931+ 1205951+ 1199102)

(35)


12 just like the



(1198641198681205951+ 120583)1015840

+ 119860119866 (1199101minus 120595) + 119898 = 0 (36a)

12058310158401015840minus [119860119866 (119910

1minus 120595)]1015840

minus 119902 = 0 (36b)

(1

21205731198601205931+ 120583)

1015840

= 0 (36c)




1


119872119898+ 120583 = 119872

119898 (37)



119898= (14)120573119860(120595

1+


tively







e = gradu +Ω

Ω119894119895= 120598119894119895119896120579119896

(39)




L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

2120573lowast1198601205932

1

minus1

4120578119860 (119910

1+ 120595)2


minus 119860120578120593 (120593 + 1199101+ 120595)

(41)


119880lowast=1

2intΩ



div 120579 = 0 (43)


1

2119888 (1199101+ 120595) (44)



1199011= 119860119866 (119910


1

2120578119860 (119910

1+ 120595) minus 119860120578120593

1199012= 119864119868120595

1

1199013= 120573lowast1198601205931

(45)


H =1199012

1

119860 (2119866 minus 120578)+1199012

2

2119864119868+1199012

3

2119860120573lowast

+2120578120593 + (2119866 + 120578)120595

2119866 minus 1205781199011+2119860120578119866 (120595 + 120593)

2

2119866 minus 120578+ 119902119910

+ 119898120595 + 119888120593

(46)


1199101015840minus 2

1199011

119860 (2119866 minus 120578)minus2120578120593 + (2119866 + 120578)120595

2119866 minus 120578= 0 (47a)

1205951015840minus1199012

119864119868= 0 (47b)

1205931015840minus1199013

119860120573lowast= 0 (47c)

1199011015840

1+ 119902 = 0 (47d)

1199011015840

2+2119866 + 120578

2119866 minus 1205781199011+ 4120578119860119866

2119866 minus 120578(120595 + 120593) + 119898 = 0 (47e)

1199011015840

3+2120578

2119866 minus 1205781199011+ 4120578119860119866

2119866 minus 120578(120595 + 120593) + 119888 = 0 (47f)



1by a


1199011(119909) = 119875

1(119909) = 119901

1+ int

119909

0

119902 (120591) 119889120591 (48)


1(0) = 119901



(1199013minus 1199012)1015840

= 119898 minus 119888 +2119866

2119866 minus 1205781198751 (49)


119875 (119909) = int

119909

0

(119898 minus 119888 +2119866

2119866 minus 1205781198751)119889119909 + 119888

0 (50)

where 1198880= 1199013(0) minus 119901


(47b) (47c) and (47e) read

1205951015840minus1199012

119864119868= 0 (51a)

1205931015840minus1199012

119860120573lowast=119875 (119909)

119860120573lowast (51b)

1199011015840

2+ 4120578119860119866


2119866 + 120578

2119866 minus 1205781198751 (51c)


2 namely

11990110158401015840

2+ 412058521199012= minus119898

1015840minus2119866 + 120578

2119866 minus 120578119902

minus 4120578119866


2119866

2119866 minus 1205781198751)

(52)


120585 = radic120578119866 (119864119868 + 119860120573

lowast)

119864119868120573lowast (2119866 minus 120578) (53)


5 Conclusions




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













e = gradu +Ω

Ω119894119895= 120598119894119895119896120579119896

(39)




L =1

21198641198681205952

1+1

2119860119866 (119910

1minus 120595)2

+1

2120573lowast1198601205932

1

minus1

4120578119860 (119910

1+ 120595)2


minus 119860120578120593 (120593 + 1199101+ 120595)

(41)


119880lowast=1

2intΩ



div 120579 = 0 (43)


1

2119888 (1199101+ 120595) (44)



1199011= 119860119866 (119910


1

2120578119860 (119910

1+ 120595) minus 119860120578120593

1199012= 119864119868120595

1

1199013= 120573lowast1198601205931

(45)


H =1199012

1

119860 (2119866 minus 120578)+1199012

2

2119864119868+1199012

3

2119860120573lowast

+2120578120593 + (2119866 + 120578)120595

2119866 minus 1205781199011+2119860120578119866 (120595 + 120593)

2

2119866 minus 120578+ 119902119910

+ 119898120595 + 119888120593

(46)


1199101015840minus 2

1199011

119860 (2119866 minus 120578)minus2120578120593 + (2119866 + 120578)120595

2119866 minus 120578= 0 (47a)

1205951015840minus1199012

119864119868= 0 (47b)

1205931015840minus1199013

119860120573lowast= 0 (47c)

1199011015840

1+ 119902 = 0 (47d)

1199011015840

2+2119866 + 120578

2119866 minus 1205781199011+ 4120578119860119866

2119866 minus 120578(120595 + 120593) + 119898 = 0 (47e)

1199011015840

3+2120578

2119866 minus 1205781199011+ 4120578119860119866

2119866 minus 120578(120595 + 120593) + 119888 = 0 (47f)



1by a


1199011(119909) = 119875

1(119909) = 119901

1+ int

119909

0

119902 (120591) 119889120591 (48)


1(0) = 119901



(1199013minus 1199012)1015840

= 119898 minus 119888 +2119866

2119866 minus 1205781198751 (49)


119875 (119909) = int

119909

0

(119898 minus 119888 +2119866

2119866 minus 1205781198751)119889119909 + 119888

0 (50)

where 1198880= 1199013(0) minus 119901


(47b) (47c) and (47e) read

1205951015840minus1199012

119864119868= 0 (51a)

1205931015840minus1199012

119860120573lowast=119875 (119909)

119860120573lowast (51b)

1199011015840

2+ 4120578119860119866


2119866 + 120578

2119866 minus 1205781198751 (51c)


2 namely

11990110158401015840

2+ 412058521199012= minus119898

1015840minus2119866 + 120578

2119866 minus 120578119902

minus 4120578119866


2119866

2119866 minus 1205781198751)

(52)


120585 = radic120578119866 (119864119868 + 119860120573

lowast)

119864119868120573lowast (2119866 minus 120578) (53)


5 Conclusions




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1by a


1199011(119909) = 119875

1(119909) = 119901

1+ int

119909

0

119902 (120591) 119889120591 (48)


1(0) = 119901



(1199013minus 1199012)1015840

= 119898 minus 119888 +2119866

2119866 minus 1205781198751 (49)


119875 (119909) = int

119909

0

(119898 minus 119888 +2119866

2119866 minus 1205781198751)119889119909 + 119888

0 (50)

where 1198880= 1199013(0) minus 119901


(47b) (47c) and (47e) read

1205951015840minus1199012

119864119868= 0 (51a)

1205931015840minus1199012

119860120573lowast=119875 (119909)

119860120573lowast (51b)

1199011015840

2+ 4120578119860119866


2119866 + 120578

2119866 minus 1205781198751 (51c)


2 namely

11990110158401015840

2+ 412058521199012= minus119898

1015840minus2119866 + 120578

2119866 minus 120578119902

minus 4120578119866


2119866

2119866 minus 1205781198751)

(52)


120585 = radic120578119866 (119864119868 + 119860120573

lowast)

119864119868120573lowast (2119866 minus 120578) (53)


5 Conclusions




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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