Exact Solution for Electro- Thermo- Mechanical Behavior of Composite Cylinder Reinforced by BNNTs under Non- Axisymmetric Thermo- Mechanical Loads

Transcription

1 mikabi Univesity of Technology (Tehan Polytechnic) Vol, No, Sping 3, pp - mikabi Intenational Jounal of Science & Reseach (Moeling, Ientification, Simulation & Contol) (IJ - MISC) Exact Solution fo Electo- Themo- Mechanical Behavio of Composite Cyline Reinfoce by s une Non- xisymmetic Themo- Mechanical Loas Ghobanpou ani *, E Haghpaast, Z Khoami Maaghi 3, S mi - Pofesso, Faculty of Mechanical Engineeing, Univesity of Kashan, Kashan, Ian - PhD Stuent Depatment of Mechanical Engineeing, Univesity of Kashan, Kashan, Ian 3- PhD Stuent Depatment of Mechanical Engineeing, Univesity of Kashan, Kashan, Ian - ssistant Potfesso, Depatment of Mechanical Engineeing, Univesity of Kashan, Kashan, Ian BSTRCT In this eseach, static stesses analysis of boon nitie nano - tube einfoce composite (RC) cyline mae of poly - vinyliene fluoie (PVDF) subecte to non - axisymmetic themo - mechanical loas an applie voltage is evelope The suoune elastic meium is moelle by Pastenak founation Composite stuctue is moele base on piezoelectic fibe einfoce composite (PFRC) theoy an a epesentative volume element has been consiee fo peicting the elastic, piezoelectic an ielectic popeties of the cyline Highe oe govening equations wee solve analytically by Fouie seies The esults emonstate that the fatigue life of RC cyline will be significantly epenent on the angle oientation an volume faction of s Results of this investigation can be use fo the optimum esign of thick - walle cylines une the multi - physical fiels KEYWORDS Composite Hollow Cyline, Non - xisymmetic Tempeatue Distibution, PFRC Theoy, Pastenak Founation, s Fibes Coesponing utho, aghoban@kashanuaci Vol, No, Sping 3

2 mikabi Intenational Jounal of Science& Reseach (Moeling, Ientification, Simulation & Contol) Ghobanpou ani, E Haghpaast, Z Khoami Maaghi, S mi - INTRODUCTION s have the same atomic stuctue as cabon nanotubes (CNTs) but thee exist many inteesting popeties incluing a moe stable electonic popety an bette esistance to oxiation at high tempeatues s potential applications consist of iffeent mechanically - einfoce mateials such as polyme, ceamic, an metal composites, an key pats of nano - electo - mechanical smat systems In this ecae, analytical themo - mechanical stess calculations fo thick - walle cyline have been investigate Dai et al [] pesente electo - magneto - themo - elastic behavio of a hollow cyline compose of functionally gae piezoelectic mateial (FGPM), place in a unifom magnetic fiel, subecte to electic, themal an mechanical loas They showe that applying suitable electic, themal an mechanical loas coul optimize the FGPM hollow cylinical stuctues In this ega, Poultangai et al [] stuie an analytical metho to obtain the solution fo the two - imensional steay state themal an mechanical stesses in a hollow thick sphee mae of functionally gae mateial (FGM) Thei esults inicate that the aial, tangential, an shea stess istibutions can be euce in magnitue when the powe law inex assumes the smalle values Electo - themo - elastic stess analysis of piezoelectic polymeic thick - walle cyline, einfoce by s subecte to electo - themo - mechanical fiels, is pesente by Ghobanpou ani et al [3] who showe that inceasing DWs content euces the stesses associate with mechanical, themal an electical fiels, in escening oe Similaly, Ghobanpou ani et al [] stuie themal stess analysis of a thick - walle cyline einfoce