Modied couple stressWave propagationSpectrum curveTwisted micro beam

the effects of the rate of twist and the material length scale on the bending wave propaga-

ted balysesat int

The focus of extensive research on twisted beams is due to their importance in a number of engineering applications.Some of the most common applications of twisted structural elements involve their use in the idealization of blades for pro-pellers, wind turbines, drill bits and gear teeth (Liao & Huang, 1995; Lin, Wang, & Lee, 2001; Yardimoglu & Yildirim, 2004).Other applications are related to their use in providing insight into wave propagation characteristics of helical waveguides,

0020-7225/$ - see front matter 2011 Elsevier Ltd. All rights reserved.

Corresponding author. Fax: +65 67911859.E-mail address: kham0002@e.ntu.edu.sg (K.B. Mustapha).

International Journal of Engineering Science 53 (2012) 4657

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International Journal of Engineering Sciencedoi:10.1016/j.ijengsci.2011.12.006cent work on the subjects includes: the studies by Subrahmanyam and Rao (1982) on the use of Reissner method for thevibration study of a twisted beam, Leung (1991) on the development of exact dynamic stiffness of a pre-twisted beam. Liaoand Dang (1992) analyzed the transverse vibration and stability of an orthotropic twisted beam. Banerjee (2001) employedthe dynamic stiffness method for the free vibration analysis of an EulerBernoulli (EB) beam. Yardimoglu and Yildirim (2004)developed a nite element model of a twisted Timoshenko beam. Leung and Fan (2010) studied the effect of multiple initialstresses on the natural frequencies of a pre-twisted Timoshenko beam. Chen (2010) investigated the parametric instability ofan axially loaded twisted beam. Sinha and Turner (2011) analyzed the dynamics of a pre-twisted beam in a centrifugal forceeld.1. Introduction

From the thorough survey, presendent that the static and dynamics antwo decades after Rosens review, thtion characteristics of the micro scale beam. Results are presented for the spectrum curve,the cut-off frequency, the phase speed and the group velocity of a propagating harmonicwave prole in the twisted micro scale beam. It is observed that the rate of twist bifurcatesthe spectrum curve of the micro scale beam within a given frequency range, while thematerial length scale improves the dispersion of the traveling wave. The cut-off frequencyis found to be independent of the material length scale, but proportional to the fourthpower of the rate of twist. Increasing the material length scale is further observed toincrease the group velocity of the wave, while a high rate of twist lowers the wave speeds.

2011 Elsevier Ltd. All rights reserved.

y Rosen (1991), of the works done by researchers up to the early nineties, it is evi-of twisted beams have long been a subject of great interest to researchers. Almosterest has not waned. A carefully selected, but obviously not exhaustive, list of re-Wave propagation characteristics of a twisted micro scale beam

K.B. Mustapha , Z.W. ZhongSchool of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Republic of Singapore

a r t i c l e i n f o

Article history:Received 28 November 2011Accepted 15 December 2011Available online 28 January 2012

Keywords:

a b s t r a c t

Twisted structural elements are inherently complex and their use in engineering systemsrequires deeper understanding. In this paper, starting with the modied couple stress the-ory, the elastodynamics governing partial differential equations of motion for the trans-verse dynamics of a twisted micro scale beam is derived. A micro scale beam ofrectangular cross-section, for which the rate of twist introduced a bendingbending cou-pling effect, is considered. The presented governing equation of motion is used to address

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twiste2009;

