Available online at ScienceDirect. Procedia Materials Science 10 (2015 )

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1 Avalable onlne at ScenceDrect Proceda Materals Scence (5 ) nd Internatonal Conference on Nanomaterals and Technologes (CNT 4) Influence of thermal and magnetc feld on vbraton of double walled carbon nanotubes usng nonlocal Tmoshenko beam theory P. Ponnusamy a, * and A. Amuthalakshm b a Assocate Professor, Department of Mathematcs, Government Arts College, Combatore 648, Tamlnadu, Inda b Research Scholar, Department of Mathematcs, Government Arts College, Combatore 648, Tamlnadu,Inda Abstract In ths paper, the nfluence of thermal and magnetc feld on transverse vbraton of double walled carbon nanotubes are nvestgated usng nonlocal Tmoshenko beam theory. The governng equaton of moton wth small scale effects are formulated by consderng thermal and Lorentz magnetc force. The dsperson relaton s derved for carbon nanotubes wth smply supported ends and the numercal smulatons are done usng MATLAB. Dsperson curves are drawn for both double and sngle walled carbon nanotubes for dfferent mode and magnetc feld strength. The numercal result shows that the strength of magnetc feld ncreases the natural frequences and the nonlocal parameter decreases the natural frequences of carbon nanotubes. Further to dscuss the accuracy of the present result, the dsperson curves of sngle walled carbon nanotubes obtaned by omttng the magnetc feld are compared wth the exstng lterature. 5 The Authors. Publshed by by Elsever Elsever Ltd. Ltd. Ths s an open access artcle under the CC BY-NC-ND lcense Peer-revew ( under responsblty of the Internatonal Conference on Nanomaterals and Technologes (CNT 4). Peer-revew under responsblty of the Internatonal Conference on Nanomaterals and Technologes (CNT 4) Keywords:Nonlocal Tmoshenko beam theory; Magnetc feld; Vbraton; Smply supported ends; Natural frequences. Introducton Carbon nanotubes (CNTs) are developed greatly due to ts outstandng mechancal, thermal, electrc, magnetc and chemcal propertes. The mechancal propertes of CNTs are smulated through two categores namely the molecular dynamcs smulaton and contnuum mechancs. Snce the molecular dynamcs smulaton are expensve * Correspondng author. Tel.: E-mal address:pponnusamy3@gmal.com The Authors. Publshed by Elsever Ltd. Ths s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of the Internatonal Conference on Nanomaterals and Technologes (CNT 4) do:.6/j.mspro

2 44 P. Ponnusamy and A. Amuthalakshm / Proceda Materals Scence ( 5 ) and takes unaffordable tme n performng experments, the contnuum mechancs have been developed for studyng the vbraton propertes of CNTs. Tmoshenko beam model s known for ts ncluson of rotary nerta and shear deformaton effect n CNT. Recently Tmoshenko beam model along wth nonlocal elastc theory of Erngen (983) s developed greatly due to ts nvestgaton of larger number of atoms wth good accuracy. Yang et al. () studed the nonlnear free vbraton of sngle walled carbon nanotubes (SWCNTs) usng von Karman geometrc nonlnearty, Erngen s nonlocal elastcty theory and Tmoshenko beam theory. Ansar and Sahman () nvestgated the free vbraton of SWCNT based on dfferent nonlocal beam theores. Yang and Lm () developed the thermal bucklng of nanobeams usng Tmoshenko beam model and nonlocal elastcty theory. Based on nonlocal Tmoshenko beam theory Pradhan and Mandal (3) analyzed the vbraton responses of CNTs under thermal envronment and transverse loadng. Ansar et al. () employed a Wnkler type elastc foundaton model and studed the dynamc stablty of embedded SWCNTs ncludng thermal effects. Maachou et al. () nvestgated the nfluence of thermal effect on the free vbraton of zgzag SWCNTs usng nonlocal Levnson beam model. Aydogdu and ECE (7) analyzed the vbraton and bucklng behavor of smply supported double walled carbon nanotubes (DWCNTs) usng Tmoshenko beam theory. Zhang et al. (4) studed the vbraton and bucklng behavor of Tmoshenko beam usng the stran gradent elastcty theory. Kan (3) examned the free transverse vbraton of elastcally supported DWCNTs subjected to axally varyng magnetc feld. The author also evaluated the flexural frequences of magnetcally affected DWCNTs wth dfferent boundary condtons. Ponnusamy (7, ) nvestgated the vbraton n a generalzed thermo elastc sold cylnder of arbtrary cross-secton and plate of polygonal cross-secton usng the Fourer expanson collocaton method. Ponnusamy and Amuthalakshm (3) studed the vbratonal frequences of DWCNTs wth smply supported ends and resonant frequences wth prestress and wthout prestress effect. Later the same authors (4) studed the thermal effect on wave propagaton of carbon nanotubes embedded n an elastc medum usng elastc shell theory. In the present paper, the effect of thermal and magnetc feld on the vbraton of DWCNTs are nvestgated. In order to analyze the small scale effects on the CNTs a nonlocal elastc theory has been proposed wth Tmoshenko beam theory. The governng dsplacement equaton of moton of CNTs s establshed by assumng the thermal and Lorentz magnetc force. The dsperson relaton s derved for DWCNTs by solvng the governng equaton for smply supported ends. The numercal values of natural frequences are computed usng MATLAB and ther characterstc behavor are dscussed. In order to valdate the accuracy of the result, a comparson of natural frequences of SWCNTs by elmnatng magnetc feld s compared wth the exstng lterature.. Formulaton of the Problem Consder a SWCNT based on Tmoshenko beam model whch ncludes both effect of shear and rotary nerta. An elementary part of smply supported SWCNT n axal magnetc feld under the effect of shear and rotary nerta s shown n Fg.. The governng equaton for the transverse vbraton of SWCNT under the effect of thermal feld s expressed as Graff (99) and Benzar et al. (8) as x t x V A w N w t q () M V I x t where x s the axal coordnate, t s the tme, Mx, tand, gven by () V x t s the bendng moment and resultant shear force M y da (3a) A