with FG single - walle cabon nanotubes (SWCNTs) The highe oe govening equation was solve in oe to obtain the istibution of isplacement an themal stesses in aial, cicumfeential an longituinal iections FG istibutions of SWCNTs have significant effect on isplacements an themal stesses of composite cyline, so that in incementally inceasing layout, the aial an cicumfeential stesses ae lowe than the othe FG stuctues Recently, Jafai Feshaaki et al [] evelope the geneal theoetical analysis of a FG piezoelectic hollow cyline subecte to the two - imensional electo - mechanical loa They use sepaation of vaiables metho an complex Fouie seies, to eive an solve the Navie equations in tems of isplacements Thei stuy eveale, using this metho an consieing the special bounay conitions an mateial popeties fo a hollow cyline, the mechanical an electical isplacements an stesses can be contolle an optimize The analytical solution of FG piezoelectic hollow cyline, which is une the aial electic potential an non - axisymmetic themo - mechanical loas, ae pesente by tian et al [6] who use complex Fouie seies to inicate the istibutions of stesses, isplacement an the effect of electic potential fiel on the cyline behavio Howeve, to ate, no epot has been foun in the liteatue on the stess analysis of composite cyline einfoce by s une non - axisymmetic themo - mechanical loas Motivate by these consieations, the nee fo the investigation of a s fibe is applie as einfocement in composite cyline whee PVDF has been selecte fo matix as piezoelectic mateial Composite cyline is subecte to non - axisymmetic themo - mechanical loas an applie voltage Highe oe govening equations ae eive analytically an effects of non - axisymmetic pessue, volume faction an oientation angle of s on the stesses ae investigate Results of this eseach coul be use fo the optimum esign of composite cyline une non - axisymmetic themo - mechanical loaings - GOVERNING EQUTIONS Heat solution Fig, illustates a thick-walle cyline subect to the tempeatue istibution T(, ) which is einfoce by fibe in aial iection Tempeatue istibution has been applie in a hollow cyline une a non-axisymmetic heat conuction cause by nonaxisymmetic themal bounay conitions This is like to having a cyline expose to a given heat flux on a pat of its oute bounay while on the othe pat a specifie tempeatue is applie The geneal istibution of tempeatue (T(, )) in hollow cyline with insie an 6 Vol, No, Sping 3

3 mikabi Intenational Jounal of Science& Reseach (Moeling, Ientification, Simulation & Contol) outsie aii a an b, espectively, can be witten as [7]: whee an B ae the abitay constants an the functions F () an G () can be etemine fo iffeent n bounay conitions The fist tem at the ight - han sie of Eq (),, pouces a unifom axial stess fo a thick cyline in the plane stain conition The secon tem causes such stesses being as a function of the aius an the tems une the summation signs pouce non - axisymmetic themal stesses ll tems automatically satisfy Michell conition an they on t contibute to the themal stesses, except fo the tems associate with n = an n = Theefoe, Eq () can be simplifie to: T, ) = F ( ) F ( )cos G ( )sin () ( whee the functions F (), F () an G () ae etemine by bounay conitions Using stess bounay conition in inne an oute aius, non - axisymmetic tempeatue istibution can be expesse as follows: B T (, ) = cos sin (3) Exact Solution fo Electo- Themo- Mechanical Behavio of Composite Cyline Reinfoce by s une Non- xisymmetic Themo- Mechanical Loas T (, ) = Tnon-axisymmetic Taxisymmetic B = cos sin T ln T a a () «Figs - a an - b,» show the istibution of tempeatue in aial an cicumfeential iections whee the tempeatue istibution is the combination of axisymmetic an non - axisymmetic tempeatue tems «Fig - b,» epicts tempeatue istibution in paticula angle whee the cuves have logaithmic istibution base on Eq () Tempatue (K) Tempatue (K) 3 3 / pi/ pi (a) /a =pi/3 =pi/ =pi/3 39 /a (b) Figue : Non - axisymmetic tempeatue istibution Figue : The schematic of composite cyline une non - axisymmetic tempeatue an pessue istibution an applie voltage In axisymmetic heat conuction, T ( ) = T ln( / a) Ta, ( T Ta Tb ) = intouces the symmetic tempeatue istibution in which T a an T b ae the insie an outsie tempeatues of the cyline, espectively Theefoe, the geneal solution fo heat conuction can be expesse as [7]: B Stess analysis Composite cyline einfoce by s une non - axisymmetic loaings embee in an elastic meium has been emonstate in Fig The stain - isplacement elations in cylinical cooinate can be witten as []: u ε =, ( - a) v u ε = ( ), ( - b) Vol, No, Sping 3 7

4 mikabi Intenational Jounal of Science& Reseach (Moeling, Ientification, Simulation & Contol) Ghobanpou ani, E Haghpaast, Z Khoami Maaghi, S mi u v v ε = ( ) ( - c) whee u an v ae the components of isplacement in aial an cicumfeential, espectively c Mico - electo - mechanical moel To moel the smat composite stuctues, a micomechanical moel known as XY PEFRC o YX PEFRC, using an appoach aopte by Tan an Tong [8, 9], is employe Base on this moeling, the einfoce s ae consiee as longituinal staight fibes place in PVDF matix In a geneal case, to evaluate the effective popeties of a PFRC unit cell in all iections, fist, the popeties of the equie stip is obtaine in which an appopiate X moel in association with the Y moel (o vice vesa), ie, XY (o YX) ectangle moel has been employe In this moeling, both matix an einfocements ae assume to be smat an the polaization has been mae along the fibe iection ccoing to the XY PEFRC micomechanical metho, the constitutive equations fo the electo - themo - mechanical behavio of the selecte RVE ae expesse as []: σ C = T D e e ε E (6) whee o', ε, D an Eae stess, stain, electic isplacement an electic fiel vectos, espectively an [ e ] an [ ] ae matices of piezoelectic an ielectic paametes espectively Eq (6) can be expane as follows: σ C σ C σ 3 C3 τ 3 τ 3 = τ D D D3 e3 C C C3 e3 C3 C3 C3 e3 C e C e C6 e e e3 ε e3 ε e3 ε 3 γ 3 γ 3 γ E E 3 E3 (7) ssuming the plane stain conition an uniiectional electic fiel along the aius of composite cyline, Eq (7) is euce to []: σ E σ = τ D εe ε γ E [ Q] (8) whee matix [ Q] is efine as []: C Ce e Ce C [ ] Q = e C e e (9) Electic fiel in the aial iection of cylinical cooinate fo piezoelectic mateials is []: φ E = () whee ϕ enotes the scala function of the electic potential To consie the effects of oientation angle of the s with espect to the longituinal axis, the following tansfomation matix can be employe as []: [ ~ ] [ T ][ Q][ T ] T Q = () The tansfomation matix [T] is: cos α sin α sinα cosα sin cos [ T ] = sinα cosα α α sinα cosα sinα cosα cos α sin α () whee α is the angle of s with espect to the longituinal axis of composite cyline D Equilibium equations The equilibium equations in the aial an cicumfeential iections, egaing the boy foces ae []: σ σ σ σ σ σ D D = σ =, Fboy foce =, (3 - a) (3 - b) (3 - c) Thick - walle cyline suoune by elastic meium that is simulate with Pastenak founation whee the effect of the elastic meium has been consiee as boy foce on the oute suface of cyline as below []: F Pastenak = K w u G = K p w u G p u u u ( ) u () Substituting Eqs () into Eq (3) an by using Eqs () an (8), the final equilibium equations, in tems of isplacements an electical potential, can be 8 Vol, No, Sping 3

