Free vibration of pre-tensioned nanobeams based. on nonlocal stress theory *

Transcription

1 3 Lim t al. / J Zhjiag Uiv-Sci A (Appl Phys & Eg) (1):3- Joural of Zhjiag Uivrsity-SIENE A (Applid Physics & Egirig) ISSN X (Prit); ISSN (Oli) jzus@zju.du.c Fr vibratio of pr-tsiod aobams basd o olocal strss thory *. W. LIM 1 hg LI 1 Ji-li YU ( 1 Dpartmt of Buildig ad ostructio ity Uivrsity of Hog Kog Kowloo Hog Kog SAR hia) ( Dpartmt of Modr Mchaics Uivrsity of Scic ad Tchology of hia Hfi 3006 hia) bccwlim@cityu.du.hk Rcivd Ja ; Rvisio accptd Mar ; rosschckd Spt Abstract: Th trasvrs fr vibratio of aobams subjctd to a iitial axial tsio basd o olocal strss thory is prstd. It cosidrs th ffcts of olocal strss fild o th atural frqucis ad vibratio mods. Th ffcts of a small scal paramtr at molcular lvl uavailabl i classical macro-bams ar ivstigatd for thr diffrt typs of boudary coditios: simpl supports clampd supports ad lastically-costraid supports. Aalytical solutios for trasvrs dformatio ad vibratio mods ar drivd. Through umrical xampls ffcts of th dimsiolss aoscal paramtr ad pr-tsio o atural frqucis ar prstd ad discussd. Ky words: Naobam Natural frqucy Nolocal strss Pr-tsiod Vibratio mod doi: /jzus.a Documt cod: A L umbr: O3 1 Itroductio Rctly rsarch o dyamic bhavior of ao-structurs has bcom a hot fild bcaus of th applicatio prospcts of ao-lctromchaical systms (NEMSs) or ao-machi compots. Although ao-structurs such as aobams ad aoblts hav b proposd to hav practical applicatios aalysis i this fild has b lackig i particular th dyamics of pr-tsiod aostructurs. Vibratio of axially movig macro-bams has b a subjct of much cocr rctly. Mot (1965) costructd mathmatical modl of axially movig bams firstly basd o th Hamilto pricipl ad also dtrmid th first thr atural frqucis ad mods. His rsults wr cofirmd by xprimt (Mot ad Nagulswara 1966). Simpso (1973) * Projct supportd by th ollaboratio Schm from Uivrsity of Scic ad Tchology of hia ad ity Uivrsity of Hog Kog Joit Advacd Rsarch Istitut ad ity Uivrsity of Hog Kog (No (B)) rsarchd th atural frqucy ad mod fuctio of axially movig bams without pr-tsio ad clampd at both ds. Oz t al. (001) itroducd axially movig bams with tim-dpdt vlocity through multipl scal aalysis. Liu ad Zhag (007) prstd th oliar vibratio of viscolastic blts. Th bifurcatio of trasvrs vibratio for axially acclratig movig strigs was ivstigatd by h ad Wu (005). Yag ad h (005) addrssd dyamic stability problm of axially movig viscolastic bams. Most classical cotiuum thoris ar basd o lastic costitutiv rlatio which assums that th strss at a poit is a fuctio of strai at oly that poit. O th othr had th olocal cotiuum mchaics assum that th strss at a poit is a fuctio of strais at all poits i th domai. Such thoris cotai iformatio about th forcs btw atoms ad th itral lgth scal is itroducd ito th costitutiv quatios as a matrial paramtr. Th olocal lasticity modl was iitiatd by Erig (197; 1983; 00) ad Erig ad Edl

