Stochastic Dynamic Analysis of Nonlinear Vibration of Fluid-conveying Double-walled Carbon Nanotubes Based on Nonlocal Elasticity Theory

Transcription

1 Stochastc Dynamc Analyss of Nonlnear Vbraton of Flud-conveyng Double-alled Carbon Nanotubes Based on Nonlocal Elastcty Theory Ta-Png Chang Department of Constructon Engneerng, Natonal Kaohsung Frst Unversty of Scence and Technology, Kaohsung, Taan Abstract - Ths study deals th the stochastc dynamc behavors of nonlnear vbraton of the flud-conveyng double-alled carbon nanotubes (DWCNTs) by consderng the effects of the geometrc nonlnearty and the nonlnearty of van der Waals (vdw) force. Besdes, the small scale effects of the nonlnear vbraton of the DWCNTs are nvestgated by usng the theory of nonlocal elastcty. Some statstcal dynamc response of the DWCNTs such as the mean values and standard devatons of the ampltude of the dsplacement are computed, meanhle the effects of the flo velocty and small scale coeffcents on the statstcal dynamc response of the DWCNTs are nvestgated. t s concluded that the mean value and standard devaton of the ampltude of the dsplacement ncrease nonlnearly th the ncrease of the frequences and change slghtly as the flo velocty ncreases. Furthermore, small scale coeffcents have sgnfcant nfluence on the mean value and standard devaton of the ampltude of the DWCNTs. Keyords: Nonlnear vbraton; Double-alled carbon nanotubes; Stochastc dynamc response; Galerkn s method; Small scale effect; Nonlocal elastcty theory. ntroducton Snce the landmark paper publshed by jma [], carbon nanotubes (CNTs) have attracted orldde attenton due to ther potental use n the felds of chemstry, physcs, nano-engneerng, electrcal engneerng, materals scence, renforced composte structures and constructon engneerng. Carbon nanotubes (CNTs) are used for a varety of technologcal and bomedcal applcatons ncludng nanocontaners for gas storage and nanoppes conveyng fluds [-8]. Some mportant applcatons of carbon nanotubes (CNTs) are such as nanotubes conveyng fluds [,7-8], dfferent types of flud flos lke ater [9], dynamc flo of methane, ethane and ethylene molecules [] and the dffusve transport of lght gases [] had been reported, and the effects of these fluds on the mechancal propertes of CNTs had been nvestgated. Generally there are to methods dely adopted to study the CNTs conveyng fluds. One s the molecular dynamcs smulatons (MDS) [-], hoever, MDS needs a tremendous amount of computatonal tme and effort so that only a very small system can be tackled. The other s the contnuum mechancs model. Natsuk et al. [] adopted a smplfed Flügge shell model to nvestgate the ave propagaton of sngle- and double-alled CNTs conveyng flud. The sngle-elastc beam model [-4] and the multpleelastc beam model [5-9] ere also broadly adopted to study the dynamc behavors of flud-conveyng sngle-alled carbon nanotubes (SWCNTs) and mult-alled carbon nanotubes (MWCNTs). The vbraton frequences of the lnear system and the system s stablty related to the nternal movng flud ere nvestgated. Moreover, the nonlocal elastcty theory as ncorporated nto the elastc beam model to study the small scale effect on the dynamcs of SWCNT conveyng flud []. Chang and Lu [-] studed small scale effects on the flo-nduced nstablty of double-alled carbon nanotubes (DWCNTs) by usng the nonlocal elastcty theory. More recently, Chang [-4] nvestgated the thermal-mechancal vbraton and nstablty of fludconveyng sngle-alled carbon nanotubes (SWCNTs) based on nonlocal elastcty theory. Generally speakng, the beam models mentoned above are lnear; hoever, the vdw forces n the nterlay space of MWCNTs are essentally nonlnear. Furthermore, the slender ratos