5 mikabi Intenational Jounal of Science& Reseach (Moeling, Ientification, Simulation & Contol) etemine as follows: u u 3 u v u 6 v φ 8 φ 7 =, v v v u 9 v 3 u φ =, φ φ u u v = Vol, No, Sping 3 ( - a) ( - b) ( - c) whee the constants to ae given in the appenix 3- SOLVING METHOD Eqs ( - a) to ( - c) ae govening the equations fo a thick - walle composite cyline une electo - mechanical fiels In geneal solution of Eqs, the aial, cicumfeential isplacements an electic potential ae pesume accoing to the complex Fouie seies fom []: in u (, ) = un ( ) e, n= - (6 - a) in v (, ) = vn ( ) e, n= - (6 - b) (, ) = in φ φn( ) e = φ( ) n= - (6 - c) whee u n () an () v n (, ae the coefficients of complex Fouie seies of u ) an v (, ), espectively, that can be expesse as: π in un ( ) = u(, ) e, π π π in vn ( ) = v(, ) e π π (7 - a) (7 - b) an electic potential ust epens on the aius of composite cyline Theefoe, Eq (6 - c) is vali only fo n = an it will be omitte fo n [6] Fo n, substituting Eq (6) into Eq () gives: un un ( 6 in v =, n 3 n ) u in n vn (8 - a) Exact Solution fo Electo- Themo- Mechanical Behavio of Composite Cyline Reinfoce by s une Non- xisymmetic Themo- Mechanical Loas 9 v n vn ( in u = n []: n ) v n in 3 un (8 - b) The solutions of the above equations ae assume as u n ( ) =, v n ( ) = B (9 - a) (9 - b) whee an Bae constants an obtaine fom the bounay conitions Substituting Eq(9) into Eqs(8), yiels: [ ( ) 3 n ] [ in in ] B =, 6 [ 9 ( ) n ] B ( - a) [ in3 in ] = ( - b) To obtain the nontivial solution of Eq (), the eteminant of the system shoul be equal to zeo [] So fou oots, n to n, ae achieve an the geneal solutions ae: u ( ) = n v ( ) = n whee: n n, ( - a) n M n n n ( - b) n n ( ) 3 n Mn =, () in in6 Fo n=, the equilibium Eqs () ae euce to: 3 8 u u u 7 φ φ =, (3 - a) 9 v v v = φ φ 8 u u =, (3 - b) (3 - c) whee the subscipt zeo inicates the solution fo n= Two equations (3 - a) an (3 - c) ae a couple oinay iffeential equations an the solution of these ae consiee as []: ( ) u, ( ) φ = C = ( - a) Substituting Eq ( - a) into Eqs (3 - a) an (3 - c) 9

6 mikabi Intenational Jounal of Science& Reseach (Moeling, Ientification, Simulation & Contol) yiels: ( ( ) 3) ( 7 ( ) 8 ) C =, ( 6 ( ) 7 ) C ( 8 ( ) 9 ) = ( - b) To obtain the nontivial solution of Eq ( - b), the eteminant of the system shoul be equal to zeo So the fou oots, to, ae achieve an the geneal solutions ae: u ( ) =, ( ) = N φ ( - a) ( ) N 3 = 7 ( ) 8 ( - b) Fo n = Eq (3 - b) is a ecouple oinay iffeential equation an the solution of this equation is consiee as: 6 v ( ) = ± 9,6, 6 = 9 (6 - a) (6 - b) Finally substituting Eqs ( - a), ( - a) an (6 - a) into Eq (6), geneal solutions fo (, ) ae expesse as: u (, ) = u, (, ) n in in n e, n=, n 6 n in in v(, ) = M n n e, n=, n φ( ) = N v an φ() (7 - a) (7 - b) (7 - c) Ghobanpou ani, E Haghpaast, Z Khoami Maaghi, S mi M ( ) ε = ( n n ) in n e n=, n ( n ) in in M n n e, n=, n ( n ) in n e M ε = in 6 n=, n ( ) ( ) n in n M n n e n=, n 6 ( ) M n n=, n ( n ) in n e (8 - b) (8 - c) Substituting Eqs (8 - a) - (8 - c) into Eq (8) an using Eqs (9) an (7 - c), the stess components an the electical isplacement in the aial iection ae calculate in ppenix B Fo n, equilibium equations equie fou bounay conitions to satisfy Expaning the given bounay conitions in complex Fouie seies gives the