2 Lim t al. / J Zhjiag Uiv-Sci A (Appl Phys & Eg) (1):3-35 (197). Th olocal lasticity thory was applid i aomchaics icludig lattic disprsio of lastic wavs wav propagatio i composits dislocatio mchaics static dflctio fractur mchaics ad surfac tsio fluids tc. (Rddy ad Wag 1998; Pddiso t al. 003; Zhag t al. 00; 005; Wag 005; Lu t al. 006; Wag ad Varada 006; Wag t al. 006; 008; Xu 006; Lim ad Wag 007; Bzair t al. 008; Kumar t al. 008; Wag ad Dua 008). Th rct work by Tousi t al. (008) cocludd that th scal cofficit was radius dpdt. Vibratio bhavior of bams has b dvlopd for a log tim. Howvr vry fw paprs cosidr aobams with olocal ffcts. Th aomchaical vibratio of a aobam is vry diffrt from th classical cotiuum mchaics thory which dals with th macroscopic scal of a bam. I this papr w attmpt to cosidr th olocal ffcts of a pr-tsiod aobam without axial motio ad subsqutly study th trasvrs vibratio of such a aobam. Th modl is dscribd by partial diffrtial quatios i dimsiolss quatitis such that th aalysis is mor gral ad distictiv to dscrib th diffrc btw aomchaics ad classical mchaics. It is foud that pr-tsio ad olocal strss play sigificat rols i th vibratio bhavior of a aobam. Thir ffcts ar aalyzd ad discussd i dtail i a fw umrical xampls. Problm dfiitio ad modlig osidr a pr-tsiod aobam with th lgth L iitial axial tsio P at th ds as illustratd i Fig. 1. Th d boudary coditios ar arbitrary ad will b spcifid i various cass of study. Th forc quilibrium for a lmt of th aobam is show i Fig.. For vibratio of a aobam th bdig rotatio agl with rspct to x-axis is dotd as θ. Bcaus oly small dformatio is cosidrd for liar vibratio w hav w cosθ 1 si θ (1) x whr w is th trasvrs dformatio. P N Q M Th quilibrium quatio of a lmt with rspct to th z-axis as show i Fig. ca b obtaid basd o th D Almbrt Pricipl (Fug 1965) as Q d w N w x N dx ρ w dx 0 t () whr ρ is th li dsity ad t is tim. Th momt balac coditio yilds M Q 0. (3) Substitutig Eq. (3) ito Eq. () ad dividig dx throughout w hav (Yag ad Lim 008) M w N w w t N ρ 0 which ca also b xprssd as z O L Fig. 1 Fr-body diagram of a aobam x M M dx x M w w t N ρ 0. () (5) For such a aobam with a costat xtral axial tsio P at th ds w hav M w w t Q Q dx x P ρ 0. N N dx x Fig. Forc quilibrium for a lmt of th aobam M: bdig momt; N: itral axial forc; Q: shar forc; x: axial coordiat; z: trasvrs coordiat (6) Accordig to Erig (1983) th olocal strss i a D domai ca b approximatly govrd by a P

3 36 Lim t al. / J Zhjiag Uiv-Sci A (Appl Phys & Eg) (1):3- scod-ordr diffrtial quatio a σ σ 1 ( 0 ) ij ij (7) whr σ ij (i j1 ) ar th olocal strsss σ ij (i j1 ) th classical local strsss 0 a costat dpdt o matrial ad a a itral charactristic lgth.g. for lattic paramtr - bod lgth. For a aobam th govrig quatio abov with rspct to th utral axis ca b rducd to a ordiary diffrtial quatio as d 1 ( a 0 ) σ σ dx (8) whr σ idicats th olocal ormal strsss whil σ th classical local ormal strsss alog th x-axis. From Eq. (8) th olocal ormal strsss ca b solvd ad xprssd i a ifiit sris as (Lim ad Wag 007) w w σ Ez Ez ( 0a) L (9) whr E is th Youg s modulus ad z is th trasvrs coordiat dfid i Fig.. Itgratig Eq. (8) abov with rspct to th distac from th utral axis ad ovr th