SCIENCE CHINA Technological Sciences. Transverse vibration of pre-tensioned nonlocal nanobeams with precise internal axial loads

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1 SCIENCE CHINA Techological Scieces RESEARCH PAPER August Vol5 No8: 7 3 doi: 7/s Trasverse vibratio of pre-tesioed olocal aobeams with precise iteral axial loads I Cheg,,3, IM C W,3*, YU Jii,3 & ZENG QigChua Departmet of Moder Mechaics, Uiversity of Sciece ad Techology of Chia, Hei 36, Chia; Departmet of Buildig ad Costructio, City Uiversity of Hog Kog, Tat Chee Aveue, Kowloo, Hog Kog SAR, Chia; 3 Chia ad USTC-CityU Joit Advaced Research Cetre, Suzhou 53, Chia; College of Water Coservacy ad Hydropower Egieerig, Hohai Uiversity, Najig 98, Chia Received March 7, ; accepted May 5, ; published olie Jue 3, This paper ivestigates the trasverse vibratio of a simply supported aobeam with a iitial axial tesio based o the olocal stress field theory with a olocal size parameter Cosiderig a axial elogatio due to trasverse vibratio, the iteral axial tesio is ot precisely equal to the exteral iitial tesio A sixth-order oliear partial differetial equatio that govers the trasverse vibratio for such olocal aobeam is derived Usig a perturbatio method, the relatio betwee atural frequecy ad olocal aoscale parameter is derived ad the trasverse vibratio mode is solved The exteral axial tesio ad olocal aoscale parameter are prove to play sigificat roles i the oliear vibratio behavior of olocal aobeams Such fects ehace the atural frequecy ad stiffess as compared to the predictios of the classical cotiuum mechaics models Additioally, the frequecy is higher if the precise iteral axial load is cosidered with respect to that whe oly the approximate iteral axial tesio is assumed olocal stress, atural frequecy, free vibratio, olocal aoscale Citatio: i C, im C W, Yu J, et al Trasverse vibratio of pre-tesioed olocal aobeams with precise iteral axial loads Sci Chia Tech Sci,, 5: 73, doi: 7/s Itroductio With rapid developmet i aotechology, aobeams have great potetial for wide applicatios as compoets i ao-electroic-mechaical systems Such aostructures have received growig iterest recetly Very ofte these compoets are subjected to exteral loadigs durig work operatio ad their resoat properties are of much cocer As a result, aotechological research o free vibratio properties of aobeams uder certai support coditios is importat because such compoets ca be used as desig compoets i ao-sesors ad ao-actuators *Correspodig author ( bccwlim@cityueduhk) At aoscale, the mechaical characteristics of aostructures are ofte sigificatly differet from their behavior at macroscopic scale due to the iheret size fects Such fects are essetial for aoscale materials or structures ad the ifluece o ao-istrumets is great [] Size fects exist ot oly for mechaical properties but also for electroic, optical ad some other fields [] Geerally, theoretical studies o size fects at aoscale are by meas of surface fects [3, ], strai gradiets i elasticity [5] ad plasticity [6], as well as olocal stress field theory [7, 8], etc Although surface fects ca be eglected at macroscale, the fects could play sigificat roles at aoscale because of the very high surface-to-volume ratio [9, ] Strai gradiet theories for elasticity ad plasticity have bee proposed i various studies followig the works of Midli [5] Sciece Chia Press ad Spriger-Verlag Berli Heidelberg techscichiacom wwwsprigerlikcom