K.B. Mustapha, Z.W. Zhong / International Journal of Engineering Science 53 (2012) 4657 47The design of micro/nano-electro-mechanical systems (MEMS/NEMS) requires widespread usage of microshatfs, micro-beams and microplates of different complexity (Sheng, Li, Lu, & Xu, 2010). Microbeams, in particular, seemed to have at-tracted the interest of researchers due to the crucial role they play in high-precision measuring instrument like atomicforce microscopy (Mahdavi, Farshidianfar, Tahani, Mahdavi, & Dalir, 2008). The use of twisted micro scale beam is foundin application areas as diverse as micro turbo machinery and micromachining (Yamashita et al., 2011; Zhang & Meng,2006), the design of efcient ultrasonic piezoelectronic motor (Liu, Friend, & Yeo, 2009; Wajchman, Liu, Friend, & Yeo,2008), and the development of a micromotor for in vivo medical procedures (Watson, Friend, & Yeo, 2009). Understandingthe mechanical behavior of a twisted micro scale beam is decisive in providing a functional guideline for the accurate designof the above devices. At present, almost all of the available studies on twisted beams, to the best our knowledge, are based onthe classical elasticity theory. A well-known deciency of the classical elasticity theory upon which those available studiesare based is the inability to characterize the size effect on the mechanical behaviors of structures within the micron or sub-micron scale (Georgiadis & Velgaki, 2003). With their sub-micron level thickness, the response of micro structural elementshas been reported to show size effect, a phenomenon that the classical elasticity theory fails to demonstrate (Zhenyu, Saif, &Yonggang, 2002). The size-free deciency of the classical elasticity theory has therefore raised concern about the legitimacyof employing the classical models to understand the intricate microscopic phenomenon in micro as well as nano structures(Exadaktylos & Vardoulakis, 2001; McFarland & Colton, 2005; Mustapha & Zhong, 2012).

The existence of size dependency in metals, anisotropic materials like human bones, and elastic-perfectly plastic mate-rials has been experimentally veried (Aifantis, 1992; Fleck & Hutchinson, 1993; Lam & Chong, 1999; Poole, Ashby, & Fleck,1996). So far the need to include size effect in structural models of micro scale structures has resulted in the emergence of anumber of higher order continuum theories with size-dependent constitutive laws. Among the higher-order continuum the-ories, Eringens nonlocal elasticity theory (Eringen & Edelen, 1972), the grade-2 strain gradient theory (Askes & Aifantis,2011), the couple stress theory and the micropolar elasticity have provided framework to capture the existence of size effectin structural elements. The distinguishing feature among the above higher-order theories is the number of material con-stants (apart from the Lam constants) that each theory comprises. The Eringens nonlocal theory (Eringen, 1972, 1983),for example, is characterized by two material constants. This theory is now broadly used to investigate the degree to whichthe size-effect phenomenon contributes to the structural response of carbon nanotubes (Mustapha & Zhong, 2010, 2012;Reddy & Pang, 2008; Wang, Zhou, & Lin, 2006). The strain gradient theory, based on the works of Mindlin and Eshel(1968) and later revisited by Aifantis (1995) and Papargyri-Beskou, Tsepoura, Polyzos, and Beskos (2003), contains fourmaterial constants, the determination of all of which is non-trivial.

The modied couple stress theory (Yang, Chong, Lam, & Tong, 2002), which is the theory upon which the current study isbased, originates from the classical couple stress theory. As well known, the classical couple stress theory is itself a variant ofthe micropolar elasticity theory (Aifantis, 1984). Furthermore, the formulation of the couple stress theory is derived from thepioneering works of Mindlin and Tiersten (1962), Toupin (1962) and Fleck and Hutchinson (1993). Under the couple stresstheory, the resolution of the applied force on an elastic material particle does not only include the force to drive the materialparticle to translate but also a couple to drive it to rotate (Yang et al., 2002). As a result, the macro-rotation of the particle isequivalent to the micro-rotation of its inner structural deformation. Based on the modied couple stress theory, Park andGao (2006) presented a size-dependent model for the static analysis of the EulerBernoulli (EB) beam. Ma, Gao, and Reddy(2008) developed a varionationally consistent size-dependent governing equation for the static and vibration analyses of theTimoshenko beam. The transverse vibration study of the EB beam, also based on the modied couple stress theory, is pre-sented by Kong, Zhou, Nie, and Wang (2008). Furthermore, the static and vibration problem of an embedded micro-beamwith a moving micro-particle, the formulation of which is based on the modied couple stress theory, is also recently pre-sented by Simsek (2010).