cross-sctioal ara th olocal bdig momt is govrd by M w M ( 0a) EI (10) whr EI is th flxural stiffss. From Eqs. (6) ad (10) th followig govrig diffrtial quatio of motio for a aobam subjctd to a iitial axial tsio P ca b drivd as ρ w P w ( 0a) ρ w P w EI w. t t (11) For grality dimsiolss formulatio is adoptd usig th followig paramtrs x x/ L w w/ L ad t t EI /( ρl ). I dimsiolss quatitis Eq. (11) th bcoms w P w τ w ( 1 Pτ ) w 0 t t whr τ a 0 / L ad (1) P PL / EI. From Eqs. (6) ad (10) th olocal bdig momt ca b xprssd as w w w M ( 0a) ρ P EI. t (13) Similarly th o-dimsioal form of th quatio abov is w w M τ ( Pτ 1 ) t (1) whr M ML / EI. For liar fr vibratio of a aobam th mods ar harmoic i tim. Hc th timdpdt trasvrs dformatio of th aobam ca b rprstd by iω t wxt ( ) W( x) (15) whr W ( x ) is th dimsiolss amplitud of vibratio ad 1 dots th vibratio mod umbr. Substitutig Eq. (15) ito Eq. (1) th govrig quatio is trasformd ito th frqucy domai as d W d W ωw ( τ ω P) (1 Pτ ) 0. (16) dx dx i j x Substitutig W( x) with as a arbitrary ozro costat ito th quatio abov w obtai a disprsio rlatio j j ω ( τ ω P) (1 Pτ ) 0. (17) Sic Eq. (17) is a fourth-ordr polyomial i trms of th four roots ar dotd by j (j1 3 ) rspctivly. Bcaus oly liar fr vibratio is cocrd th suprpositio of th four solutios with rspct to ach root j is also a solutio of Eq. (16). Hc

4 Lim t al. / J Zhjiag Uiv-Sci A (Appl Phys & Eg) (1):3-37 W ( x) ( ) (18) i 1 x i x i 3 x i x 1 3 whr j (j1 3 ) ar four arbitrary costats of itgratio associatd with Eq. (16) which is a fourth-ordr ordiary diffrtial quatio. 3 Exampls ad discussio To illustrat th ffcts of olocal strss ad iitial axial tsio o th fr vibratio frqucy of a aobam th followig xampls for various boudary coditios ar prstd ad discussd. 3.1 Simply supportd aobams For a aobam simply supportd at both ds th boudary coditios for th bdig momts ad displacmts ar M(0 t ) 0 M(1 t ) 0 w(0 t ) 0 w(1 t ) 0. (19) Substitut Eqs. (1) ad (15) ito Eq. (19) ad simplify th rsults. Furthr substitutig Eq. (18) ito th rsults obtaid abov yilds ( Pτ 1) 1 ( 1 33 ) 0 1 ( Pτ 1) 1 ( ) 0 (0) 1 (1 3 ) 0 ( ) For Pτ 1 0 or quivaltly P EI /( 0a) th amplitud solutio W ( x ) is a arbitrary fuctio ad this is appartly ot a solutio of itrst. O th othr had for Pτ 1 0 Eq. (0) ca b xprssd i a matrix form as (1) For a arbitrary 1 0 th cofficits i Eq. (0) ca b obtaid as ( )( ) ( )( ) 1 ( )( ) ( )( ) ( )( 1 ) 3 3 ( )( 3) 1 ( )( 1 3) ( )( ) 3. () Thrfor th -mod amplitud of vibratio from Eqs. (18) ad () is 1 ( )( ) x ( ) 1 ( )( 3) 1 ( )( 1 ) x 1 3 ( )( 3) 1 ( )( 1 ) 3 3 x ( )( 3) 1 ( )( 1 3) x ( )( 3) W x (3) ad th corrspodig tim-dpdt displacmt from Eq. (15) is wxt ( ) 1 ( )( ) x 1 ( )( 3) 1 ( )( 1 ) x 1 3 ( )( 3) 1 ( )( 1 ) 3 3 x ( )( 3) 1 ( 1 3 x iωt ( )( 3) )( ) () whr 1 0 is a arbitrary costat. For otrivial solutio of Eq. (1) th dtrmiat of th cofficit matrix must b zro which yilds a charactristic quatio as 0 (5)

