Vibration Analysis Of Carbon Nanotube Using Continuum Model And Finite Element Model

Transcription

1 Univerity of Central Florida Eletroni Thee and Diertation Mater Thei Open e Vibration nalyi Of Carbon Nanotube Uing Continuum Model nd Finite Element Model 005 Hari Subramaniam Univerity of Central Florida Find imilar ork at: Univerity of Central Florida Librarie Part of the Mehanial Engineering Common STRS Citation Subramaniam, Hari, "Vibration nalyi Of Carbon Nanotube Uing Continuum Model nd Finite Element Model" 005. Eletroni Thee and Diertation Thi Mater Thei Open e i brought to you for free and open ae by STRS. It ha been aepted for inluion in Eletroni Thee and Diertation by an authorized adminitrator of STRS. For more information, pleae ontat lee.doton@uf.edu.

2 VIBRTION NLYSIS OF CRBON NNOTUBES USING CONTINUUM MODEL ND FINITE ELEMENT MODEL by HRI SUBRMNIM B.E. Bharathair Univerity, India 00 thei ubmitted in partial fulfillment of the requirement for the degree of Mater of Siene from the Department of Mehanial, Material and eropae Engineering in the College of Engineering and Computer Siene at the Univerity of Central Florida Orlando, Florida Fall Term 005

3 005 Hari Subramaniam ii

4 BSTRCT The main objetive of the thei i to propoe the method of determining vibration behavior of arbon nanotube CNT uing ontinuum model and finite element model. Seondary objetive i to find the effet of van der aal fore on vibration of multialled arbon nanotube. The tudy of vibration behavior of CNT i important beaue of their potential engineering appliation uh a nano-mehanial reonator and tip of anning probe intrument here they are ubjeted to mehanial vibration. Continuum modeling i baed on an elati beam model. The interlayer van der aal interation are repreented by Lennard- Jone potential. In finite element modeling, ingle alled nanotube SNT are modeled a finite beam element and multi-alled nanotube MNT a finite olid element. The interlayer van der aal interation are imulated by ditributed pring. The propoed finite element approah and ontinuum approah for vibration analyi of CNT are verified by omparing the reult ith eperimental and analytial reult available in the literature. The reult from both ontinuum and finite element modeling ho that the effet of van der aal fore on vibration of MNT are high for maller apet ratio irrepetive of boundary ondition and number of layer; fied nanotube than antilever nanotube for the ame dimenion ; five-alled nanotube than a double alled nanotube for the ame apet ratio. iii

5 CKNOLEDGMENTS I ould like to take thi opportunity to epre my inere appreiation to my profeor and advior, Dr. Quan. ang, ho ha helped and guided me throughout thi reearh. I am grateful to Dr. David Niholon ho taught me finite element tehnique and Dr. Rihard P. Zarda ho taught me the kill in finite element modeling. I ould like to thank Dr. Ranganathan Kumar ho helped ith finanial aitanthip for my graduate tudie. I ould alo like to thank my thei defene ommittee member; Dr. Kuo-Chi Kurt Lin and Dr. Gangyi Zhou. Lat but not leat, I ould like to thank my parent, my family and my friend for their enduring upport in my aademi puruit. iv

6 TBLE OF CONTENTS LIST OF FIGURES... vii LIST OF TBLES...viii CHPTER INTRODUCTION.... Struture of CNT..... Type of CNT.... Manufature of CNT ppliation of CNT... 6 CHPTER LITERTURE REVIE Literature Revie on Modeling Tehnique Need for Vibration nalyi.... Literature Revie on Vibration nalyi of CNT.... Propoed Reearh... CHPTER CONTINUUM MODELING.... Introdution.... Bulk Propertie of CNT Van der aal Fore Bending Rigidity..... Ma Denity Continuum Model of CNT Continuum Model of Single alled CNT... 0 v

7 .. Continuum Model of Double alled CNT..... Continuum Model of Five alled CNT... 5 CHPTER FINITE ELEMENT NLYSIS.... Introdution.... Bulk Propertie of Nanotube in FE Modeling Finite Element Modeling..... FE Model of Single alled CNT..... FE Model of Double alled CNT..... FE Model of Five alled CNT Eigen Value Etration Frequeny Calulation Frequeny of Single alled CNT Frequeny of Double alled CNT... 9 CHPTER 5 RESULTS ND DISCUSSION Continuum Model Reult Finite Element Model Reult... 5 CHPTER 6 CONCLUSION... 6 CHPTER 7 FUTURE ORK REFERENCES vi

8 LIST OF FIGURES Figure I Single alled CNT... Figure II Single and Multi alled Nanotube... Figure III Chiral Vetor of a Nanotube... Figure IV Type of Single alled CNT... 5 Figure V Type of Modeling of CNT... 0 Figure VI beam in bending... Figure VII Interlayer Spaing of djaent Nanotube... 0 Figure VIII B Beam Element... Figure IX Finite Element Model of SNT... Figure X CD Linear Brik Element... Figure XI ial Spring Element... Figure XII Finite Element Model of DNT... Figure XIII Finite Element Model of Five alled Nanotube... 5 Figure XIV FE model for Single and Multi alled nanotube... 6 Figure XV Firt five mode of antilevered SNT diameter=.50 nm and length = 6. nm. 5 Figure XVI Firt three mode of fied - fied double alled arbon nanotube Figure XVII Firt three mode of antilevered double alled arbon nanotube... 5 Figure XVIII Frequenie of Fied DNT ith inner diameter 0.7 nm. and the outer diameter. nm Figure XIX Frequenie of Cantilever DNT ith inner diameter 0.7 nm. and the outer diameter. nm vii