In the absence of available study on the modeling of twisted micro scale beam based on the above higher order theories,we make effort to present a size-dependent governing equation for the dynamics of a twisted micro scale beam on the basisof the modied couple stress theory. The governing equation, thus derived, is then employed to address the effect of both thematerial length scale and the rate of twist on the wave propagation characteristics of a micro scale beam with a rectangularcross section. Propagation of elastic waves in structures is crucial for most of the well-known non-destructive tests. For frag-ile micro scale structures, especially those that require non-invasive modal analysis, the presence of the small-scale effect isexpected to affect the wave propagation characteristics (Georgiadis & Velgaki, 2003).

2. Fundamental relation of the modied couple stress theory

The generalized constitutive model based on the nonlocal modied couple stress theory is rst presented for an isotropic,Hookean material. From this constitutive model, the coupled bendingbending deformation energy of the twisted microbeam is derived. According to the modied couple stress theory, the strain in an isotropic linear elastic material occupyinga volume V can be written as (Ma et al., 2008; Park & Gao, 2006; Wang, Zhou, Zhao, & Chen, 2011):

Ps 12Z8r : em : v dV 1d hollow waveguides and twisted quantum waveguides (Kovak & Sacchetti, 2007; Wilson, Cheng, Fathy, & Kang,Yabe, Nishio, & Mushiake, 1984).

where e is the dilatation strain tensor, r is the Cauchy stress tensors,m is the deviatoric component of the couple stress andv is the symmetric curvature tensors. If the displacement and rotation vectors are both represented by u and h, respectively,then the tensors e and v in Eq. (1) satisfy the following geometric relations:

e 12ru ruTh i

2

v 12rh rhTh i

3

where r is the Laplacian notation. The rotation vector h is related to the displacement vector in the form:

h 12curl u 4

The dilatation strain and the symmetric curvature tensors in Eqs. (2) and (3) are related to the Cauchy stress and the devi-atoric component of the couple stress to form the constitutive relations as (Wang et al., 2011):

r ktreI 2Ge 5m 2l2Gv 6

where k and G are the bulk and shear modulus respectively. An additional material constant in the form of lwould be noticedin Eq. (6). This parameter is used to denote the material length scale and is physically a property measuring the effect of thecouple stress (Kahrobaiyan, Tajalli, Movahhedy, Akbari, & Ahmadian, 2011; Kong, Zhou, Nie, & Wang, 2008). In terms of itsmathematical equivalent, the material length scale l, according to Park and Gao (2006) and Aifantis and Willis (2005), isfound to be analogous to the square root of the ratio of the modulus of curvature of a deformed body to the modulus of shear.Experimental determination of the value l has been carried out through the bending tests (Stlken & Evans, 1998) of thin

48 K.B. Mustapha, Z.W. Zhong / International Journal of Engineering Science 53 (2012) 4657Fig. 1. Schematic of a uniformly twisted micro scale beam and coordinate systems.beams or torsion tests of slim cylinders (Chong, Yang, Lam, & Tong, 2001). In subsequent derivations, k and G are writtenin terms of the conventional Poissons ratio (#) and the Youngs modulus (E). That is, k = E#/(1 + #) (1 2#) and G = E/2(1 + #).

3. Governing equation of a twisted micro scale beam

For the problem at hand, the schematic shown in Fig. 1, of a twisted micro scale beam of total length L, thickness h andwidth b is considered.

In deriving the governing equation of the twisted micro scale beam, two fundamental assumptions are made. The rst isbased on the postulate of the EulerBernoulli beam theory, while the second is the assumption of a uniform rate of twistalong the length of the micro scale beam. With these assumptions, a sets of kinematically admissible exible displacementsand rotations is employed to form the system of direct and shearing strains. In Fig. 2, the two sets of coordinate systemsneeded (inertia frame of reference and body coordinates) to derive the governing equation of this geometry are shown. Alongthe neutral surface of the undeformed micro beam cross section is the inertial frame of reference XYZ located at origin O. CGis taken to be the centroid at a distance z from the origin of the inertial frame, while CGx and CGy are the principal axes in

bendi

the mLet

axial s

Fig. 2. Coordinate systems and eld variables of the twisted micro scale beam.