5 38 Lim t al. / J Zhjiag Uiv-Sci A (Appl Phys & Eg) (1): ( )( ) ( )( ( )( ) ( )( ) ( )( ) ( )( ) ) 0. (6) By combiig Eqs. (17) ad (6) th fiv ukow quatitis j (j1 3 ) ad ω ca b solvd. Subsqutly substitutig th rsults ito Eqs. (3) ad () th -mod vibratio mod ad trasvrs dformatio ca b solvd to th xtt of a arbitrary costat 1 0. Th aalysis abov ca b dscribd clarly through umrical xampls. For istac takig τ0.3 ad P th roots for ω (1 ) satisfyig Eqs. (17) ad (6) ca b obtaid as th itrcpts of th horizotal axis i Fig. 3 whr th dtrmiat Eq. (5) vaishs. Dtrmiat valu ( ) Natural frqucy Fig. 3 Th rlatioship btw dtrmiat valu ad atural frqucy It is obvious that thr ar ifiit mods of frqucy which mak th dtrmiat zro. Th first itrcpt with th horizotal axis is th fudamtal frqucy; th scod itrcpt is th scod mod frqucy ad so o. Followig th umrical procdur abov th rlatioship btw ω 1 ω ad th dimsiolss aoscal paramtr τ ca b obtaid as show i Fig.. Natural frqucis W ca fid that th fudamtal ad th scod mod frqucis rduc with th icrasig τ. Hc th atural frqucis rduc wh th strogr olocal strss ffct is prst. It is also obvious that th frqucis icras with th dimsiolss pr-tsio P. Obviously τ ad P affct vry much th atural vibratio frqucis. 3. lampd aobams Th problm of a clampd pr-tsiod aobam is prstd i th followig xampl. Th clampd boudary coditios ar w w w(0 t ) 0 w(1 t ) 0 (0 t ) 0 (1 t ) 0. (7) From th abov quatio th rsult ca b dducd by Eqs. (15) ad (18) which yilds ω (P0.) ω 1 (P0.5) ω 1 (P0.) ω (P0.5) Nolocal aoscal paramtr Fig. Naoscl ffcts o th first two mod frqucis for simply supportd aobams 0 (8) ( 3 )( ) ( )( ) ( 1 )( ) 3 (9) ( )( ) ( 3 )( ) ( )( ) ( )( ) ( )( ) ( )( ). )( ) ( )( ) ( )( ) i 1 i i i ( )( ) ( 3 3

6 Lim t al. / J Zhjiag Uiv-Sci A (Appl Phys & Eg) (1):3-39 Hc th -mod amplitud of vibratio is ( )( ) ( )( ) ( )( ) x 3 1 x ( ) 1 3 ( )( ) ( 3 )( ) W x ( )( ) ( )( ) i ( )( ) ( )( ) x 1 3 ( 3 )( ) ( 1 )( ) 3 x ( )( ) ( 3 )( ) (30) ad th corrspodig tim-dpdt displacmt is ( )( ) ( )( ) ( )( ) wxt ( ) x 3 1 x 1 3 ( )( ) ( 3 )( ) ( )( ) ( )( ) i ( )( ) ( )( ) ( 3 )( ) ( 1 )( ) 3 x iωt. ( )( ) ( 3 )( ) 3 x (31) For otrivial solutio of matrix Eq. (8) th dtrmiat of th cofficit matrix must b zro or ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 0. i 3 i 1 i i i i 1 i i (3) Aalogously from Eqs. (17) ad (3) w ca solv th ukow quatitis i Eqs. (30) ad (31). To solv th problm umrically th rlatioship btw ω 1 ω ad τ is prstd i Fig. 5 for two valus of P. Agai it is obvious that frqucy dcrass with a icras i τ whil it icrass with a icras i P. Natural frqucis ω 1 ω ω 1 ω (P0.) (P0.) (P1.0) (P1.0) Nolocal aoscal paramtr Fig. 5 Naoscal ffcts o th first two mod frqucis for clampd aobams 3.3 Naobams with lastically costraid ds I this xampl w cosidr a spcial supportig coditio for aobams with lastically costraid ds (Xi 007). Th support coditios may b formulatd with th followig boudary coditios w(0 t ) 0 w(1 t ) 0 w(0 t ) w(0 t ) k 0 w(1 t ) w(1 t ) k 0 (33) whr k k/(pl) is th dimsiolss stiffss of th lastically costraid ds i which k is th physical stiffss of th lastic costrait. If k approachs 0 ths ds dgrat to simply supports as discussd i subsctio 3.1 whil if k approachs ifiity thy dgrat to clampd os i subsctio 3..