3 P. Ponnusamy and A. Amuthalakshm / Proceda Materals Scence ( 5 ) V da (3b) A n whch and are the normal and shear stress respectvely, A s the area of cross-secton, I s the moment of nerta, s the mass densty, q wxts, the transverse deflecton of nanotube and xt, s the pressure exerted on nanotube, s the slope of cross secton due to the bendng of the nanotube and Nt denotes the addtonal axal force gven by Nt EA (4) where, E and denotes the thermal expanson coeffcent, Young s modulus and temperature change respectvely. Fg. An elementary part of SWCNT wth smply supported ends under the effect of magnetc feld and external force based on Tmoshenko beam model Applyng a longtudnal magnetc feld vector HH,, x wth magnetc feld permeablty on the nanotube, a Lorentz force exerts on nanotube. We assume that the Lorentz force acts n the z drecton only, therefore the pressure exerted on nanotube due to Lorentz magnetc feld can be expressed as Karlcc et al. (4) as w q fzda AHx (5) x A where f z denotes the Lorentz force along z drecton. In nonlocal contnuum theory, the stress at a pont s assumed to be a functon of stran feld at every pont n the body. Based on the nonlocal theory of Erngen (983) and generalzed Hooke s law, the consttutve relaton between the stress and stran n one-dmensonal case s gven by ea E (6a) x ea G (6b) x where eadenotes the nonlocal parameter whch reveals the small scale effects, G denotes the shear modulus, and respectvely denotes the normal and shear stran gven by y x (7a) w x. (7b) Substtutng Equatons (7) n Equaton (6) and then usng Equaton (3) n the resultng equaton, we obtan the expresson for bendng moment and shear force for a nonlocal model as M M ea EI x x (8a) V w x V ea GA x (8b)