9 LIST OF TBLES Table I Frequenie of fied DNT ith inner diameter 0.7nm, outer diameter.nm and =e0 N/m... Table II Frequenie of fied DNT ith inner diameter 0.7nm and outer diameter.nm... Table III Frequenie of fied DNT ith inner diameter 0.7nm and outer diameter.nm. Table IV Frequenie of antilever DNT ith inner diameter 0.7nm, outer diameter.nm and =e0 N/m... Table V Frequenie of antilever DNT ith inner diameter 0.7nm and outer diameter.nm... 5 Table VI Frequenie of antilever DNT ith inner diameter 0.7nm and outer diameter.nm... 6 Table VII Frequenie of fied five alled nanotube ith inner diameter 0.7 nm, outer diameter.5 nm and L/d 5 = 0... Table VIII Frequenie of fied five alled nanotube ith inner diameter 0.7 nm, outer diameter.5 nm and L/d 5 = Table IX Frequenie of fied five alled nanotube ith inner diameter 0.7 nm, outer diameter.5 nm and L/d 5 = Table X Frequenie of fied five alled nanotube ith inner diameter 0.7 nm and the outer diameter.5 nm... 5 Table XI Frequeny of antilever SNT... 5 Table XII Frequeny of antilever SNT... 5 Table XIII Frequeny of antilever SNT... 5 Table XIV Frequenie of antilever SNT diameter =.50 nm and length = 6. nm... 5 viii

10 Table XV Frequenie of fied DNT ith inner diameter 0.7 nm. and the outer diameter. nm Table XVI Frequenie of fied DNT ith inner diameter 0.7 nm. and the outer diameter. nm Table XVII Frequenie of antilever DNT ith inner diameter 0.7nm and outer diameter.nm Table XVIII Frequenie of Cantilever DNT ith inner diameter 0.7 nm. and the outer diameter. nm Table XIX Frequenie of five alled nanotube ith inner diameter 0.7 nm and the outer diameter.5 nm... 6 Table XX Frequenie of five alled nanotube ith inner diameter 0.7 nm and the outer diameter.5 nm... 6 Table XXI Frequenie of five alled nanotube ith inner diameter 0.7 nm and the outer diameter.5 nm... 6 i

11 CHPTER INTRODUCTION DREXLER ay, Coal and diamond, and and omputer hip, aner and healthy tiue: throughout the hitory, variation in the arrangement of atom have ditinguihed the heap from the herihed, the dieaed from the healthy. rranged one ay the atom make up oil, air and ater: arranged other they make ripe traberrie. rranged one ay they make up home and freh air, arranged another they make up ah and moke! Our ability to arrange atom lie at the foundation of nanotehnology Iijima diovered CNT in 99. The prefi nano orrepond to a bai unit on a length ale, meaning 0 9 meter, hih i a hundred to a thouand time maller than a typial biologial ell or baterium. t the nanometer length ale, the dimenion of the material and devie begin to reah the limit of 0 to 00 of atom, herein entirely ne phyial and hemial effet are oberved; and poibilitie arie for the net generation of utting-edge produt baed on the ultimate miniaturization or o alled nanoization of the tehnology. CNT have a variety of appliation beaue of their ditintive moleular truture and their fainating mehanial and eletrial propertie.

12 . Struture of CNT The truture of CNT an be onidered a ariing from the folding of one or more layer of graphite to form a ylinder ompoed of arbon heagon. Thee nanotube have a hemipherial "ap" at eah end of the ylinder a hon in the figure I. They are light, fleible, thermally tabile, and are hemially inert. Figure I Single alled CNT Nanotube are ompoed entirely of p bond, hih are tronger than the p bond found in diamond. Thi bonding truture provide them ith their unique trength. Nanotube align themelve into rope held together by van der aal fore. Under high preure, nanotube an merge together, trading ome p bond for p bond, giving great poibility for produing trong, unlimited-length ire through high-preure nanotube linking.

13 .. Type of CNT Baially nanotube are to type depending upon the layer a hon in figure II. They are. Single alled nanotube SNT. Multi alled nanotube MNT Figure II Single and Multi alled Nanotube Nanotube an be ategorized into three type aording to their truture. They are. rm-chair. Chiral. Zigzag Nanotube form different type, hih an be deribed by the hiral vetor n, m, here n and m are integer of the vetor equation C h = na ma. The hiral vetor i determined a hon by the figure III.

14 Figure III Chiral Vetor of a Nanotube Imagine that the nanotube i unraveled into a planar heet. Dra to ontinuou line along the tube ai here the eparation take plae. In other ord, if e ut along the to ontinuou olid line and then math their end together in a ylinder, you get the nanotube that you tarted ith. No, find any point on one of the red line that interet one of the arbon atom point. Net, dra the rmhair line n, n hih travel aro eah heagon, eparating them into to equal halve. No that you have the armhair line dran, find a point along the other tube ai that interet a arbon atom nearet to the rmhair line point B. No onnet and B ith our hiral vetor C h.the rapping angle; not hon i formed beteen C h and the rmhair line. If C h lie along the rmhair line =0, then it i alled an "rmhair" nanotube. If =0, then the tube i of the "zigzag" type. Otherie, if 0 <<0 then it i a "hiral" tube. The

15 value of n and m determine the hirality, or "tit" of the nanotube. The hirality in turn affet the ondutane of the nanotube, it denity, it lattie truture, and other propertie. Figure IV ho the different type of arbon nanotube aording to their truture. Figure IV Type of Single alled CNT. Manufature of CNT CNT an be manufatured by the folloing method a. r diharge b. Laer blation. Chemial Vapour Depoition 5

16 r diharge involve an eletrial diharge from a arbon-baed eletrode in a uitable atmophere to produe both ingle and multi-all tube of high quality but in lo quantitie. Laer ablation ue a high-poer laer to vaporie a graphite oure loaded ith a metal atalyt. The arbon in the graphite reform a predominantly ingle-all nanotube on the metal atalyt partile. Chemial vapour depoition CVD i here a hydroarbon feedtok i reated ith a uitable metal-baed atalyt in a hot furnae to gro nanotube hih are ubequently removed from the ubtrate and atalyt by a imple aid ah. In a typial hemial vapor depoition proe the ubtrate i epoed to one or more volatile preuror, hih reat and/or deompoe on the ubtrate urfae to produe the deired depoit. The hemial vapor depoition method ha hon the mot promie in being able to produe larger quantitie of nanotube ompared to the other method at loer ot.. ppliation of CNT CNT ehibit unique eletroni and mehanial propertie beaue of their urvature. Beaue of their inimitable propertie, CNT find a number of intereting appliation in different field of engineering. Some of the appliation diued by jayan et al. 00 are a follo 6