7 0 Lim t al. / J Zhjiag Uiv-Sci A (Appl Phys & Eg) (1):3- Substitutig Eqs. (15) ad (18) ito Eq. (33) yilds (3) 1 ik 1 ik 3 ik 3 ik ( 1 ik 1 ) ( ik ) ( 3 ik 3) ( ik ) For 1 0 th followig solutios of cofficits ar obtaid by solvig Eq. (3): ( )( ) ik ( ) ( )( ) ik ( ) ( )( ) 1 ik ( 1 ) ( )( ) 3 ik ( 3 ) ( ) ik ( ) ( )( ) 3 ik ( 3 ) 1 3 ( ) 3 ik ( 3 ) ( ) 1 ik ( 1 ). (35) 3 ( ) ik ( ) ( ) 3 ik ( 3 ) ( ) ik ( ) ( )( ) 3 ik ( 3 ) 3 i i i i i i i i i i Th th -mod amplitud of vibratio ca b obtaid as W ( x) 1 x ( )( ) 3 ik ( 3 ) ( )( ) 1 ik ( 1 ) 1 3 i ( ) ik ( ) ( )( ) 3 ik ( 3 ) 3 1 ( )( ) 1 ik ( 1 ) ( )( ) 3 ik ( 3 ) 1 i ( ) ik ( ) ( )( ) 3 ik ( 3 ) 1 3 ( ) 3 ik ( 3 ) ( ) 1 ik ( 1 ) x (36) 3 ( ) ik ( ) ( ) 3 ik ( 3 ) x 3 x ad th corrspodig tim-dpdt displacmt is show i Eq. (37): wxt ( ) 1 x ( )( ) 3 ik ( 3 ) ( )( ) 1 ik ( 1 ) 1 3 i ( ) ik ( ) ( )( ) 3 ik ( 3 ) 3 1 ( )( ) 1 ik ( 1 ) ( )( ) 3 ik ( 3 ) 1 3 x ( ) ik ( ) ( )( ) 3 ik ( 3 ) (37) 1 3 ( ) 3 ik ( 3 ) ( ) 1 ik ( 1 ) x iωt. 3 ( ) ik ( ) ( ) 3 ik ( 3 ) For otrivial solutio of Eq. (3) th dtrmiat of th cofficit matrix must b zro or Eq. (38) x

8 Lim t al. / J Zhjiag Uiv-Sci A (Appl Phys & Eg) (1):3-1 ( ) ( ik )( ik ) ( ik )( ik ) ( ) ( ik )( ik ) ( ik )( ik ) ( ) ( ik )( 3 ik 3) ( 3 ik 3)( ik ) ( ) ( ik )( ik ) ( ik )( ik ) ( ) ( ik )( ik ) ( ik )( ik ) i i 1 ( ) ( 1 ik 1)( ik ) ( ik )( 1 ik 1 ) 0. (38) Th rlatioship btw th atural frqucis ω 1 ω ad aoscal paramtr τ is prstd i Fig. 6 for k 0.. Agai w obsrv similar ffcts of τ ad P whr icrass i τ ad P caus th frqucis to dcras ad icras rspctivly. Natural frqucis oclusio ω 1 ω (P0.) (P0.) ω 1 (P1.5) (P1.5) Nolocal aoscal paramtr Fig. 6 Naoscal ffcts o th first two mod frqucis for aobams with lastically costraid ds I this papr w cocludd that th trasvrs fr vibratio of a aobam is sigificatly iflucd by th xistc of a pr-tsio ad th dimsiolss aoscal paramtr. Thr umrical xampls ar prstd which iclud simply supportd aobams clampd aobams ad aobams with lastically costraid ds. I th umrical xampls w fid that th first two mod frqucis drop quickly with icrasig dimsiolss aoscal paramtr. O th cotrary th first two mod frqucis icras with icrasig pr-tsio. Th ffcts ar similar for th thr xampls ivstigatd. ω Rfrcs Bzair A. Tousi A. Bssghir A. Hirch H. Moulay N. Boumia L Th thrmal ffct o vibratio of sigl-walld carbo aotubs usig olocal Timoshko bam thory. Joural of Physics D: Applid Physics 1():50. [doi: /00-377/1// 50] h L.Q. Wu J Bifurcatio i trasvrs vibratio of axially acclratig viscolastic strigs. Acta Mchaica Solida Siica 6(1): Erig A Nolocal polar lastic cotiua. Itratioal Joural of Egirig Scic 10(1):1-16. [doi: /000-75(7) ] Erig A O diffrtial quatios of olocal lasticity ad solutios of scrw dislocatio ad surfac wavs. Joural of Applid Physics 5(9): [doi: / ] Erig A Nolocal otiuum Fild Thoris. Sprigr-Vrlag Nw York USA p Erig A.. Edl D.G.B O olocal lasticity. Itratioal Joural of Egirig Scic 10(3):33-8. [doi: /000-75(7) ] Fug Y Foudatios of Solid Mchaics. Prtic-Hall Eglwood liffs NJ USA p Kumar