2 8 i C, et al Sci Chia Tech Sci August () Vol5 No8 ad Aifatis [6] The strai gradiet theory cosists of two groups icludig the higher-order ad lower-order theories I the higher-order theory, higher-order stresses are died to be the work-cojugate to strai gradiet, thus leadig to the ecessity of usig higher-order goverig equatios ad boudary coditios The olocal elasticity theory was first developed by Erige [7] ad it assumes that the stress tesor at a poit is a fuctio of strais at all poits i the cotiuum It is differet from the classical cotiuum theory because the latter is based o costitutive relatio which states that the stress at a poit is a fuctio of strai at that particular poit This olocal theory is proved to be i accordace with atomic model of lattice dyamics ad with experimetal observatios o phoo dispersio [8] I olocal theory, the olocal aoscale i the costitutive equatio could be cosidered simply as a material-depedet parameter The ratio of iteral characteristic scale (such as lattice parameter, C-C bod legth, graular distace, etc) to exteral characteristic scale (such as crack legth, wave legth, etc) is died withi a olocal aoscale parameter If the iteral characteristic scale is much smaller tha the exteral characteristic scale, the olocal aoscale parameter approaches zero ad the classical cotiuum theory is recovered Otherwise, the olocal aoscale parameter is fiite ad the correspodig solutios differ greatly from the classical solutios For aostructures, the fects of olocal aoscale parameter o the mechaical behaviors are sigificat because the exteral characteristic scale is relatively at the same order of magitude as the iteral characteristic scale At preset, the olocal elasticity theory has bee used extesively to study lattice dispersio of elastic waves, wave propagatio i composites, dislocatio mechaics, static dlectio, damage ad fracture mechaics, surface tesio fluids, piezoelectric materials, etc [ 5] The mechaical or thermal properties of aostructures such as carbo aotube, aowire, aoswitch, aofilm, aobeam, aoplate, etc are sought [, 6 9] I the applicatio of olocal elasticity models for aostructure, im [, ] recetly showed that the classical goverig equatios ad equilibrium equatios caot be directly applied to olocal stress models eve with the relevat quatities replaced by the correspodig olocal quatities He further derived via the variatioal priciple ew olocal stress models ad a fective olocal bedig momet is derived as a ifiite series of higher-order olocal bedig momets The aalysis were applied to bedig [, ], vibratio [] ad wave propagatio [3] for aobeams The coclusios reached were cotradictory to those based o other olocal approaches [, 6, 7] Based o the previous coclusio that a fective olocal bedig momet, istead of the olocal bedig momet, should replace the classical bedig momet i the equilibrium coditio, a exact olocal stress model for a simply supported aobeam subject to a exteral axial tesio is formulated i this paper Subsequetly, the model is applied usig perturbatio aalysis to free vibratio study of a aobeam with a iteral axial tesio Here, the iteral axial load is ot approximated as equal to the exteral (iitial) axial load i the presece of axial elogatio The precise iteral axial load is derived with a additioal term due to axial elogatio iduced by trasverse vibratio With the cosideratio of olocal fects, the free vibratio frequecies of aostructures are expected to be higher tha the correspodig classical cotiuum solutios The solutios disagree with the other published results which predict lower free vibratio frequecy Numerical examples for a silico beam are preseted ad ew results discussed It is cocluded that a exteral axial tesio ad the olocal aoscale parameter play sigificat roles i the free vibratio of a simply supported aobeam New olocal elasticity model Cosider a aobeam with a exteral (iitial) axial tesio ad simply supported at both eds, as show i Figure, where the axial ad trasverse directios are deoted by x ad y coordiates, respectively The equilibrium aalysis for a elemet of the aobeam is illustrated i Figure, where M is the fective olocal bedig momet [, ], N the iteral axial tesio, V the shear force ad the bedig agle with respect to the x-axis Here, the Euler-Beroulli beam model is adopted ad shear dormatio is thus igored With rerece to Figure, the elemet equilibrium equatio with respect to the y -axis based o the D Alembert priciple is as follows: V w V V dx N N dx A dx, x x () t where is the mass desity, w the trasverse displacemet, A the cross sectioal area, ad t the time From the momet Figure Force equilibrium of a elemet of the aobeam