17 a. Carbon nanotube have the right ombination of propertie nanometer ize diameter, trutural integrity, high eletrial ondutivity, and hemial tability that make good eletron emitter. b. Prototype matri-addreable diode flat panel diplay an be fabriated uing CNT a the eletron emiion oure.. Nanotube an be ued a reinforement in ompoite material. Nanotube reinforement ill inreae the toughne of the ompoite by aborbing energy during their highly fleible elati behavior. d. Nanotube filled polymer an be ued in eletromagneti indution EMI hielding appliation. e. Beaue of it hollo geometry and nano ale diameter, it ha been predited that the arbon nanotube an tore liquid and ga in the inner ore through a apillary effet. f. CNT beaue of their etremely mall ize, high ondutivity, high mehanial trength and fleibility, they are ued in STM, FM intrument a ell a other anning probe intrument, uh a an eletrotati fore miroope. g. MNT and SNT tip ere ued in a tapping mode to image biologial moleule uh a amyloid-b-protofibril ith reolution never ahieved before. h. Nanotube ith ontrolled heliitie ould be ued a unique probe for moleular reognition, baed on the heliity and dimenion, hih are reognized by organi moleule of omparable length ale. i. CNT have relatively traight and narro hannel in their ore hih an be filled ith foreign material to fabriate one-dimenional nanoire. 7

18 j. Sine the eletrial reitivitie of SNT ere found to hange enitively on epoure to gaeou ambient, hene CNT an be ued a hemial enor. They an be ued to diipate heat from tiny omputer hip. Nanotube ompoite motor bruhe are better lubriated, tronger and more aurately moldable. CNT have already been ued a ompoite fiber in polymer and onrete to improve their mehanial, thermal and eletrial propertie of the bulk produt. Nanotube are ritial material that enable ontrution of pae elevator from earth to geoynhronou orbit. Gao et al., 000 found that CNT have the highet reverible apaity of any arbon material for ue in lithium-ion batterie. Beaue of their negligible eight, they find appliation in pae appliation. Sine nanotube are imilar ale ize of DN, promiing poibilitie an be epeted by introduing them to reinfore tiue affold. CNT have a high urfae area and their ability to attah to any hemial peie to their ideall provide an opportunity for unique atalyt upport.

19 CHPTER LITERTURE REVIE. Literature Revie on Modeling Tehnique Etenive reearh ha been arried out in the field of nanotube. The firt method that ha been ued by reearher to tudy the propertie of CNT i eperimental method Ponharal et al., 999; Krihnan et al., 99. Though the eperimental method gave aurate reult, it i very time onuming and otlier. The eond method that ha been ued by reearher i atomi modeling like moleular dynami Iijma et al., 996 ine nanoeffet and aurate olution for nano-ale ize problem are poible by thi modeling. But Moleular Dynami MD imulation are only limited to ytem ith a maller number of moleule and atom. So only ingle alled nanotube ith mall ize an be imulated uing moleular dynami. The third method that ha been ued by reearher in tudying CNT i ontinuum modeling Yakobon et al., 996; Ru et al., 000; Yoon et al., 00; ang et al., 005. The main iue in tudying CNT uing ontinuum model a determining the bulk propertie at maroopi level orreponding to moleular propertie of CNT. ang et al. 005 eplored and developed benhmark for the appliability of hell model and beam model in CNT tability reult. They alo propoed an independent fleural tiffne ontant of a nanotube hen elati beam model i ued. Ru et al. 000 ued a ontinuum beam model to tudy the olumn bukling of nanotube. Yakobon et al. 996 ued a ontinuum hell model to tudy bukling of nanotube. 9

20 Different type of CNT model uh a moleular dynami model, ontinuum hell model and ontinuum olid model are hon in the figure V. Figure V Type of Modeling of CNT Harik et al. 00 tudied about the appliability of ontinuum beam model for tudying the behavior of CNT. He onluded that ontinuum beam model an be ued for the qualitative analyi of CNT hen the ratio of length and diameter of nanotube i greater than 0. the number of all of CNT inreae it beome omple to tudy it behavior uing eperiment, moleular dynami and ontinuum model. The fourth method that ha been ued by reearher in tudying CNT i finite element method. Very little ork had been done in the field of finite element analyi of CNT. Pantano et al. 00 modeled individual tube a hell finite element and the effet of van der aal fore ere imulated uing peial interation element. He tudied the mehani of 0

21 rinkling of multialled CNT demontrating the role of multialled hell truture and interall van der aal interation in governing buking and potbukling behavior. Liu et al. 00 ued a olid ylindrial finite element model for tudying the rippling of a nanobeam under pure bending.. Need for Vibration nalyi CNT are idely ued a tip of FM, STM intrument a ell a other anning probe intrument. the nanotube tip touhe the ample, it indue mehanial vibration. Hene there i a greater need to tudy the vibration behavior of ingle and multi alled CNT. CNT are alo ued a nanomehanial reonator. Sno et al 00 tudied the tability of imaging uing ingle-all CNT a probe for atomi fore miroopy. They uggeted that thiker, multialled probe or very hort ingle-all probe etending from nanotube bundle might be better for imaging highly tetured urfae.. Literature Revie on Vibration nalyi of CNT Reearher have been uing to method to tudy the vibration behavior of CNT namely eperimental method and ontinuum modeling method. Ponharal et al., 999 and Krihnan et al., 99 ued eperimental method. Ponharal et al. 999 meaured the fundamental reonane frequeny of multialled CNT indued by an alternating eletri field in a tranmiion eletron miroope and they then alulated the aial elati modulu uing the modulu frequeny

22 relation reulting from the laial analyi of linear elatiity for antilevered beam. Krihnan et al. 99 etimated the tiffne of ingle alled CNT by oberving their free tanding room temperature vibration in a tranmiion on eletron miroope. The nanotube dimenion and vibration amplitude ere meaured from eletron mirograph and it a aumed that the vibration mode ere driven tohatially and ere hoen of a lamped antilever. Yoon et al. 00 and ang et al. 005 ued ontinuum model for tudying vibration behavior of CNT. Yoon et al. 00 ued a multiple elati model to tudy vibration behavior of double and five-alled nanotube. They alulated non-oaial reonant frequenie and the aoiated non-oaial vibration mode. They found that the firt fe nonoaial reonant frequenie are found to be inenitive to vibration mode, length of MNT and the end ondition, hile they dereae ith the number of neted layer. They alo found that internal non-oaial reonane ill be eited at the high natural frequenie, and MNT annot maintain their onentri truture at ultra high frequenie. ang et al. 005 found the frequenie of antilever SNT and ompared the reult ith the eperimental reult of Krihnan et al. 99.