K.B. Mustapha, Z.W. Zhong / International Journal of Engineering Science 53 (2012) 4657 49Vz; t sinu cosu vz 9

where

bx dvzdz

dudz

uz; by duzdz

dudz

vz 10

Putting Eq. (10) into Eq. (7) results in:

W y dv du u

x du duv

11From

With

By taktrain effect), the following relations hold (Banerjee, 2001; Rosen, Loewy, & Mathew, 1987):

Wz; t ybx xby 7

x

y

cosu sinu

sinu cosu

X

Y

8

Uz; t cosu sinu uz tively and the rotations about the x and y axes to be bx and by respectively. Also, let the corresponding displacement along theinertia frame of reference be U, V andW along X, Y and Z respectively. From the kinematics of the deformation (neglecting theicro scale beam is du/dz.the admissible exible displacement of a material point along the body coordinates x, y and z be u, v and w respec-right-handed rotation about OZ. This right-handed rotation ensures the angle of rotation (u) between CGx and OZ andCGy and OZ to be equal. At the centroidal point CG, u is the angle of twist. Therefore the rate of twist along the length ofng of the cross section. Following the local coordinates derivation in Banerjee (2001), the CG(x,y) axis system has adz dz dz dz

Eq. (8), we substitute the expression for y and x in Eq. (11) to have:

W X sinu Y cosu dvdz du

dzu

X cosu Y sinu du

dz du

dzv

12

the displacement functions dened, the normal component of the Cauchy strain (Mase & Mase, 1991) is dened as:

ezz oWoz dudz

Y cosu X sinu ouoz

w ouoz

ou

ozX cosu Y sinu ou

ozu ov

oz

X cosu Y sinu ouoz

ovoz o

2uoz2

! Y cosu X sinu ou

ozouoz o

2voz2

!13

ing note of the transformation matrix in Eq. (8), the expression for the normal strain (ezz) is further simplied as:

ezz x ouoz 2

u 2 ouoz

ovoz o

2uoz2

" # z ou

oz

2w 2 ou

ozouoz o

2voz2

" #14

eyy exx exy eyz exz 0 15

From

The nsubstiponent of the couple stress:

2 2

Asreplacmicro

oz oz oz oz oz oz

wherearea of the cross section of the micro scale beam. Using the allowable displacement components, u and v, the total kinetic

2 ot ot

wherewith the Neumann boundary conditions, are now obtained with the help of the extended Hamiltons principle, which allows

Negleinto E

The ab

50 K.B. Mustapha, Z.W. Zhong / International Journal of Engineering Science 53 (2012) 4657 E#rIXX oz4 2c oz3 2c oz2 2c oz c v 2cE#rIYY c oz 2c oz2 oz3l2AG o4uoz4

2c o3voz3

2 ocoz

o2voz2

3 o2coz2

ovoz v o

3coz3

c2 o2uoz2

c ocoz

ouoz uc o

2coz2

u ocoz

2" # qA o

2uot2

#du

o4v o3u 2 o2v 3 ou 4

" #2 ou o

2v o3u" #"Z T0

Z L0

E#rIYYo4uoz4

2c o3voz3

2c2 o2uoz2

2c3 ovoz c4u

" # 2cE#rIXX c2 ovoz 2c

o2uoz2

o3voz3

" #"( d o2voz2

c oduoz

du ocoz

" # qA od

otouot qA od

otovot

!dzdt 0 22

ove equation is integrated by part to give: c2dv 2c oduoz

d o2voz2

" # l2AG o

2uoz2

c ovoz v oc

oz

" #d o

2uoz2

c odvoz

dv ocoz

" # l2AG o

2voz2

c ouoz u oc

oz

" #0

cting the external work done Pw, we substitute the expressions for Ps and Pt from Eq. (19) and Eq. (20), respectively,q. (21) and apply the criterion of variational calculus to obtain:

Z T0

Z L0

E1 #

1 #1 2# IYY c2u 2c ov

oz o

2uoz2

" #c2du 2c odv

oz d o

2uoz2

" # E 1 #1 #1 2# IXX c

2v 2c ouoz o

2voz2

" # the derivation of equations of motion of a conservative, holonomic system in the form (Goldstein, 1980):

dZ T

Ps Pt Pw dt 0 210

q is the density of the beammaterial and A is the area. The governing equations of the twisted micro scale beam, alongenergy (Pt) of the micro scale beam is given as:

Pt 1Z L

qAou 2

ov 2" #

dz 20the denitionsRA y

2dA IXX andRA x

2dA IYY have been introduced in Eq. (19) to represent the principal moment ofl2AG o2u2 c

ov v oc" #2

l2AG o2v2 c

ou u oc" #21Adz 19oz 1 #1 2# oz oz oz oz oz oz oz oz

mzy l2G ooz ouoz ou

ozv

; mzx l2G ooz

ovoz ou

ozu

18

earlier pointed out ouoz is the rate of twist along the length of the micro scale beam. For convenience this term will beed by c from now on. By substituting Eqs. (14)(18) into Eq. (1), the total strain energy (in bending) of the twistedscale beam is obtained as follows:

Ps 12Z L0

E1 #

1 #1 2# IYY c2u 2c ov

oz o

2uoz2

" #2 E 1 #1 #1 2# IXX c

2v 2c ouoz o

2voz2

" #20@rzz o E 1 # x ou 2

u 2 ou ov o2u

" # y ou

2w 2 ou ou o

2v" # !

17Eqs. (3) and (10), we obtain the components of the symmetric curvature tensor as:

vij v11 v12 v13v21 v22 v23v31 v32 v33

264

375 1

2

0 0 oozovoz ouoz u

0 0 ooz ouoz ouoz v

0 0 0

264

375 16

on-zero components of the Cauchy strain and the symmetric curvature tensors in Eqs. (14)(16) are appropriatelytuted in Eqs. (5) and (6), to arrive at the following non-zero components of the Cauchy stress and the deviatoric com-

2 o4v o3u oc o2u o2c ou o3c oc ov o2v o2c oc

2" # o2v# )

T

0

where r 1#12#

" #

4.1. W

Tovidesto invby wr

wheredirectoscilla

wherein thethe poleads

K.B. Mustapha, Z.W. Zhong / International Journal of Engineering Science 53 (2012) 4657 51U(z,x) and V(z,x) are the frequency domain amplitudes of the exural deformation of the twisted micro scale beamxz and yz planes respectively. k represents the wavenumbers (which is equal to the inverse of the waves wavelength insitive z direction). x denotes the frequency of the motion of the wave. Substituting Eq. (28) into Eqs. (24) and (25),to two homogenous algebraic equations in terms of U and V as:

M11 M12M21 M22

U

V

0

0

29wave prole in the xz and yz planes:

uz; t Uz;xejkzxt; vz; t Vz;xejkzxt 28ave dispersion relation

determine the phase speed and phase velocity of a wave guided through a medium, the dispersion relation, which pro-the relationship between the frequency of the traveling wave and the wavenumbers, is needed (Doyle, 1997). In orderestigate the propagation of the bending elastic wave in the twisted micro scale beam the dispersion relation is derivediting the general solution of Eqs. (24) and (25) in the form:

uz; t Q1z cst Q2z cst 26vz; t Q3z cst Q4z cst 27

Qi(i = 1 . . .4) are used to represent wave proles. Q1(z cst) and Q3(z cst) represent waves moving in the positive zion with a velocity of cs, while Q2(z + cst) and Q4(z + cst) are the waves moving in the opposite direction. Considering antory motion of the wave, which is typical in dynamics analyses (Virgin, 2007), we consider the following travelingE#rIXXo4voz4