4 46 P. Ponnusamy and A. Amuthalakshm / Proceda Materals Scence ( 5 ) n whch s the shear coeffcent dependng on the shape of cross-secton. Substtutng Equatons () and () n Equaton (8) and usng Equaton (5), the expresson for nonlocal bendng moment and shear force can be obtaned as 3 w w w M EI ea I A N t AH x x xt t x x 3 3 w w w V GA ea A N t 3. () x xt x By substtutng Equatons (9) and () n Equatons () and (), the equatons of moton for a nonlocal Tmoshenko beam model under the effect of thermal and magnetc feld can be obtaned as w EI GA I ea x x t x w w w w GA N t w ea AH x w ea A w ea. () x x x x x x t x Assume that the nanotube s smply supported, and then the deflecton and rotaton of the nanotube can be expressed n terms of perodc soluton as, wxt We t sn x (3a), xt e t cos x (3b) where n n L, W the ampltude of the deflecton of the beam, s the ampltude of the slope of beam due to bendng deformaton and s the frequency of the wave moton. Substtutng Equatons (3) nto Equatons () and (), the wave dsperson relaton can be obtaned as n an an 4bn (4) Nt AHx n GEIn AG where an (5) A I ea b n EG Nt AHx n EIn AG 4 n ea IA ea n n From Equaton (4), t s known that there are two modes of vbraton for Tmoshenko beam model n whch one mode of deformaton s the transverse deflecton and the other mode s the transverse shearng deformaton. Further the Tmoshenko beam model can be reduced to Euler beam model by elmnatng the effect of rotary nerta and shear force. 3. Double walled carbon nanotube subjected to thermal and magnetc feld Consder a DWCNT under the effect of thermal and longtudnal magnetc feld subjected to rotary nerta and shear deformaton. Then the governng equaton of moton usng nonlocal Tmoshenko beam theory can be wrtten as w EI GA I ea x x (7) t x w w w GA Nt A H x w ea p A w ea (8) x x x x t x n (9) () (6)

5 P. Ponnusamy and A. Amuthalakshm / Proceda Materals Scence ( 5 ) w EI GA I ea x x t x w w w GA Nt A H x w ea p A w ea x x x x t x where w,, A, and Nt, moment of nerta of nner and outer tube respectvely and p, I der Waals (vdws) nteracton between adjacent tube and s gven by j j (9) () are the deflecton, rotaton angle of cross-secton, area of cross-secton and s the pressure exerted on th tube due to van p c w w () where s the vdw nteracton force between the adjacent tubes. Snce the nteracton between the tubes s neglgble, the pressure s proportonal to the dfference n deflecton between the two tubes. Snce the ends of the tube are smply supported, the deflecton and rotaton of DWCNT wth respect to perodc soluton can be wrtten as, w x t We t sn x, (a) c, t cos, (b) xt e x where W and denote the ampltude of deflecton and slope of nner and outer tube. Substtutng Equatons () n Equatons (7) () wth Equaton (), we obtan a set of coupled dfferental equaton of the form EI GA I ea GAW (3) t x GA GA N A H e a c A e a W cw EI GA I e a GA W t x GA GA N AH ea c A ea W cw (6) Equatons (3) (6) are homogeneous equaton whch gves a trval soluton. To obtan non-trval soluton we equate the determnant of the coeffcent matrx to zero as follows g g g c 3 g g g c g g g g g g 8 9 t x g GA N AH e a c, g A e a n whch g I e a g EI GA 4, 7 g A e a, g8 GA, g EI GA (4) (5) (7), g3 GA, 5, 6 t x and g I e a 9 g GA N A H e a c,. Evaluatng the determnant gven n Equaton (7), we obtan a polynomal equaton of degree eght n terms of frequency as B B B B B (8) B g g g g B g g5 gg 7 9 gg 6 ggg, 4 7 gggg 5 7 where 5 7,