are normally large f the beam models are adopted, that s, the large deformaton ll occur. Therefore, t s qute essental to consder to types of nonlnear factors, namely, the geometrc nonlnearty and the nonlnearty of vdw force n nvestgatng the dynamc behavors of flud-conveyng MWCNTs. Kuang et al. [5] nvestgated the dynamc behavors of double-alled carbon nanotubes (DWCNTs) conveyng flud by consderng to types of nonlneartes mentoned above. Salvetat et al. [6] measured the flexural Young s modulus and shear modulus usng AFM test on clamped clamped nanoropes, gettng values th 5% of error. nformaton related to statstcal dstrbutons of expermental data s also rare, and the mportant study from Krshnan et al. [7] provdes one of the fe examples avalable of hstogram dstrbuton of the flexural Young s modulus derved from 7 CNTs. The Young s modulus as estmated observng free-standng vbratons at room temperature usng transmsson electromcroscope (TEM), th a mean value of. TPa -.4 TPa/+.6 TPa. Pronouncedly, n [8], stochastcally averaged probablty ampltude for the vbraton modes s computed to obtan the rms vbraton profle along the length of the tubes. Uncertanty s also assocated to the equvalent atomstc-contnuum models adopted extensvely n partcular by the engneerng and materals scence communtes. Hence, to be realstc, the Young s modulus of elastcty of carbon

2 nanotube (CNTs) should be consdered as stochastc th respect to the poston to actually descrbe the random property of the CNTs under certan condtons. n the present study, e nvestgate the stochastc dynamc behavors of nonlnear vbraton of the double-alled carbon nanotubes (DWCNTs) conveyng flud by consderng the effects of the geometrc nonlnearty and the nonlnearty of van der Waals (vdw) force. n addton, the small scale effects on the nonlnear vbraton of the DWCNTs are studed by usng the theory of nonlocal elastcty. Based on the Hamlton s prncple, the nonlnear governng equatons of the fludconveyng double-alled carbon nanotubes are formulated. The Young s modulus of elastcty of the DWCNTs s consdered as stochastc th respect to the poston to actually characterze the random materal propertes of the DWCNTs. The effects of the flo velocty and small scale coeffcents on the statstcal dynamc response of the DWCNTs are nvestgated. Nonlnear beam model for flud-conveyng DWCNTs ux, z, t u x, t z x x z t,, x, t u x x here x s the axal coordnate, t s tme, u and () denote the total dsplacements of the th tube along the x coordnate drectons, u and defne the axal and transverse dsplacements of the th tube on the neutral axs, the correspondng total stran, and the subscrpt = and =. Notce that tube s the nner tube hle tube s the outer tube. Based on Eq. (), the potental energy V stored n a DWCNTs and the vrtual knetc energy T n the DWCNTs as ell as the flud nsde the DWCNTs can be ndvdually determned. Based on Hamlton s prncple, the varatonal form of the equatons of moton for the DWCNTs can be gven by t t V T dt () Fg.. Double-alled carbon nanotubes conveyng flud. n Fg., the double-alled carbon nanotubes (DWCNTs) s modeled as a double-tube ppe hch s composed of the nner tube of radus R and the outer tube of radus R. The thckness of each tube s h, the length s L, and Young s modulus of elastcty s E. t s noted that the Young s modulus of elastcty E s assumed as stochastc th respect to the poston to actually descrbe the random materal property of the DWCNTs. The nternal flud s assumed to flo steadly through the nner tube th a constant velocty U. Besdes, the boundary condtons of the DWCNTs are assumed as smply-supported at both ends. Based on the theory of Euler Bernoull beam and a nonlnear stran dsplacement relatonshp