2c o3uoz3

2c2 o2voz2

2c3 ouoz c4v

" # 2cE#rIYY c2 ouoz 2c

o2voz2

o3uoz3

" #

l2AG o4voz4

2c o3uoz3

2 ocoz

o2uoz2

3 o2coz2

ouoz u o

3coz3

c ocoz

ovoz c2 o

2voz2

vc o2coz2

v ocoz

2" # qA o

2vot2

0 25

Before moving onto the analysis of the bending waves in the twisted micro scale beam, a couple of remarks about the derivedgoverning equations (that is, Eqs. (24) and (25)) is in order. First, if the rate of twist c is set to zero, IXX = IYY = I and G = l, thesize-dependent governing equation of the micro scale beam presented by Kong et al. (2008) is obtained. Secondly, if thematerial length scale parameter l and Poissons ratio are set to zero, then the governing equation of the classical twistedEB beam presented by Banerjee (2001) is retrieved. Thirdly, it is observed that the modied couple stress theory has intro-duced higher derivatives of the rate of twist c. The higher derivatives of the rate of twist must be considered for a non-uni-formly twisted micro scale beam. However, due to the assumption of a constant rate of twist, the higher derivatives of therate of twist introduced by the couple stress will be neglected in analyses that follow.

4. Bending wave analysis and numerical discussion l2AG o4uoz4

2c o3voz3

2 ocoz

o2voz2

3 o2coz2

ovoz v o

3coz3

c2 o2uoz2

c ocoz

ouoz uc o

2coz2

u ocoz

2 qA o

2uot2

0 24the elastodynamics governing equation of a twisted micro scale beam as:

E#rIYYo4uoz4

2c o3voz3

2c2 o2uoz2

2c3 ovoz c4u

" # 2cE#rIXX c2 ovoz 2c

o2uoz2

o3voz3

" #

# is used to represent 1# . From Eq. (23), the fundamental lemma of variational calculus is invoked to arrive at0

E#rIXX c2ovoz 2c o u

oz2 o v

oz3dv E#rIXX c2v 2c ouoz

o voz2

odvoz

E#rIXX2c c2v 2c ouoz o voz2

du

E#rIYY c2 ouoz 2co2voz3

o3voz3

" #du E#rIYY c2u 2c ovoz

o2uoz2

" #oduoz

2cE#rIYY c2u 2c ovoz o2uoz2

" #dv#Ldt 0 23l AGoz4

2coz3

2oz oz2

3oz2 oz

uoz3

coz oz

c2oz2

voz2

voz

qAot2

dv dzdt

Z 2 3" # 2" # 2" #"

The ze

In spitpointi

Thscalelength

In

decreathe ca

microthe afof thefurthe

Eacmater

52 K.B. Mustapha, Z.W. Zhong / International Journal of Engineering Science 53 (2012) 4657(i) that the bifurcation of the spectrum curve does occur in the frequency range 0.2 MHz 6x 6 0.4 MHz and (ii) that thewave prole of the couple stress theory travels at a faster frequency than that of the classical theory. Furthermore, it is ob-scale beam due to the presence of the material length scale is observed to narrow the spectrum curve. In addition toorementioned effect, bifurcations of the spectrum curves for the TCEB, h = 4 l, and h = 4 l is also noticed. The bifurcationspectrum curve is observed to occur in the frequency range 0.2 MHz 6x 6 0.4 MHz. To probe the bifurcation pointr, a plot of the real and imaginary wavenumbers is given in Fig. 5.h of the plots in the positive region of Fig. 5 is the real component of the dispersion curve at the different values of theial length scale, while those in the negative region are the imaginary equivalent. Fig. 5 conrms two important points:The combined effect of the rate of twist and the material length scale is shown in Fig. 4, where TCEB in the gure repre-sents twisted classical EulerBernoulli beam. Here, the rate of twist is kept at a value of c = 45 = p/4 radian, while the mate-rial length scale is varied for the spectrum curve. Similar to the case of the untwisted beam, the increase of the stiffness of these in the propagating space of the bending wave. The pattern of the dispersion curve of the propagating wave mode, inse of the CEB, agrees with those predicted by Doyle (1997) and Vinod, Gopalakrishnan, and Ganguli (2006).classical EulerBernoulli beam. The wavenumber-frequency relationship is almost linear in the absence of the rate of twist.Besides, as the thickness h decreases from 6 l to around 2 l (l being the material length scale parameter), there is a gradualFig. 3, the spectrum curve of a micro scale beam with a zero rate of twist is shown, where CEB in the gure denotesrial properties (Kong et al., 2008): E = 1.44 GPa, # = 0.38, q = 1.22 103 kg/m3. Furthermore, the cross section of the beam issuch that L = 20 h, b = 2 h, 17.6 lm 6 h 6 100 lm, and a uniform rate of twist c = 45 = p/4 radian.q4 S2S1