23 . Propoed Reearh The reearh in thi thei propoe to effetive method of determining natural frequenie of SNT and MNT. The firt method i uing ontinuum model. The propoed ontinuum model unlike ontinuum model of Yoon et al. 00 doen t ue traight normal potulate for finding the bending rigidity of CNT intead it alulate the bending rigidity by the method propoed by ang et al The eond method propoed for determining natural frequenie of CNT i finite element modeling, here ingle alled nanotube SNT are modeled a finite beam element and multi-alled nanotube MNT a finite olid element. The propoed finite element model for MNT i imple and effetive. The model i imple beaue the van der aal fore i imulated uing ditributed pring and the model i effetive beaue it alo predit the effet of van der aal fore on vibration of MNT.

24 CHPTER CONTINUUM MODELING. Introdution CNT an be onidered a Euler-Bernoulli beam to tudy it vibration behavior. ording to Euler-Bernoulli beam theory, the equation of motion for a fored vibration of a uniform beam a hon in figure VI i given by Figure VI beam in bending,,, t f t t I E b b b b b b b =

25 here E b i the young modulu, I b i the moment of inertia of the beam ro-etion about y- ai, i the ma denity, f b, t i the eternal fore, b, t i the fleural defletion of b the beam and i the area of the ro-etion of the beam. The governing equation for a SNT to tudy it natural frequenie i given from Euler-Bernoulli beam model hen f, t = 0 no eternal fore i ubtituted in eq, t, t EI = 0. here E i the young modulu of SNT, I i the moment of inertia of SNT, i the ma denity of SNT,, t i the fleural defletion of SNT and i the area of the roetion of SNT For a MNT, a multiple elati beam model i uggeted in hih eah of the neted nanotube i deribed a an individual elati beam and the defletion of all neted tube are oupled through the van der aal interation beteen any to adjaent tube. The fore term i given by produt of van der aal interation o-effiient and the defletion. The governing equation for a MNT from Euler-Bernoulli beam model are given by = EI a t = EI b t 5

26 . N N N N N = EI N N t here ubript, and N tand for the variable in firt innermot tube, eond tube and the N th tube. here E i the young modulu of CNT, i the ma denity of CNT, I N i the moment of inertia of N th CNT, N i the fleural defletion of N th CNT and N i the area of the ro-etion of N th CNT, N i the van der aal fore beteen N th CNT and N th CNT.. Bulk Propertie of CNT.. Van der aal Fore The multiple layer of graphite heet in a multi alled CNT are held together by van der aal fore. The van der aal fore i a non-bonded interation, and it an be an attration fore or a repulion fore. The attration our hen a pair of atom approahe eah other ithin a ertain ditane. The repulion our hen the ditane beteen the interating atom beome le than the um of their ontat radii. The van der aal fore i modeled uing Lennard-Jone potential. 6

27 van der aal fore beteen firt innermot tube and eond tube i given by V r = ri r here r i i the inter-atomi ditane, V r i Lennard-Jone Potential and i the denity. Lennard-Jone Potential give the potential energy beteen to phyially interating nonbonded arbon atom in graphite. Lennard-Jone Potential i given by 6 σ σ V r = ε 5 r r here σ,ε are Lennard-Jone ontant ε =.96E 0ev σ =.07 0 r i the ditane beteen to atom having the Lennard-Jone interation. The Lennard-Jone Potential i differentiated tie ith repet to inter-atomi ditane r V r = ε 6 7 [ σ r 6σ r ] 6 V r = ε 6 [ 56σ r σ r ] 7 6 σ σ ε 56 r r = 6 7

28 V r= 0.E09.07E 0 =.96E E 09.07E E =.05E9 ev/m 0 The denity i given by qrt = 9r Subtituting eq 0 and eq in eq, van der aal fore beteen innermot tube d i and outer tube d o i given by V qrt 0 d o d i = ri =.0670 r 9r N/m.. Bending Rigidity Krihnan et al., 99 ued the traight normal potulate to alulate the bending tiffne of nanotube. Straight normal potulate tate that the longitudinal deformation at any point in the fleural diretion i proportional to the ditane beteen that point to the mid-plane of midurfae of the truture. Hoever, the atomi layer in a SNT annot be divided into different layer and the fleural train or deformation are atually onentrated on a narro region around the enter-line of the atom layer, rather than ditributed linearly over the thikne diretion.

29 Hene it i inappropriate to aume traight normal potulate for CNT. Hene ang et al. 005 propoed that ine the repreentative thikne of the nanotube layer i nm Yakobon et al., 996 hih i muh maller than the diameter of the tube, the tiffne of the nanotube beam truture for a SNT an be epreed a follo EI E = d o d i 6 Subtituting the equality d d = h and d d d o i o i EI Et d C d = = here d o i the outer diameter, d i i the inner diameter, tiffne of CNT. Et C = = 60J / m i the in-plane.. Ma Denity Yoon et al., 00 propoed =.g / m. The ro area of the nanotube i given by = dt. Sine a SNT i rolled up from a heet of graphite, the value of thikne in alulating the ro area of the CNT hould be ued from the equilibrium interlayer paing of adjaent nanotube, i.e. t = 0.nm, hih i hon in figure VII. 9

30 0 Figure VII Interlayer Spaing of djaent Nanotube. Continuum Model of CNT.. Continuum Model of Single alled CNT Equation for SNT i given by 0,, = t t EI 5 Subtituting the bending tiffne propoed by ang et al. 00, here d C EI = 0,, = t t t Cd 6 The free vibration olution an be found uing the method of eparation of variable, t T t = 7

31 Subtituting in the eq 7 in eq 6 Cd d d T t T t = 0 d dt Cd d d = ω 9 Cd d ω = 0 d 0 d ω By ubtituting β = 0 ; here β = & d q Cd β ω = 0 Cd q = ω Cd = β Subtituting rea, = dt and β = β L β Cd ω = L dt The natural frequeny for a SNT i given by β Cd ω = 5 L t here β depend on end ondition and mode number. For fied end ondition β i.700, 7.505, ,.765 and 7.7 for firt five mode hape repetively.