6 48 P. Ponnusamy and A. Amuthalakshm / Proceda Materals Scence ( 5 ) B g g g g g g g g g g g g g g g g g g g g g g cg g , B gg gg g g g gg g g gg gg g gg gg c gg gg , B g g g g g g g g g cg g Dsperson relaton of DWCNT subjected to thermal feld Consder a DWCNT wth thermal feld alone under the nfluence of rotary nerta and shear force. By neglectng the magnetc terms and followng the same procedure as dscussed n secton 3, we obtan the frequency equaton of DWCNT under the effect of thermal feld as C C C C C (9) 3 5 where C gg5g7g, C g g5gg 7 9 ggggg, 4 7 gggg 5 7 C3 g g5 gg9 g8 g4 g7g9 gg g g5 g7g9 gg g4g7g g3 g7g cg5g, C gg g g g g g gg g g gg g g g gg g g c gg gg C g g g g g g g g g cg g , , n whch g GA N e a c, t 5. Numercal results and dscusson t g GA N e a c. In ths paper, the thermal and magnetc feld vbraton propertes of both SWCNT and DWCNT are studed. The numercal value of magnetc feld strength and magnetc permeablty are taken from L et al. (8) as 7 7 Hx A mand 4 H m. Further the coeffcent of thermal expanson s negatve at low temperature and postve at hgher temperature. Here we have consdered the low temperature n whch the value of thermal 6 expanson s.6 K and temperature s 4K.The numercal parameters used n the calculatons are 3 consdered from Benzar et al. (8) as Young s modulus E TPa, mass densty.3 gcm, dameter d.7nmand thckness to be.3nm. The value of nonlocal parameter ea.5nmand the aspect rato Ld 4. A graph s drawn between mode n n,,3... and natural frequency of a DWCNT under the effect of thermal and magnetc feld and s shown n Fg.. From Fg., t s observed that the natural frequency ncreases as modes of vbraton ncreases. Here the numercal value of mode and mode 3 occurs n complex values, therefore the real part of natural frequences are same. Further the negatve frequences are gnored as t gves waves n opposte drectons. A varaton graph s drawn between the mode n and nonlocal frequency rato of a DWCNT under the effect of thermal feld at dfferent nonlocal parameter and s shown n Fg. 3. From Fg. 3, t s observed that the nonlocal frequency rato decreases as nonlocal parameter ncreases. A dsperson curve s drawn between the mode n and natural frequency of thermally affected DWCNT at dfferent modes of vbraton and s shown n Fg. 4. From Fg. 4, t s observed that the natural frequency ncreases as modes of vbraton ncreases. A dsperson curve s drawn between the mode n and natural frequency of a SWCNT under the effect of thermal and longtudnal magnetc feld at dfferent nonlocal parameter and s shown n Fg. 5. From Fg. 5, t s observed that as mode ncrease the natural frequences ncreases. Also t s observed that the natural frequency decreases as nonlocal parameter ncreases for every modes of vbraton. A graph s drawn between the mode n and nonlocal frequency rato of DWCNT under the effect of thermal and longtudnal magnetc feld at dfferent nonlocal parameter and s shown n Fg. 6. From Fg. 6, t s observed that as n ncreases the nonlocal frequency rato decreases. Also t s observed that the nonlocal frequency rato decreases as nonlocal parameter ncreases for every modes of vbraton.

7 P. Ponnusamy and A. Amuthalakshm / Proceda Materals Scence ( 5 ) Fg. Dsperson curve between the mode and natural frequency of DWCNT under the effect of thermal and magnetc feld Fg. 3 Varaton of nonlocal frequency rato of a thermal effect on DWCNT at dfferent nonlocal parameter

8 5 P. Ponnusamy and A. Amuthalakshm / Proceda Materals Scence ( 5 ) Fg. 4 Dsperson curve between the mode and natural frequency of the effect of thermal on DWCNT at dfferent modes of vbraton Fg. 5 Dsperson curve between the mode and natural frequency of a SWCNT under the effect of thermal and magnetc feld at dfferent nonlocal parameter

9 P. Ponnusamy and A. Amuthalakshm / Proceda Materals Scence ( 5 ) Fg. 6 Varaton of nonlocal frequency rato of DWCNT under the effect of thermal and magnetc feld at dfferent nonlocal parameter Graphs are drawn between the mode n and natural frequency of a DWCNT and SWCNT under the effect of thermal and magnetc feld at dfferent magnetc feld strength and are shown n Fgs. 7 and 8. From Fgs. 7 and 8, t s observed that the natural frequences n terahertz range ncreases as mode ncreases. Also t s observed that the natural frequency ncreases as magnetc feld strength ncreases for every modes of vbraton. Fg. 7 Dsperson curve between mode and natural frequency of a DWCNT under the effect of thermal and magnetc feld at dfferent magnetc feld strength

10 5 P. Ponnusamy and A. Amuthalakshm / Proceda Materals Scence ( 5 ) Fg. 8 Dsperson curve between mode n and natural frequency of a SWCNT under the effect of thermal and magnetc feld at dfferent magnetc feld strength A comparson graph s drawn between the nonlocal frequency rato of SWCNT and DWCNT at dfferent nonlocal parameter and s shown n Fg. 9. From Fg. 9, t s observed that the nonlocal frequency rato decreases as n ncreases. Also t s observed that the nonlocal frequency rato of SWCNT s lower than DWCNT. The nonlocal frequency rato curves of SWCNT wth marked lne matches wth the curves of Fgure of Benzar et al. (8). Ths shows the accuracy of the present result. Fg. 9 Comparson of nonlocal frequency rato between SWCNT and DWCNT under the effect of thermal at dfferent nonlocal parameter