of Von Karman type, the dsplacement feld and stran dsplacement relaton can be rtten as follos: here s the vrtual ork due to the vdw nteracton and the nteracton beteen tube and the flong flud. Based on Eq. () and the formulatons derved by Chang [, 4], the coupled nonlnear governng equatons for the free vbraton of DWCNTs conveyng flud based on nonlocal elastcty theory are gven as follos: 4 4 ( ) ( ) E x e a MU eoa MU eoa M m x x x t x t E( x) A MU MU MU L + dx + x x L L x x x ( M+ m ) +MU MU () t xt t x x = -(e a) c ( ) c -(e a) -(e a) x x x 4 ( ) Ex m ea m dx x x t x t x L x = -(-(ea) )( c ( )) c-(e a) -(ea) (4) x x x L E( x) A t s noted that the scale ea n the Eq. (-4) ll lead to small scale effect on the response of structures n nano-sze. n Eqs. (-4), t s assumed that the small scale effects on the

3 nonlnear terms due to geometrcal nonlnearty are neglected snce they are normally small compared th those on the lnear terms. Stochastc dynamc analyss of nonlnear vbraton of DWCNTs n the present study, the Young s modulus of elastcty E(x) s consdered as stochastc th respect to the poston to actually characterze the random propertes of the DWCNTs and t s assumed as Gaussan dstrbuted. Applyng the perturbaton technque on the Young s modulus of elastcty E(x), the follong equatons can be rtten: E( x) E ( x) E ( x) (5) here E ( x ) s the mean value of the Young s modulus of elastcty E(x), s a zero-mean small parameter, and E ( x) s the frst varaton of the Young s modulus of elastcty E(x). Smlarly, the dsplacement ( x), ( x) of the DWCNTs can be rtten as follos: ( x) ( x) ( x) (6) ( x) ( x) ( x) (7) here ( x), ( x ) are the mean values of dsplacement of the nner and outer tubes separately. Substtutng Eqs. (5-7) nto Eqs. (-4), e can obtan the follong coupled equatons based on the zero order of : 4 MU E A L 4 ( ) ( ) ( ) ( ) E MU dx x x L L x x MU ( ) ( M m ) (8) x x t MU MU ( )( )( ) xt t x x c ( ) c ( ) 4 E A L 4 ( ) ( ) ( ) E m dx x t L x x (9) c ( ) c ( ) Frst of all, e have to solve, n Eqs. (8-9). By applyng the harmonc balance method and Galerkn s method and substtutng A ( x)sn( t) (,) nto Eqs. (8-9), after some tedous dervatons the relatonshp beteen the ampltude A and the resonant frequency ω of the loestorder mode ( x) can be acheved as follos GA GA G( A A) G4( A A) () GAGA G( AA) G( AA) () here G are constants hch can be determned by performng the ntegraton. After solvng coupled Eqs. (8-9) for the ampltudes A, A, e can obtan, readly. Substtutng, nto nonlnear coupled dfferental equatons based on the frst order of, and adoptng the same technque for solvng,, fnally e can obtan, thout any dffcultes except the dervatons are somehat lengthy. 4 Numercal examples and dscusson n the numercal computatons, the smply supported boundary condton s consdered for the DWCNTs conveyng flud. The nner and the outer tubes are assumed to have the same Young s modulus, the same thckness and the same mass densty. The numercal values of the parameters are adopted as follos: Mean value of Young s modulus E= Tpa, tube thckness h=.4 nm, mass densty Kg / m, the mass densty of ater flo s f Kg / m, the nner radus R. 7nm and the outer radus R. 4 nm and mean square values of s assumed as E.. The velocty of the flud s assumed as U=4 m/sec unless t s specfed otherse. Frst of all, e examne the effect of the nonlnearty on the ampltude-frequency propertes of the nonlnear vbraton. The relatons of the mean value of ampltude versus frequency are depcted n Fg.. t can be seen that the mean value of the ampltude ncreases th the ncrease of the frequences. t s completely reasonable that the relaton beteen the mean value of the ampltude and the frequency s nonlnear; n addton, the