following equation: ( ) = in in g G ( n) e, =,,6, n= whee: π in G ( n) = g ( ) e, =,,6 π π (9 - a) (9 - b) Constants n ae calculate using Eqs (8) an (9) Substituting Eqs (7 - a) - (7 - c) into Eq (), the stains ae obtaine as: ε M = n=, n ( ) ( n ) in in n n e, (8 - a) - NUMERICL RESULTS ND DISCUSSION Consie a thick - walle cyline with inne aius a=m an oute aius b=m [] The cyline is embee in a Pastenak founation with G p =773 N/m [] In this stuy, the PVDF an have been taken into account fo smat matix an fibe of the composite cyline, espectively In the pat of axisymmetic of tempeatue istibution, the tempeatue at the inne an oute sufaces have been assume to be T a =3K an T b =3K, espectively The themal expansion coefficients of the composite cyline in, z an Vol, No, Sping 3

7 mikabi Intenational Jounal of Science& Reseach (Moeling, Ientification, Simulation & Contol) Exact Solution fo Electo- Themo- Mechanical Behavio of Composite Cyline Reinfoce by s une Non- xisymmetic Themo- Mechanical Loas iections can be witten as [3]: α = ( υ ( υ ) c m z m m α z ) c α υξ α z, m (3 - a) α = c α c α, (3 - a) z m α = α, (3 - a) υ ξ = c υz cm υm (3 - a) whee α z an α ae themal expansion coefficients of s in longituinal an aial iections, espectively, an α m is the themal expansion coefficient of PVDF an assume to be α m =7 - (K) [] Howeve, α z an α ae not expecte to vay significantly fo the tempeatue ange of K<T<K consiee in this wok Hence, thei aveage values ae taken at 3KTheefoe, they ae assume to be α z = -6 / K an α =6-6 /K [] υ an υ m ae Poisson s atios of s an matix an ae taken as υ z = 3 an υ m = 39 [] s mentione in the pevious sections, in oe to achieve the total stesses, the themal an mechanical stesses ae calculate sepaately an finally supepose The complete solution fo stesses, ue to the tempeatue istibution (Eq ()), is obtaine though the supeposition of the axisymmetic an non - axisymmetic stesses Hence, these stesses ae calculate as [7]: Eα T b a b σ = ln n ( ) ln n ( ν ) a b a a Eα a b ( )( )( cos B sin ), ), ( ν ) ( a b ) Eα T b a b σ = n ln ( ) ln n ( ν ) a b a a Eα a b a b (3 )( cos B sin ), ), ( ν ) ( a b ) σ Eα a b ( )( ) ( sin B cos ) ( ν ) ( a b ) (3 - a) (3 - b) = (3 - c) Theefoe, the themal stesses in aial an cicumfeential iections ae shown in «Figs 3 an,» an themal shea stess is epicte in «Fig,» base on Eqs 3 These figues confim the hamonic patten pesente in the above equations Themmal - / pi/ /a pi Figue 3: Raial themal stess in composite cyline une non - axisymmetic loaings σ x 7 x / pi/ /a pi Figue : Cicumfeential themal stess in composite cyline une non - axisymmetic loaings Themmal σ Themmal σ x 7 - / pi/ pi /a Figue : Shea themal stess in composite cyline une non - axisymmetic loaing In oe to examine the popose solution, the effect of two - imensional themo - mechanical behavio of piezoelectic composite cyline is consiee The insie an outsie cyline an the bounay conitions ae expesse as [6]: S S ( = a) = cos Mpa, ( = b) = 3 cos Mpa, S ( = a) =, S ( = b) =, φ( = a) =, φ( = b) = 3 (3) Vol, No, Sping 3