11 P. Ponnusamy and A. Amuthalakshm / Proceda Materals Scence ( 5 ) Conclusons The nonlocal Tmoshenko beam theory has been proposed to study vbraton of DWCNT under the nfluence of thermal and magnetc feld. The dsperson relatons are derved for both SWCNT and DWCNT. The numercal smulatons are done usng MATLAB and the computed natural frequences are shown n the form of dsperson curves. The numercal result shows that the strength of magnetc feld ncreases the natural frequences whle the nonlocal parameter decreases the natural frequences of CNTs. Further the curves of nonlocal frequency rato of SWCNT obtaned by omttng the magnetc feld matches well wth the exstng lterature. References Ansar, R., Gholam, R., Sahman, S.,. On the dynamc stablty of embedded sngle-walled carbon nanotubes ncludng thermal envronment effects. Scenta Iranca F 9(3), Ansar, R., Sahman, S.,. Small scale effect on vbratonal response of sngle-walled carbon nanotubes wth dfferent boundary condtons based on nonlocal beam models. Commun Nonlnear Sc Numer Smulat 7, Aydogdu, M., ECE, M.c., 7. Vbraton and bucklng of n-plane loaded double-walled carbon nanotubes. Turksh Journal of Engneerng and Envronmental Scence 3, Benzar, A., Touns, A., Bessegher, A., Hereche, H., Moulay, N., Bouma, L., 8. The thermal effect on vbraton of sngle-walled carbon nanotubes usng nonlocal Tmoshenko beam theory. Journal of Physcs D: Appled Physcs 4, Erngen, A.C., 983. On dfferental equatons of nonlocal elastcty and solutons of screw dslocaton and surface waves. Journal of Appled Physcs 54, Graff, K.F., 99. Wave moton n elastc solds, Dover publcatons, New York. Karlcc, D., Murmu, T., Caj, M., Koz, P., Adhkar, S., 4. Dynamcs of multple vscoelastc carbon nanotube based nanocompostes wth axal magnetc feld. Journal of Appled Physcs 5, Kan, K., 3. Characterzaton of free vbraton of elastcally supported double-walled carbon nanotubes subjected to a longtudnally varyng magnetc feld. Acta Mech 4, L, Y., Chen, C., Zhang, S., N, Y., Huang, J., 8. Electrcal conductvty and electromagnetc nterference sheldng characterstcs of multwalled carbon nanotube flled polyacrylate composte flms. Appled Surface Scence 54, Maachou, M., Zdour, M., Baghdad, H., Zane, N., Touns, A.,. A nonlocal Levnson beam model for free vbraton analyss of zgzag sngle-walled carbon nanotubes ncludng thermal effects. Sold State communcaton 5, Ponnusamy, P., 7. Wave propagaton n a generalzed thermo elastc sold cylnder of arbtrary cross-secton. Internatonal Journal of Sold and Structure 44, Ponnusamy, P.,. Dsperson analyss of generalzed thermo elastc plate of polygonal cross-sectons. Appled Mathematcal Modellng 36, Ponnusamy, P., Amuthalakshm, A., 3. Free vbraton analyss of double-walled smply supported carbon nanotubes. Internatonal Journal of Mechancs and Applcatons 3(3), Ponnusamy, P., Amuthalakshm, A., 4. Thermal effect on wave propagaton of carbon nanotubes embedded n an elastc medum. Journal of Computatonal and Theoretcal Nanoscence (6), Pradhan, S.C., Mandal, U., 3. Fnte element analyss of CNTs based on nonlocal elastcty and Tmoshenko beam theory ncludng thermal effect. Physca E 53, 3 3. Yang, J., Ke, L.L., Ktporncha, S.,. Nonlnear free vbraton of sngle walled carbon nanotubes usng nonlocal Tmoshenko beam theory. Physca E 4, Yang, Q., Lm, C.W.,. Thermal effects on bucklng of shear deformable nanocolumns wth von Karman nonlnearty based on nonlocal stress theory. Non lnear Analyss : Real world Applcaton 3, Zhang, B., He, Y., Lu, D., Gan, Z., Shen, L., 4. Non-classcal Tmoshenko beam element based on the stran gradent elastcty theory. Fnte elements n analyss and desgn 79, 39.