mean value of the ampltude of the outer tube s larger than that of the nner tube. Furthermore, t s noted that the mean value of the ampltude gets smaller as the small scale coeffcent eancreases for the fxed frequency. n Fg., the standard devaton of the ampltude s plotted th respect to the frequency. As t can be found from the fgure that the standard devaton of the ampltude ncreases nonlnearly th the ncrease of the frequences, and t s noted that the standard devaton of the ampltude of the outer tube s larger than that of the nner tube. n Fg. 4, the coeffcent of varaton (COV) of the ampltude s depcted th respect to the frequency. t s notced that the

4 coeffcent of varaton of the ampltude of the nner tube s around., hoever, the coeffcent of varaton of the ampltude of the outer tube s around.. Fnally, Fg. 5 presents the coeffcent of varaton of ampltude versus frequency th ea / L. for dfferent values of flo velocty. t s found that the coeffcent of varaton of ampltude fluctuates beteen. and. and no specfc relaton beteen COV and flo velocty can be establshed despte they are correlated. Therefore, based on the results from Fgs. -5, t can be concluded that the small scale coeffcent has sgnfcant nfluence on the mean value, standard devaton and coeffcent of varaton of the ampltude of the DWCNTs, hoever, the flo velocty has only a lttle effect on the stastcal dynamc response of the DWCNTs. COV of ampltude COV of A a/l=..6 COV of A a/l=..4 COV of A a/l=. COV of A a/l=.. COV of A a/l=. COV of A a/l= x Mean value of ampltude (nm) A a/l=. A a/l=. A a/l=. A a/l=. A a/l=. A a/l= x Fg.. Mean value of ampltude versus frequency for dfferent values of ea / L. Standard devaton of ampltude (nm) SD of A a/l=. SD of A a/l=. SD of A a/l=. SD of A a/l=. SD of A a/l=. SD of A a/l= x Fg.. Standard devaton of ampltude versus frequency for dfferent values of ea / L. Fg. 4. Coeffcent of varaton of ampltude versus frequency for dfferent values of ea / L. COV of ampltude A, U=4 m/s.6 A, U=4 m/s.4 A, U=8 m/s A, U=8 m/s. A, U=6 m/s A, U=6 m/s x Fg. 5. Coeffcent of varaton of ampltude versus frequency th ea / L. for dfferent values of flo velocty. 5 Conclusons n the present study, e nvestgate the stochastc dynamc behavors of nonlnear vbraton of the doublealled carbon nanotubes (DWCNTs) conveyng flud by consderng the effects of the geometrc nonlnearty and the nonlnearty of van der Waals (vdw) force. n addton, the small scale effects of the nonlnear vbraton of the DWCNTs are studed by usng the theory of nonlocal elastcty. Based on the Hamlton s prncple, the nonlnear governng equatons of the flud-conveyng double-alled carbon nanotubes are formulated. The Young s modulus of elastcty of the DWCNTs s consdered as stochastc th respect to the

5 poston to actually characterze the random materal propertes of the DWCNTs. By usng the perturbaton technque, the nonlnear governng equatons of the fludconveyng double-alled carbon nanotubes can be decomposed nto to sets of nonlnear dfferental equatons nvolvng the mean value of the dsplacement and the frst varaton of the dsplacement separately. Then the harmonc balance method and Galerkn s method are adopted to solve the nonlnear dfferental equatons successvely. Some statstcal dynamc response of the DWCNTs such as the mean values and standard devatons of the ampltude of the dsplacement are calculated, meanhle the effects of the flo velocty and nonlocal scale coeffcents on the statstcal dynamc response of the DWCNTs are nvestgated. t can be concluded that the mean value and standard devaton of the ampltude of the dsplacement ncrease nonlnearly th the ncrease of the frequences. Besdes, these stochastc dynamc responses change