8 mikabi Intenational Jounal of Science& Reseach (Moeling, Ientification, Simulation & Contol) Ghobanpou ani, E Haghpaast, Z Khoami Maaghi, S mi «Figs 6-8,» emonstate the istibution of aial, cicumfeential an shea total stesses It is evient that all components of stesses an electic isplacement follow fom a hamonic patten Likewise, it can be seen fom Fig 8, that the shea total stess is zeo at the inne an oute sufaces ue to the assume bounay conitions σ x 8 - / pi/ /a pi Figue 6: Raial total stess in composite cyline une electo - themo - mechanical loaings x 8 - / pi/ /a pi Figue 7: Cicumfeential total stess in composite cyline une electo - themo - mechanical loaings σ σ x 8 - / pi/ /a pi Figue 8: Shea total stess in composite cyline une electo - themo - mechanical loaings Effective stess as Von - Mises stess has been illustate in «Fig 9,» s expecte; the effective stess follows fom a hamonic patten σ Effective x 8 / pi/ /a pi Figue 9: The effective stess in composite cyline une electo - themo - mechanical loaings Fig, shows electic isplacement istibution along the aius an cicumfeential iections It is also note that the pattens of Figs 6 -, ae consistent with efeence [] D x / pi/ pi /a Figue : Raial electic isplacement in composite hollow cyline ccoing to geneal bounay conitions (Eq 3), the insie an outsie bounay conitions have been simplifie to stuy the effect of volume faction, elastic meium an Loentz foce on Von - Mises stess in composite cyline, as follows: S ( = a) = Mpa, S ( = b) = Mpa, S ( = a) =, S ( = b) =, φ( = a) =, φ( = b) = 3 (33) Themal stesses fo =π ae ae to mechanical stesses Fig, shows the effect of volume faction on Von - Mises stess in composite cyline It can be seen fom this figue that inceasing volume content of s causes to ecease Von - Mises stess an leas to incease the stength of composite cyline Von - Mises stess is also euce when the aspect atio inceases Vol, No, Sping 3

9 mikabi Intenational Jounal of Science& Reseach (Moeling, Ientification, Simulation & Contol) Exact Solution fo Electo- Themo- Mechanical Behavio of Composite Cyline Reinfoce by s une Non- xisymmetic Themo- Mechanical Loas Designes coul meet thei puposes by selecting the suitable pecent of fibe in composite cyline stength of composite cyline is euce with eceasing angle of s with espect to the longituinal axis σ Effective x /a Figue : The effect of volume faction of s on effective stess in composite cyline The effect of elastic meium on effective stess in composite cyline is investigate in Fig, whee the pesence of elastic meium has been consiee as a facto fo inceasing the stability of composite cyline Hee, the Pastenak founation has been selecte ue to consieing the shea foce s can be seen, inceasing Pastenak moulus eceases the effective stess σ Effective x /a Figue : The effect of elastic meium on effective stess in composite cyline Fig 3, illustates the effect of oientation angle of s on effective stess in the composite cyline The oientation angle of fibe in composite is vey impotant because it can be affecte the mechanical behavio of composite This composite has been einfoce by s that can be aligne in iffeent iection Since the pupose is stess analyzing, the angle is the best at which stess is euce an subsequently isplacement slake It can be conclue that ue to the significant aial piezoelecticity effect of, the Vol, No, Sping 3 Figue 3: The effect of oientation angle of s on aial isplacement in composite cyline - CONCLUSIONS nalytical solution, fo two - imensional static stesses fo RC cyline, was evelope une multi - physical fiels, whee s ae aange in a longitiuinal iection insie of PVDF matix The composite cyline is embee in an elastic meium as Pastenak founation which is consieing the shea effect Highe oe govening equations wee solve analytically by Fouie seies an the esults illustate the istibution of themo - mechanical stesses 3D figues show a hamonic patten in stess istibution in sipte of non - axisymmetic themo - mechanical loas that featue the behavio of composite cyline in iffeent an