3 i C, et al Sci Chia Tech Sci August () Vol5 No8 9 balace coditio, the followig coditio ca be derived M w w x x t where V dm dx ad w x N A, For trasverse free vibratio, the trasverse dormatio yields a axial elogatio Thus, the iteral axial tesio ca be expressed as, () N P A (3) where P is the exteral axial tesio, is the fective axial ormal stress which is give by [] d, () ea dx ad is the olocal stress give by d e a E, (5) dx where E is Youg s modulus, e is a olocal costat depedet o specific material, ad a is the iteral characteristic legth scale Note that A i eq (3) represets the force due to trasverse dormatio The ormal strai ca be deduced from a elastic dormatio aalysis as dxdx w d x, (6) x where is the legth of aobeam ad dx the dormatio of uit legth dx Applyig Taylor s expasio to eq (6) ad substitutig eqs () (6) ito eq (3) yield EA w N P d, x x (7) where terms of order O[(e a) ] are retaied while other higher-order terms are eglected Substitutig eq (7) ito eq (), oe obtais M EA w w w P dx A x x x t (8) Based o the variatioal priciple, the exact relatioship betwee bedig momet ad trasverse displacemet based o olocal stress theory is give by [, ] w x M EI EI e a, (9) x where I is the secod momet of area From eqs (8) ad (9), the followig oliear partial differetial equatio goverig the trasverse vibratio ca be obtaied: w x x t EA w w dx w w w EI EI e a A P () x x It is obvious to observe i eq () the differece betwee the olocal ad classical cotiuum models The trasverse free vibratio of a beam or tube based o classical cotiuum theory is described by a well-kow fourth-order differetial equatio which ca be recovered by droppig the oliear terms (axial elogatio iduced by trasverse dormatio) ad the olocal terms cotaiig ea i eq () Accordig to some previous olocal models [, 6, 7], a fourth-order goverig equatio was preseted However, the goverig equatio derived from the variatioal priciple turs out to be a ifiite higher-order differetial equatio i terms of e a The major differece betwee this ew olocal model ad the previous olocal models was highlighted by im [, ] Because it is almost impossible to solve the ifiite higher-order differetial equatio, oly terms of order O[(e a) ] ad lower i the ifiite series i eq () are retaied This term is the most sigificat term which cotais olocal fect i the goverig equatio Hece, eq () becomes a sixth-order partial differetial-itegral equatio as 6 w w w EI ea A 6 x x t EA w w P x x dx () For trasverse free vibratio of a simply supported aobeam cosidered here, the trasverse displacemet ca be assumed as πx wx, t Wtsi, () where =,,3, ad W t idicates the time-depedet vibratio mode Substitutig eq () ito eq (), oe has the goverig equatio EI π W π ea EA π W π W d W P A dt (3) Itroduce the o-dimesioal variables ad parameters as follows: A EI P ea W W, t t, P,, () I A EI

4 i C, et al Sci Chia Tech Sci August () Vol5 No8 where is the o-dimesioal olocal aoscale parameter By usig these dimesioless variables ad parameters, eq (3) ca be cast ito the dimesioless form as follows: where ad d W 3, W W (5) dt P π π π, (6a) π, (6b) where =,,3, The dimesioless frequecy i eq (6a) is related to the dimesioal frequecy via A (7a) EI 3 Because W i eq (5) is origiated from A i eq (3) it ca be cosidered as a small parameter compared with P Thus the iteral axial force N fluctuates with respect to P by a small force A due to additioal extesio caused by trasverse dormatio Eq (5) is a Duffig-type oliear equatio with dimesioless oscillatio frequecy died as A (7b) EI If A is eglected, N=P ad the oliear term i eq (5) disappears The exact frequecy i this case is π P π π O the other had, if the aobeam is subjected ito a compressio P com, the liear atural frequecy is give by π Pcom π π Thus, the iitial compressio should satisfy P (8) com π π, Newto-harmoic balace [7], idstedt Poicaré method [8], multiple scales [9], etc Accordig to the method of perturbatio aalysis, the secod-order approximatio for eq (5) ca be expressed as [6] where a a W a cos cos 3 cos a cos5, () 3a 5 a t 3, 8 56 () where a ad are itegral costats which ca be determied from the iitial coditios The frequecy ca be approximated as 3a 5 a 8 56 () 3 The fudametal frequecy, =, based o the ew olocal stress model is show i Figure, where umerical solutios for without oliear fect from eq (6a) ad with oliear fect from eq () are preseted, ad a =5 is assumed It is see that the presece of a olocal aoscale icreases the atural frequecy The result here is i cotradictio to the coclusio of may previous studies [, 6, 7] The critical differeces have bee fully described i detail by im [, ] I bri, as metioed i the itroductio, the disagreemet is due to the adoptio of a partial olocal model i the previous works, i which the classical goverig equatios of motio were directly applied with relevat classical quatities replaced by the correspodig olocal quatities Such direct replacemet is ivalid because the goverig equatio obtaied is a mixture of classical ad olocal models, thus ad the critical compressio, or the critical bucklig load, ca be approximated as P cr (9) Eq (9) implies that the olocal aoscale fects ted to icrease the critical compressio for bucklig 3 Solutio ad discussio Sice eq (5) is a stadard Duffig-type equatio [5], its solutio ca be obtaied via a umber of approximate methods such as perturbatio [6], harmoic balace [5], Figure Compariso of approximate atural frequecies for without oliear fect ad with oliear fect for icreasig aoscale, a =5

5 i C, et al Sci Chia Tech Sci August () Vol5 No8 they are termed partial olocal models The ew coclusio is cosistet with the outcome of other o-olocal approaches, eg Ma et al [3] It is also observed from Figure that a larger exteral tesio P causes higher ad as expected Moreover, cosiderig precise iteral axial tesio ehaces the stiffess, ie, oliear fects cause a further icrease of frequecy ad stiffess, as ca be observed by comparig with The dimesioal atural frequecy of a simply supported beam with exteral axial tesio based o the classical cotiuum theory was give by Thomso [9] ad Guede ad Elishakoff [3] as π EI P cla π,,,3, A A (3) Cosiderig eqs (7b) ad (), the dimesioal form of olocal frequecy with oliear fect ca be derived Takig a simply supported silico beam subjected to exteral axial tesio as a example, the umerical solutios for oliear olocal (solid curves) ad liear classical or local (dotted curves) models are show i Figures 3 ad Figure 3 The fect of e a o the fudametal frequecies ad for a silico aobeam The cross sectio of beam is assumed rectagular (a circular cross sectio is equally valid) with width 3 m, thickess m, A=3 8 m, ad I=5 37 m Note that the Youg s modulus is size-depedet at aoscale I this example, E=83 GPa which is depedet o the thickess [3], ad other parameters are = kg/m 3, P= N ad a =5 I these figures, ( ) cla are the classical cotiuum results which have o olocal fect ad the values are ot affect by e a Values for are higher tha ( ) cla ad the differece is much larger for a smaller or a higher e a The differece caot be igored especially for beam at aoscale For istace, for =8 m ad e a=5 m, ( ) cla is % ad 5% lower tha for the fudametal (=) ad the secod mode (=) frequecies, respectively The olocal model preseted i this study ca be applied to vibratio of carbo aotubes Cosider aother example: a sigle-walled carbo aotube with ier ad outer radii 8 ad 3 m, respectively, I=3 37 m, A=3 8 m, =3 kg/m 3, P= N ad e a= m Note that Youg s modulus