slghtly as the flo velocty ncreases, furthermore, they are smaller as the small scale coeffcents get larger. Hoever, as the values of flo velocty or small scale coeffcents ncrease, the coeffcents of varaton (COV) of the ampltude of the dsplacement reman almost constant and stay thn certan range th respect to the frequency. t s noted that the computed stochastc dynamc response plays an mportant role n estmatng the structural relablty of the DWCNTs. Acknoledgments Ths research as partally supported by the Natonal Scence Councl n Taan through Grant NSC-99--E-7-. The author s grateful for ths support. 6 References [] S. jjma, Helcal mcrotubules of graphtc carbon, Nature 54 (99) [] E. Evans, H. Boman, A. Leung, D. Needham, D. Trrell, Bomembrane templates for nanoscale conduts and netorks, Scence 7 (996) [] G.E. Gadd et. al., The World s Smallest Gas Cylnders?, Scence 77 (997) [4] G. Che et. al., Carbon nanotubule membranes for electrochemcal energy storage and producton, Nature 9 (998) [5] J. Lu et. al., Fullerene Ppes, Scence 8 (998) [6] A. Karlsson et. al., Netorks of nanotubes and contaners, Nature 49 () 5-5. [7] Y. Gao,Y. Bando, Carbon nanothermometer contanng gallum, Nature 45 () 599. [8] G. Hummer, J.C. Rasaah, J.P. Nooryta, Water conducton through the hydrophobc channel of a carbon nanotube, Nature 44 () [9]. Hanasak, A. Nakatan, Water flo through carbon nanotube junctons as molecular convergent nozzles, Nanotechnology 7 (6) [] Z. Mao, S.B. Snnott, A computatonal study of molecular dffuson and dynamc flo through carbon nanotubes, J. Phys. Chem. B 4 () [] A. Skouldas, D.M. Ackerman, K.J. Johnson, D.S. Sholl, Rapd transport of gases n carbon nanotubes, Phys. Rev. Lett. 89 () [] T. Natsuk, Q.Q. N, M. Endo, Wave propagaton n sngle- and double- alled carbon nanotubes flled th fluds, J. Appl. Phys. (7) [] J. Yoon, C.Q. Ru, A. Moduchosk, Flo-nduced flutter nstablty of cantlever CNTs, nt. J. Solds Struct. 4 (6) [4] L. Wang, Q. N, On vbraton and nstablty of carbon nanotubes conveyng flud, Comput. Mater. Sc. 4 (8) [5] X.Q. He, C.M. Wang, Y. Yan, L.X. Zhang, G.H. N, Pressure dependence of the nstablty of multalled carbon nanotubes conveyng fluds, Arch. Appl. Mech. 78 (8) [6] Y. Yan, X.Q. He, L.X. Zhang, C.M. Wang, Dynamc behavor of trple-alled carbon nanotubes conveyng flud, J. Sound Vb. 9 (9) 8. [7] L. Wang, Q. N, M. L, Q. Qan, The thermal effect on vbraton and nstablty of carbon nanotubes converyng flud, Physca E 4 (8) [8] Y. Yan, W.Q. Wang, L.X. Zhang, Dynamcal behavors of flud-conveyed mult-alled carbon nanotubes, Appl. Math. Modell. (9) [9] L. Wang, Q. L, M. L, Bucklng nstablty of doubleall carbon nanotubes conveyng flud, Comput. Mater. Sc. 44 (8) [] H. Lee, W. Chang, Comment on Free transverse vbraton of the flud-conveyng sngle-alled carbon nanotube usng nonlocal elastc theory, J. Appl. Phys. (8) 4-4. [] T.P. Chang, M.F. Lu, Flo-nduced nstablty of double-alled carbon nanotubes based on nonlocal elastcty theory, Physca E 4 () [] T.P. Chang, M.F. Lu, Small scale effect on flo-nduced nstablty of double-alled carbon nanotubes, Eur. J. Mech. A. Solds () [] T.P. Chang, Thermal-nonlocal vbraton and nstablty of sngle-alled carbon nanotubes conveyng flud, J. Mech. 7 () [4] T.P. Chang, Thermal-mechancal vbraton and nstablty of a flud-conveyng sngle-alled carbon nanotube embedded n an elastc medum based on nonlocal elastcty theory, Appl. Math. Model. 6 () [5] Y.D. Kuang et al, Analyss of nonlnear vbratons of double-alled carbon nanotubes conveyng flud, Comput. Mater. Sc. 45 (9) [6] J.P. Salvetat, J.A.D. Brggs, J.M. Bonard, R.R. Bacsa, A.J. Kulk, T. Stöckl, N.A. Burnham, L. Forró, Elastc

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