It can be obseve that using s as a fibe an its oientation angle with espect to the composite cyline axis have significant effects on the mechanical behavio of the RC cyline Similaly, it has been foun that the magnitue of the effective stess was stongly epenent on the volume faction of so that inceasing the volume content of s significantly leas to incease the stength of composite cyline 6- CKNOWLEDGEMENT The authos ae gateful to Univesity of Kashan fo suppoting this wok by Gant No 67/6 They woul like also to thank the Ianian Nanotechnology Development Committee fo thei financial suppot 3

10 mikabi Intenational Jounal of Science& Reseach (Moeling, Ientification, Simulation & Contol) Ghobanpou ani, E Haghpaast, Z Khoami Maaghi, S mi ppenix ppenix B 7- REFERENCES [] HL Dai, L Hong, YM Fu an X Xiao, nalytical solution fo electo - magneto - themo - elastic behavios of a functionally gae piezoelectic hollow cyline, pplie Mathematical Moelling, vol 3, pp 33-37, Febuay, [] R Poultangai, M Jabbai an MR Eslami, Functionally gae hollow sphees une non - axisymmetic themo - mechanical loas, Intenational Jounal of Pessue Vessels an Piping, vol 8, pp 9-3, May, 8 [3] Ghobanpou ani, Haghshen, S mi, M zami an Z Khoami Maaghi, Electo - Themo - Mechanical Response of Thick - Walle Piezoelectic Cyline Reinfoce by s, Jounal of Nanostuctues, vol, pp Vol, No, Sping 3

11 mikabi Intenational Jounal of Science& Reseach (Moeling, Ientification, Simulation & Contol) Exact Solution fo Electo- Themo- Mechanical Behavio of Composite Cyline Reinfoce by s une Non- xisymmetic Themo- Mechanical Loas 3 -, pil, [] Ghobanpou ani, S mi, V Saooghi an M Mohammaimeh, Themal Stess nalysis of a Composite Cyline Reinfoce with FG SWCNTs, Jounal of Soli Mechanics, vol 3, pp 3 -, pil, [] J Jafai Feshaaki, V Jafai Feshaaki, M Yazipoo an B Razavian, Two - imensional solution fo electo - mechanical behavio of functionally gae piezoelectic hollow cyline, pplie Mathematical Moelling, vol 36, pp - 33, Novembe, [6] tian, J Jafai Feshaaki, GH Mazoobi an M Sheiaee, Effects of Electic Potential on Themo - Mechanical Behavio of Functionally Gae Piezoelectic Hollow Cyline une Non - xisymmetic Loas, Wol caemy of Science, Engineeing an Technology, vol 9, pp , pil, [7] RB Hetnaski an MREslami, Themal Stesses - vance Theoy an pplications, vol I New Yok: Spinge, 9 [8] P Tan an L Tong, Mico - electo - mechanics moels fo piezoelectic - fibe - einfoce composite mateials, Composite Science Technology, vol 6, pp , pil, [9] P Tan an L Tong, Micomechanics moels fo nonlinea behavio of piezoelectic fibe einfoce composite mateials, Intenational Jounal of Solis an Stuctues, vol 38, pp , Decembe, [] Ghobanpou ani, R Shaai, S mi an Loghman, Electo - themo - mechanical nonlinea nonlocal vibation an instability of embee mico - tube einfoce by conveying flui, Physica E, vol, pp - 3, ugust, [] M Jabbai, S Sohabpou an MR Eslami, Geneal Solution fo Mechanical an Themal Stesses in a Functionally Gae Hollow Cyline ue to Non - axisymmetic Steay - State Loas, Jounal of pplie Mechanics, vol 7, pp - 8, Januay 3 [] Z Khoami Maaghi, Ghobanpou ani, R Kolahchi, S mi an MR Baghei, Nonlocal vibation an instability of embee DW conveying viscose flui, Composites: Pat B, vol, pp 3-3, Febuay, 3 [3] Ghobanpou ani, MR Mozianfa, V Saooghi, M Mohammaimeh an R Kolahchi, Magneto - themo - elastic Behavio of Cyline Reinfoce with FG SWCNTs une Tansient Themal Fiel, Jounal of Soli Mechanics, vol 3, pp -, Febuay, [] JE Mak, Polyme Data Han Book, vol I New Yok: Oxfo Univesity Pess, 999 Vol, No, Sping 3