varies for aotubes with differet legths while it teds to be uchaged for sufficietly log aotubes I this example, E=35 TPa is adopted for the legth which exceeds 5 m [33], where a =5 Table shows umerical solutios for the first two mode frequecies for the classical, liear olocal ad oliear olocal models, respectively It is observed i Table that ( ) cla, ad decrease with icreasig I additio, the olocal frequecies are higher tha the classical frequecies, of which is the highest The differece is particularly sigificat for a small For example, for =5 m, ( ) cla is 8% ad 36% lower tha ad, respectively, ad for the secod mode, the differeces are 77% ad 3%, respectively Furthermore, the oliear fect causes the fudametal ad secod mode frequecies to icrease by 8% ad 5%, respectively, with respect to the liear olocal frequecy for =5 m These differeces should be take ito cosideratio i aalyzig ad desigig aoscale compoets for NEMS With icreasig legth, the differece becomes Table Compariso of classical ( ) cla, liear olocal ad oliear olocal frequecies for a sigle-walled carbo aotube Frequecy Naotube legth (m) (GHz) Figure The fect of e a o the secod model frequecies ad for a silico aobeam ( ) cla ( ) cla

6 i C, et al Sci Chia Tech Sci August () Vol5 No8 smaller ad the frequecies have a good agreemet To aalyze vibratio of olocal aotubes, it is ecessary to determie the iteral scale parameter e a ad this ca be doe by comparig with some other aomechaical studies, such as the scaig force microscopy study of Garcia-Sachez et al [33] I the study, a aotube with =93 m, outer radius R =3 m, ier radius R =7 m, desity of graphite = kg/m 3, Youg s modulus E= 3 TPa was ivestigated The first mode frequecy was reported as f = /=573 MHz Therore, the iteral scale parameter is matched as e a=96 m If the iteral characteristic scale is assumed as the crystal lattice parameter of carbo, ie, a= m, the material costat is obtaied as e =676 i this example The axial displacemet which varies with time is show ext The o-dimesioal form of eq () is give by w x, t W t siπ x, () where w w is the dimesioless trasverse displacemet ad x x is the dimesioless coordiate Applyig eq (), we obtai the time-depedet oliear trasverse vibratio from eq () The relatios of dimesioless displacemet, coordiate ad time are show i Figures 5 ad 6, for the first ad the secod modes, respectively, where a =5, =5, = ad P are assumed Coclusios Figure 6 The secod vibratio mode Two major poits are cocluded i this work Firstly, the olocal fects ehace atural frequecy of a aobeam based o the ew olocal stress model preseted here, or the aobeam stiffess is stregtheed i compariso with the classical cotiuum theory This is a ew coclusio because the previous olocal aobeam studies cocluded with reduced stiffess ad thus lower vibratio frequecies Secodly, the precise iteral axial tesio is ot equal to the exteral tesio The oliear vibratio is equivalet to a Duffig-type oscillatio ad the cosideratio of oliearity results i higher atural frequecies Exteral axial tesio also has sigificat ifluece o the vibratio behaviors ad the frequecy is higher for a larger tesio For a aobeam with sufficiet legth, the olocal aoscale fect becomes isigificat ad thus the goverig equatio ca be reduced to the classical beam vibratio equatio Both olocal ad classical solutios are i good agreemet, ad validity of the olocal model developed here for vaishig olocal fect is established Figure 5 The first vibratio mode This work was supported by a collaboratio scheme from Uiversity of Sciece ad Techology of Chia-City Uiversity of Hog Kog Joit Advaced Research Istitute ad by City Uiversity of Hog Kog of Chia (Grat No 7699 (BC))