D. Hirich. Wass A.M Bucklig aalysis of carbo aotubs modld usig olocal cotiuum thoris. Joural of Applid Physics 103(7): [doi: / ] Lim.W. Wag.M Exact variatioal olocal strss modlig with asymptotic highr-ordr strai gradits for aobams. Joural of Applid Physics 101(5):0531. [doi: / ] Liu Y.Q. Zhag W Trasvrs oliar dyamical charactristic of viscolastic blt. Joural of Bijig Uivrsity of Tchology 33(11): Lu P. L H.P. Lu. Zhag P.Q Dyamic proprtis of flxural bams usig a olocal lasticity modl. Joural of Applid Physics 99(7): [doi: / ] Mot.D.Jr A study of bad saw vibratios. Joural of th Frakli Istitut 79(6):30-. [doi: / (65)9073-5]

9 Lim t al. / J Zhjiag Uiv-Sci A (Appl Phys & Eg) (1):3- Mot.D.Jr. Nagulswara S Thortical ad xprimtal bad saw vibratios. ASME Joural of Egirig Idustry 88(): Oz H.R. Pakdmirli M. Boyaci H No-liar vibratios ad stability of a axially movig bam with tim-dpdt vlocity. Itratioal Joural of No-Liar Mchaics 36(1): [doi: / S000-76(99) ] Pddiso J. Buchaa G.R. McNitt R.P Applicatio of olocal cotiuum modls to aotchology. Itratioal Joural of Egirig Scic 1(3-5): [doi: /s000-75(0)0010-0] Rddy J.N. Wag.M Dflctio rlatioships btw classical ad third-ordr plat thoris. Acta Mchaica Siica 130(3): Simpso A Trasvrs mods ad frqucis of bams traslatig btw fixd d supports. Joural of Mchaical Egirig Scic 15(3): [doi:10.13/jmes_jour_1973_015_031_0] Tousi A. Hirch H. Brrabah H.M. Bzair A. Boumia L Effct of small siz o wav propagatio i doubl-walld carbo aotubs udr tmpratur fild. Joural of Applid Physics 10(10): [doi: / ] Wag.M. Dua W.H Fr vibratio of aorigs/archs basd o olocal lasticity. Joural of Applid Physics 10(1): [doi: /1.9516] Wag.M. Zhag Y.Y. Ramsh S.S. Kitiporchai S Bucklig aalysis of micro- ad ao-rods/tubs basd o olocal Timoshko bam thory. Joural of Applid Physics 39(17): [doi: /00-377/39/ 17/09] Wag.M. Kitiporchai S. Lim.W. Eisbrgr M Bam bdig solutios basd o olocal Timoshko bam thory. Joural of Egirig Mchaics ASE 13(6): [doi: /(ase) (008)13:6(75)] Wag Q Wav propagatio i carbo aotubs via olocal cotiuum mchaics. Joural of Applid Physics 98(1):1301. [doi: / ] Wag Q. Varada V.K Vibratio of carbo aotubs studid usig olocal cotiuum mchaics. Smart Matrials ad Structurs 15(): [doi: / /15//050] Xi G.M Vibratio Mchaics. Natioal Dfs Idustry Prss Bijig hia p (i his). Xu M.T Fr trasvrs vibratios of ao-to-micro scal bams. Procdigs of th Royal Socity A: Mathmatical Physical ad Egirig Scics 6(07): [doi: /rspa ] Yag X.D. h L.Q Dyamic stability of axially movig viscolastic bams with pulsatig spd. Applid Mathmatics ad Mchaics 6(8): Yag X.D. Lim.W Noliar Vibratios of Nao-bams Accoutig for Nolocal Effct. Fourth Jiagsu-Hog Kog Forum o Mchaics ad Its Applicatio Suzhou hia. Jiagsu Socity of Mchaics Najig p Zhag Y.Q. Liu G.R. Wag J.S. 00. Small-scal ffcts o bucklig of multiwalld carbo aotubs udr axial comprssio. Physical Rviw B 70(0):0530. [doi: /PhysRvB ] Zhag Y.Q. Liu G.R. Xi X.Y Fr trasvrs vibratios of doubl-walld carbo aotubs usig a thory of olocal lasticity. Physical Rviw B 71(19): [doi: /physrvb ]