32 For antilever end ondition β i.750,.700, 7.505, and.765 for firt five mode hape repetively... Continuum Model of Double alled CNT Equation for vibration of double alled CNT are t EI = 6 t EI = 7 Subtituting the bending tiffne propoed by ang et al. 00, here d C EI = t Cd = t Cd = 9 The free vibration olution an be found uing the method of eparation of variable, t T t I I = 0

33 Subtituting in the eq 0 into eq & 9 [ ] 0 = d d Cd ω [ ] 0 = d d Cd ω uming that all the neted tube have the ame vibration mode i, determined by 0 = d d i i β uming a = b = 5 Subtituting the eq to 5 in eq and eq, e have an Eigen value problem [ ] 0 = b a Cd ω β 6 [ ] 0 = b Cd a ω β 7 For a non-trivial olution of a and b, the determinant of their o-effiient mut be zero. 0 = ω β ω β Cd Cd

34 Epanding the determinant give the frequeny equation 0 6 = d d t L d d C d d C L td L L Cd td L L Cd β β ω β β ω 9 Let td L L Cd td L L Cd β β ξ = 9a 6 d d t L d d C d d C L β β ζ = 9b Subtituting eq 9a and eq 9b, eq 9 beome 0 = ζ ξω ω 0 The frequenie are given by 0 ζ ξ ξ ω = n & ζ ξ ξ ω = n

35 5.. Continuum Model of Five alled CNT The governing equation for five alled arbon nanotube i given by t EI = t EI = t EI = 5 t EI = t EI = 6 Subtituting the bending tiffne propoed by ang et al. 00, here d C EI = t Cd = 7 t Cd = t Cd = 9 5 t Cd = 50

36 t Cd = 5 The free vibration olution an be found uing the method of eparation of variable, t T t I I = 5 Subtituting in the eq 5 into eq 7 to 5 [ ] 0 = d d Cd ω 5 [ ] [ ] 0 = d d Cd ω 5 [ ] [ ] 0 = d d Cd ω 55 [ ] [ ] 0 5 = d d Cd ω 56 [ ] = d d Cd ω 57 uming that all the neted tube have the ame vibration mode i, determined by 0 = d d i i β 5 uming a = 59 b = 60

37 7 = 6 d = 6 5 e = 6 Subtituting the eq 5 to 6 in eq 5 to 57, e have an Eigen value problem [ ] 0 = b a Cd ω β 6 [ ] [ ] 0 = b Cd a ω β 65 [ ] [ ] 0 = d Cd b ω β 66 [ ] [ ] 0 = e d Cd ω β 67 [ ] = e Cd d ω β 6 For a non-trivial olution of a, b,, d and e the determinant of their o-effiient mut be zero. Epanding the determinant ill give the frequeny equation = ω β ω β ω β ω β ω β Cd Cd Cd Cd Cd 69

38 CHPTER FINITE ELEMENT NLYSIS. Introdution The finite element method FEM ha beome a poerful numerial method for analyzing phyial phenomena in the field of trutural, olid and fluid mehani. In the lat almot four deade, the finite element method FEM ha beome the prevalent tehnique ued for analyzing phyial phenomena in the field of trutural, olid and fluid mehani a ell a for olution of field problem. The FEM i a ueful tool beaue one an ue it to find out fat or tudy the proe in a ay not poible ith any other tool. The omputational approah i an important tool in the development of nano ompoite and their propertie. It help to undertand and deign thee novel material. By mean of finite element method, it i poible to identify mehanial, thermal and eletrial propertie and to tudy their trutural repone under variou load. Finite element analyi enable to reate parameter and boundary ondition, hih are not aeible eperimentally, or analytially to be invetigated. Finite element analyi ha three bai tage. They are Pre-Proeing, nalyi, Pot-Proeing Baially pre-proeing involve diretization of domain of interet into finite element. The element are onneted to eah other through node. Diretization i one of the bai and

39 important tep in finite element analyi. The quality of output ill depend upon the quality of the element and quality of mehing. The bai unknon parameter i the diplaement at the nodal point. The net tep i defining phyial and material propertie for the element. ll thee tep ome under mehing. Then load and boundary ondition are applied. The analyi tage involve tiffne generation, ma generation, tiffne modifiation and olution of equation reulting in the evaluation of nodal variable. Other derived quantitie uh a gradient or tree may be evaluated at thi tage. tiffne and ma matri are formed for all the element and are aembled to obtain the global matrie. Then by applying the boundary ondition, the global matrie are modified and the diplaement are alulated by olving the finite element equation, hih in turn i ued to alulate the tree and train. The potproeing tage i onerned ith interrogating the reult of the analyi. In ae of free vibration thi ill be the natural frequenie and mode of free vibration. Pre-proeing for all FE model of SNT and MNT are done uing finite element oftare I-DES. Preproeing inlude diretization of the truture, applying material and phyial propertie, applying load and boundary ondition. nalyi and Pot-proeing, here done uing poerful finite element oftare alled BQUS. 9

40 . Bulk Propertie of Nanotube in FE Modeling Material propertie of CNT hould be hoen orretly to imulate the atual nano-effet. In ontinuum modeling, e ue thikne t of 0. hile alulating the area of the nanotube and thikne t of 0.066nm hile alulating bending rigidity. The young modulu ued in ontinuum modeling i 5 Tpa. In finite element modeling e an t ue to thiknee o e have to ue a ontant thikne t of 0. nm. The young modulu of nanotube i modified to Tpa o that in-plane tiffnec of nanotube i maintained a 60 J/m. Sine finite element ode annot be diretly applied to a nano-ale beaue of their maller dimenion, o the nano-ale problem ha to be aled up for olving uing finite element ode. The nano model i aled to a model, hih i in meter-ale. 0

41 . Finite Element Modeling.. FE Model of Single alled CNT The finite element modeling of ingle alled nanotube i eay ine there i no van der aal fore. baqu Euler-Bernoulli beam B noded ubi beam hon in figure VIII ere ued to model ingle alled nanotube. Figure VIII B Beam Element Thi beam element i a one-dimenional line element that ha tiffne aoiated ith deformation of the line Beam ai. Thee deformation onit of aial treth; urvature hange Bending and torion. Thi element doe not allo tranvere hear deformation; plane etion initially normal to the beam ai remain plane and normal to the beam ai. The Euler- Bernoulli beam element ue ubi interpolation funtion. The main advantage of thi element i that they are geometrially imple, fe degree of freedom and omputational time i le.