7 i C, et al Sci Chia Tech Sci August () Vol5 No8 3 Maragati R, Sharma P egth scales at which classical elasticity breaks dow for various materials Phys Rev ett, 7, 98: 955 iao G, Zuo H B, Cao Y B, et al Optical properties of the micro/ao structures of Morpho butterfly wig scales Sci Chia Tech Sci,, 53: Cammarata R C Surface ad iterface stress fects i thi films Prog Surf Sci, 99, 6: 38 Zhu H X, Wag J X, Karihaloo B Effects of surface ad iitial stresses o the bedig stiffess of trilayer plates ad aofilms J Mech Mater Struct, 9, : Midli R D Micro-structure i liear elasticity Arch Ratioal Mech Aalysis, 96, 6: Aifatis E C O the microstructural origi of certai ielastic models ASME J Eg Mater Tech, 98, 6: Erige A C iear theory of olocal elasticity ad dispersio of plae waves It J Eg Sci, 97, : Erige A C O differetial equatios of olocal elasticity ad solutios of screw dislocatio ad surface waves J Appl Phys, 983, 5: im C W, He H Size-depedet oliear respose of thi elastic films with ao-scale thickess It J Mech Sci,, 6: im C W, i Z R, He H Size depedet, o-uiform elastic field iside a ao-scale spherical iclusio due to iterface stress It J Solids Struct, 6, 3: Wag Q Wave propagatio i carbo aotubes via olocal cotiuum mechaics J Appl Phys, 5, 98: 3 Zhag Y Q, iu G R, Xie X Y Free trasverse vibratios of double-walled carbo aotubes usig a theory of olocal elasticity Phys Rev B, 5, 7: 95 3 Zhag P W, Zhou Z G, Wu Z The o-local theory solutio of a Griffith crack i fuctioally graded materials subjected to the harmoic ati-plae shear waves Sci Chia Ser E-Tech Sci, 7, 5: 5 65 Hu Y G, iew K M, Wag Q, et al Nolocal shell model for elastic wave propagatio i sigle- ad double-walled carbo aotubes J Mech Phys Solids, 8, 56: iag J The olocal theory solutio of a Mode-I crack i fuctioally graded materials Sci Chia Ser E-Tech Sci, 9, 5: 6 Dua W H, Wag C M Exact solutios for axisymmetric bedig of micro/aoscale circular plates based o olocal plate theory Naotechology, 7, 8: im C W, Wag C M Exact variatioal olocal stress modelig with asymptotic higher-order strai gradiets for aobeams J Appl Phys, 7, : 53 8 Yag X D, im C W Noliear vibratios of ao-beams accoutig for olocal fect usig a multiple scale method Sci Chia Ser E-Tech Sci, 9, 5: im C W Equilibrium ad static dlectio for bedig of a olocal aobeam Adv Vib Eg, 9, 8: 77 3 im C W O the truth of aoscale for aobeams based o olocal elastic stress field theory: equilibrium, goverig equatio ad static dlectio Appl Math Mech,, 3: 37 5 im C W, i C, Yu J Dyamic behaviour of axially movig aobeams based o olocal elasticity approach Acta Mech Si,, 6: im C W, Yag Y Nolocal elasticity for wave propagatio i carbo aotubes: the physics ad ew predictio of aoscale i olocal stress field J Comput Theor Naosci,, 7: im C W A aorod (or aotube) with lower Youg s modulus is stiffer? Is ot Youg s modulus a stiffess idicator? Sci Chia Phys Mech Astro,, 53: 7 7 Nayfeh A H, Mook D T Noliear Oscillatios New York: Wiley, Nayfeh A H Itroductio to Perturbatio Techiques New York: Wiley, 98 6 ai S K, im C W, Wu B S, et al Newto-harmoic balacig approach for accurate solutios to oliear cubic-quitic Duffig oscillators Appl Math Mod, 9, 33: Su W P, Wu B S, im C W A modified idstedt Poicaré method for strogly mixed-parity oliear oscillators ASME J Comput Noliear Dy, 7, : 5 8 Thomso W T Theory of Vibratio with Applicatios Eglewood Cliffs: Pretice-Hall, 98 9 Ma H M, Gao X, Reddy J N A microstructure-depedet Timosheko beam model based o a modified couple stress theory J Mech Phys Solids, 8, 56: Guede Z, Elishakoff I Apparetly first closed-form solutios for ihomogeeous vibratig beams uder axial loadig Proc R Soc od A,, 57: Bao F, Yu H, Huag Q A Elastic modulus of aometer silico membrae I: the 6 IEEE Iteratioal Coferece o Iformatio Acquisitio, Weihai, Chia, 6 3 Cai J, Wag Y D, Wag Y C Effect of edig surface o eergy ad Youg s modulus of sigle-walled carbo aotubes studied usig liear scalig quatum mechaical method Phys B: Codes Matter, 9, : Garcia-Sachez G, Sa Paulo A, Espladiu M J, et al Mechaical detectio of carbo aotube resoator vibratios Phys Rev ett, 7, 99: 855