42 Firt tep in finite element model of SNT i to reate a et of 0 node. Then baqu pipe etion i eleted a the beam ro-etion. Seond tep i reating 9 beam element beteen the node. The third tep i to aign appropriate material propertie to the element. The fourth tep i fiing the left mot node of the beam o that it imulate antilever boundary ondition. The fifth tep i to elet Lanzo Eigen value etration method. The ith tep i to olve for Eigen value. The final tep i to pot proe the reult obtained. Figure IX ho the finite element model of SNT. Figure IX Finite Element Model of SNT

43 .. FE Model of Double alled CNT For double alled nanotube, noded linear olid brik element hon in figure X ere ued for modeling the to tube. Figure X CD Linear Brik Element From trutural mehani viepoint, the effet of van der aal fore i like a ditributed pring ith tiffne N attahed at the interfae of inner and outer tube. peial pring element alled aial pring element are ued to imulate the effet of van der aal fore. Figure XI ho the aial pring element. Thi peial pring element line of ation i the line joining the to node. Thi pring element introdue tiffne beteen to degree of freedom ithout introduing an aoiated ma. Figure XI ial Spring Element

44 Firt a quarter of the model i modeled and then it i refleted to make the omplete model, by thi ay the model i ymmetri. One the model i refleted, then the proedure i ame a SNT. ppropriate material propertie and boundary ondition are applied and then the model i olved for natural frequenie. Figure XII ho the finite element model of SNT. Figure XII Finite Element Model of DNT

45 .. FE Model of Five alled CNT For multi alled nanotube, noded olid brik element ere ued for modeling the individual tube of MNT modeling i imilar to DNT. ial pring element are ued to imulate the effet of van der aal fore. Firt a quarter of the model i modeled and then it i refleted to make the omplete model. One the model i refleted, then the proedure i ame a DNT. ppropriate material propertie and boundary ondition are applied and then the model i olved for natural frequenie. Figure XIII ho the finite element model of five alled nanotube Figure XIII Finite Element Model of Five alled Nanotube 5

46 Figure XIV ho finite element model for ingle, double and five-alled nanotube. It alo ho the aial pring beteen inner and outer tube of double alled nanotube Figure XIV FE model for Single and Multi alled nanotube. 6

47 . Eigen Value Etration The Eigen value problem for the natural frequenie of an undamped finite element model i given by y y M K φ = 0 ω 70 y M - Ma Matri y K - Stiffne Matri φ - Eigen Vetor Mode of Vibration ω - Frequeny and y Degree of freedom BQUS offer to method for eigenvalue etration. They are Lanzo and ubpae eigenvalue etration method. I have ued Lanzo method beaue it i generally fater hen a large number of eigenmode i required for a ytem ith many degree of freedom. 7

48 .5 Frequeny Calulation.5. Frequeny of Single alled CNT The frequenie obtained uing finite element model ha to be aled don uing aling fator to get the natural frequenie of the nano-ize problem. Frequeny of atual model, Frequeny of aled Model β Cd ω = 7 L t β E I ω = 7 L here ω i the frequeny, d i the diameter, i the denity, L i the length of ingle alled nanotube atual model. ω i the frequeny, i the denity, L i the length, the young modulu, I i the moment of inertia of aled finite element model. i area, E i Ratio of frequenie of atual and aled model i given by ω ω β Cd L t = 7 β E I L Frequenie of atual model in term of aled model i given by L ω = ω * * L EI E I 7

49 9.5. Frequeny of Double alled CNT Frequeny of atual model i given by 0 ζ ξ ξ ω = n & ζ ξ ξ ω = n 75a here td L L Cd td L L Cd β β ξ = 75b 6 d d t L d d C d d C L β β ζ = 75 here 0 n ω and n ω are the frequenie, d and d are the diameter of inner and outer tube, C i the in-plane tiffne, L i the length, i the ma denity, i the van der aal fore beteen inner and outer tube of the DNT atual model. Frequeny of aled model i given by 0 n ζ ξ ξ ω = & n ζ ξ ξ ω = 76a here EI I E β β ξ = 76b I E I E I E I E β β β ζ = 76 here n0 ω and n ω are the frequenie, E i the young modulu, i the ma denity, I i the moment of inertia of, and are the area of the ro-etion of inner and outer tube, i the van der aal fore beteen inner and outer tube of aled DNT finite element model.

50 0 Ratio of frequenie of atual and aled model i given by 0 0 n n ζ ξ ξ ζ ξ ξ ω ω = & n n ζ ξ ξ ζ ξ ξ ω ω = 77 Frequenie of atual model in term of aled model i given by * 0 0 n n ζ ξ ξ ζ ξ ξ ω ω = & * n n ζ ξ ξ ζ ξ ξ ω ω = 7

51 CHPTER 5 RESULTS ND DISCUSSION 5. Continuum Model Reult Table I Frequenie of fied DNT ith inner diameter 0.7nm, outer diameter.nm and =e0 N/m Cae Mode Continuum Model THz Single Beam Theory THz % of Error ω ω L/d =0 ω ω ω ω ω L/d =0 ω ω ω ω ω L/d =50 ω ω ω If the value of i large, it an be verified that the effet of van der aal fore on MNT i negligible and thu loer order frequeny given by eq ill be equal to the frequeny of ingle elati beam model given by C d d β ω = 79 L Table I ho the natural frequenie of fied double alled nanotube and an equivalent ingle beam. Fied double alled CNT hould behave a a fied ingle beam hen the vanderaal

52 fore i very large e0 N/m. From table I, it i lear that the propoed ontinuum model for a fied double alled nanotube i valid, ine the perentage of error i zero. Table II Frequenie of fied DNT ith inner diameter 0.7nm and outer diameter.nm Cae L/d =0 L/d =0 L/d =50 Mode Continuum Model THz J. Yoon[9] THz % of Error ω ω ω ω ω ω.5..7 ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω

53 Table III Frequenie of fied DNT ith inner diameter 0.7nm and outer diameter.nm Cae Mode Continuum Model THz Single Beam Theory THz % of Error ω ω L/d =0 ω ω ω ω ω L/d =0 ω ω ω ω ω L/d =50 ω ω ω

54 Table IV Frequenie of antilever DNT ith inner diameter 0.7nm, outer diameter.nm and =e0 N/m Cae Mode Continuum Model THz Single Beam Theory THz % of Error ω ω L/d =0 ω ω ω ω ω L/d =0 ω ω ω ω ω L/d =50 ω ω ω Table IV ho the natural frequenie of antilevered double alled nanotube and an equivalent ingle beam. Cantilevered double alled CNT hould behave a an equivalent antilevered ingle elati beam model hen the vanderaal fore i very large e0 N/m. Sine the perentage of error i zero it prove that the propoed ontinuum model for a fied DNT i valid.

55 Table V Frequenie of antilever DNT ith inner diameter 0.7nm and outer diameter.nm Cae L/d =0 L/d =0 L/d =50 Mode Continuum Model THz J. Yoon[9] THz % of Error ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω

56 Table VI Frequenie of antilever DNT ith inner diameter 0.7nm and outer diameter.nm Cae Mode Continuum Model THz Single Beam Theory THz % of Error ω ω L/d =0 ω ω ω ω ω L/d =0 ω ω ω ω ω L/d =50 ω ω ω It i een from table II, III, V and VI that the loet intertube frequenie are almot ame for fied DNT and antilever DNT indiating that they are inenitive to the end ondition a predited by Yoon et al. 00. The firt three intertube frequenie are around 0 THz a predited by Yoon et al. 00. The intertube reonant frequeny ω n i inenitive to the mode number and i muh higher than the loet natural frequeny ω n0 for larger apet ratio a predited by Yoon et al. 00. The perentage of error beteen ontinuum model and Yoon et al. 00 and for loet natural frequeny ω n0 i almot negligible. But for ae L/d >0 the intertube frequeny ω n i higher than the frequeny alulated by Yoon et al. 00 for both fied and antilever ae. 6

57 The ontinuum model i alo ompared ith ingle beam theory. The frequeny of ingle beam theory i given by C d d β ω = 0 L The differene beteen frequenie of ontinuum model and ingle beam theory an be een from the table III and VI. The differene learly ho the effet of van der aal fore on vibration of nanotube. There i no effet of van der aal fore on vibration if L/d >=50. The highet effet of van der aal fore on vibration i 7 % for the fifth mode of fied DNT L/d =0. The effet of van der aal fore i high for maller apet ratio L/d <=0 and thi effet dereae a the apet ratio inreae. 7

58 Table VII Frequenie of fied five alled nanotube ith inner diameter 0.7 nm, outer diameter.5 nm and L/d 5 = 0 Cae Mode Continuum J. Yoon[9] Model THz THz % of Error ω ω ω ω.5..5 ω ω.6..9 ω ω ω.6..7 ω ω ω L/d 5 = 0 ω ω.7..5 ω ω ω ω ω ω ω ω ω ω.6..5 ω

59 Table VIII Frequenie of fied five alled nanotube ith inner diameter 0.7 nm, outer diameter.5 nm and L/d 5 = 0 Cae Mode Continuum J. Yoon[9] Model THz THz % of Error ω ω ω ω.5..0 ω ω ω ω ω.5..0 ω ω ω L/d 5 = 0 ω ω.5..5 ω ω ω ω ω.5..5 ω ω ω ω ω.6..7 ω

60 Table IX Frequenie of fied five alled nanotube ith inner diameter 0.7 nm, outer diameter.5 nm and L/d 5 = 50 Cae Mode Continuum J. Yoon[9] Model THz THz % of Error ω ω ω ω.5..0 ω ω ω ω ω.5..0 ω ω ω L/d 5 = 50 ω ω.5..0 ω ω ω ω ω.5..0 ω ω ω ω ω.5..0 ω

61 Table X Frequenie of fied five alled nanotube ith inner diameter 0.7 nm and the outer diameter.5 nm Cae Mode Continuum Model THz Single Beam Theory THz % of Error ω ω L/d 5 =0 ω ω ω ω ω L/d 5 =0 ω ω ω ω ω L/d 5 =50 ω ω ω Table VII, VIII and IX ho the frequeny of fied five alled CNT ith apet ratio of 0, 0 and 50 repetively. From table VII to IX it an be een that the frequeny of propoed ontinuum model i in agreement ith the frequenie of Yoon et al. 00. The perentage of error beteen thee to model i almot negligible for the ae L/d 5 =50. Table X ho the frequenie of fied five alled nanotube and fied equivalent ingle beam. The perentage of differene beteen frequenie of ontinuum model and ingle beam i very large for the ae L/d 5 =0, hih i % for the fifth mode. Thi effet dereae a the apet ratio inreae. 5

62 5. Finite Element Model Reult Table XI Frequeny of antilever SNT Length nm Diameter nm FE Model Frequeny THz Krihnan[]- Method-I THz % of Error Table XII Frequeny of antilever SNT Length nm Diameter nm FE Model Frequeny THz Krihnan[]- Method-II THz % of Error Table XIII Frequeny of antilever SNT Length nm Diameter nm FE Model Frequeny THz ang THz % of Error Table XIV Frequenie of antilever SNT diameter =.50 nm and length = 6. nm Mode FE Model Frequeny THz

63 The validation of the propoed FE model for SNT i done by omparing the reult ith Krihnan et al. 99 and ang et al. 005 and for DNT and MNT i done by omparing the reult ith Yoon et al. 00. Table XI to XIII ho the omparion of finite element reult of antilever SNT ith the eperimental reult Method I & II of Krihnan et al. 99 and analytial reult of ang et al The maimum perentage of error hen ompared ith eperimental reult Method I of Krihnan et al. 99 i 7 %. The maimum and minimum perentage of error hen ompared ith eperimental reult Method II of Krihnan et al. 99 i % and %. The maimum perentage of error hen ompared ith analytial reult of ang et al. 005 i 0.55 %. Hene the uggeted finite element model for SNT loely agree ith ontinuum model of ang et al Figure XV ho the firt five mode hape of antilever SNT predited by propoed FE model. The mode hape are imilar to the mode hape of any antilever beam. 5

64 Figure XV Firt five mode of antilevered SNT diameter=.50 nm and length = 6. nm 5

65 Table XV Frequenie of fied DNT ith inner diameter 0.7 nm. and the outer diameter. nm Cae Mode FE Model J. Yoon[9] Frequeny THz THz % of Error ω ω L/d =0 ω ω ω ω ω ω L/d =0 ω ω ω ω ω ω L/d =50 ω ω ω ω Table XVI Frequenie of fied DNT ith inner diameter 0.7 nm. and the outer diameter. nm Cae Mode FE Model Frequeny THz Single Beam Theory THz % of Error ω L/d =0 ω ω ω L/d =0 ω ω ω L/d =50 ω ω

66 Figure XVI Firt three mode of fied - fied double alled arbon nanotube 56

67 Table XVII Frequenie of antilever DNT ith inner diameter 0.7nm and outer diameter.nm Cae Mode FE Model J. Yoon[9] Frequeny THz THz % of Error ω ω L/d =0 ω ω.. -. ω ω ω ω L/d =0 ω ω ω ω ω ω L/d =50 ω ω ω ω Table XVIII Frequenie of Cantilever DNT ith inner diameter 0.7 nm. and the outer diameter. nm Cae Mode FE Model Frequeny THz Single Beam Theory] THz % of Error ω L/d =0 ω ω ω L/d =0 ω ω ω L/d =50 ω ω

68 Figure XVII Firt three mode of antilevered double alled arbon nanotube 5

69 Frequeny THz 0 6 n-nalytiall/d=0 n-nalytiall/d=0 n-nalytiall/d=0 n-nalytiall/d=0 n-nalytiall/d=50 n-nalytiall/d=50 n-finite ElementL/d=0 n-finite ElementL/d=0 0 Mode n-finite ElementL/d=0 n-finite ElementL/d=0 n-finite ElementL/d=50 n-finite ElementL/d=50 Figure XVIII Frequenie of Fied DNT ith inner diameter 0.7 nm. and the outer diameter. nm 59

70 Frequeny THz 0 6 n-nalytiall/d=0 n-nalytiall/d=0 n-nalytiall/d=0 n-nalytiall/d=0 n-nalytiall/d=50 n-nalytiall/d=50 n-finite ElementL/d=0 n-finite ElementL/d=0 n-finite ElementL/d=0 0 Mode n-finite ElementL/d=0 n-finite ElementL/d=50 n-finite ElementL/d=50 Figure XIX Frequenie of Cantilever DNT ith inner diameter 0.7 nm. and the outer diameter. nm Table XVIII ho the frequenie of fied DNT ith d = 0. 7nm and d =. nm for different apet ratio. Table XIX ho the frequenie of antilever DNT ith d = 0. 7nm and d =. nm for different apet ratio. It i een from table XVIII and XIX that the loet intertube frequenie are almot ame for fied DNT and antilever DNT indiating that they are inenitive to the end ondition a predited by Yoon et al. 00. The firt three intertube frequenie are around 0 THz a predited by Yoon et al. 00. The intertube reonant frequeny ωn i inenitive to the mode number and i muh higher than the loet natural frequeny ωn0 for larger apet ratio a predited by Yoon et al. 00. The perentage of error beteen ontinuum method and Yoon et al. 00 and for loet natural frequeny ωn0 i almot negligible. The perentage of error beteen finite element 60

71 reult for both antilever DNT and fied DNT ith J. Yoon [9] i very high [up to 0%] for mall apet ratio L d 0. The perentage of error for larger apet ratio are ithin %.Thi reult i alo in agreement ith Harik et al. 00 ho aid that beam model an be ued only if L d > 0.The finite element frequeny for both antilever DNT and fied DNT here almot the ame frequeny of Yoon et al. 00 for firt mode for all apet ratio. The FE model reult are alo ompared ith ingle beam theory. The frequeny of ingle beam theory for a DNT i given by equation 50. Table XVI and XVIII ho the differene beteen DNT finite element and ingle beam theory frequenie. The differene learly ho the effet of van der aal fore on vibration of nanotube and the effet i high up to 7 % for maller apet ratio. For apet ratio L d 50, the differene beteen DNT frequeny and ingle beam theory frequeny i almot zero indiating that effet of van der aal fore on vibration of nanotube i negligible for larger apet ratio. The other intereting reult i there i no effet for the firt mode hape in all apet ratio. Figure XVIII ho the omparion of ontinuum model and finite element model frequenie of fied DNT. It an be een that finite element model frequenie and ontinuum model frequenie are in good agreement for inner tube. Figure XIX ho the omparion of ontinuum model and finite element model frequenie of antilever DNT. It an be een that finite element model frequenie and ontinuum model frequenie are in good agreement for inner tube. 6

72 Table XIX Frequenie of five alled nanotube ith inner diameter 0.7 nm and the outer diameter.5 nm Cae Mode FE Model Frequeny THz J. Yoon 00 THz % of Error ω ω L/d 5 = 0 ω ω ω ω ω L/d 5 = 0 ω ω ω ω ω L/d 5 =5 0 ω ω...7 ω Table XX Frequenie of five alled nanotube ith inner diameter 0.7 nm and the outer diameter.5 nm Cae Mode FE Model Frequeny THz Single Beam Theory THz % of Error ω ω.9 L/d 5 = 0 ω.0 ω.76 ω 5. ω ω 5. L/d 5 = 0 ω 9. ω.55 ω 5.96 ω ω 5. L/d 5 = 50 ω 9. ω. ω 5.7 6

73 Table XXI Frequenie of five alled nanotube ith inner diameter 0.7 nm and the outer diameter.5 nm Cae Mode Continuum FE Model Model Frequeny Frequeny THz THz % of Error ω ω L/d 5 = 0 ω ω ω ω ω L/d 5 = 0 ω ω ω ω ω L/d 5 =5 0 ω ω ω Table XIX ho the finite element reult for fied five-alled nanotube in omparion ith Yoon et al. 00. The perentage of error i high up to % for maller apet ratio L d 5 = 0. The perentage of error dereaed a apet ratio inreaed. Table XX ho the differene in frequenie of fied five-alled nanotube and ingle beam theory. The van der aal fore ha dereaed the frequeny of nanotube by % for apet ratio L d 5 = 0. Thi effet dereae a the apet ratio inreae. There i no effet of van der aal fore on vibration of nanotube of apet ratio L d Table XXI ho the omparion of ontinuum model frequenie and finite element model frequenie. The finite element model i in good agreement ith ontinuum model for apet ratio L d 5 = 0 and L d 5 = 50. 6