q3 S3S1

q2 S4S1

q1 S5S1 0 37

ros of Eq. (37) are then obtained by applying the normal root-nding technique to nd the wavenumbers in the form:

k1;2 q1p ; k3;4 q2p ; k5;6 q3p ; k7;8 q4p 38e of the above reduction, the expressions for qj(j = 1,2,3,4) are too unwieldy to be included here. However, it is worthng out that, four of the eight wavenumbers are real, while the other four are imaginary.e effects of the material length scale and the rate of twist on the fundamental wave propagation behavior of a microbeam are reported from here on. For the purpose of numerical analysis and illustration of the effects of the materialscale on the wave propagation characteristics, the beam considered in this study is taken to have the following mate-The zeros of Eq. (31) are the wavenumbers of the traveling bending wave in the twisted micro scale beam and from Eq.(31), bearing in mind Eqs. (32)(36), the dependence of the dispersion relation on the material length scale parameter l andthe rate of twist c is easily noticed. One of the easiest way to render the eight order polynomial that denes the dispersionrelation tractable is to reduce it to its quartic form (see e.g. Press (2007)):S5 E2#2r IXXIYYc8 E#rIXXc4qAx2 E#rIYYc4qAx2 qA2x4 36S4 E#rIXXGAl2c6 E#rIYYGAl2c6 4E2#2r IXXIYYc6 2GA2l2c2qx2 6E#rIXXc2qAx2 6E#rIYYc2qAx2 35where

M11 GAl2 k4 k2c2

4E#rIXXk2c2 E#rIYY k4 2k2c2 c4

qAx2

M12 i2GAl2k3c 2E#rIXXc ik3 ikc2

E#rIYY 2ik3c 2ikc3

M21 i2GAl2k3c 2E#rIYYc ik3 ikc2

E#rIXX 2ik3c 2ikc3

M22 GAl2 k4 k2c2

4E#rIYYk2c2 E#rIXX k4 2k2c2 c4

qAx2

The dispersion relation of the bending wave is now determined by the condition for the non-trivial solution of Eq. (29)given as the following form:

M11 M12M21 M22

0 30

The resulting dispersion relation from Eq. (30) is found as:

k8S1 k6S2 k4S3 k2S4 S5 0 31where

S1 G2A2l4 E#rIXXGAl2 E#rIYYGAl2 E2#2r IXXIYY 32

S2 2G2A2l4c2 E#rIXXGAl2c2 E#rIYYGAl2c2 4E2#2r IXXIYYc2 33

S3 G2A2l4c4 E#rIXXGAl2c4 E#rIYYGAl2c4 6E2#2r IXXIYYc4 2GA2l2qx2 E#rIXXqAx2 E#rIYYqAx2 34

K.B. Mustapha, Z.W. Zhong / International Journal of Engineering Science 53 (2012) 4657 53served that the imaginary parts of the wave prole are spatially decaying waves and hence can actually not be used to effec-tively transport energy from point to point within the beam material.

4.2. Cut-off frequency and wave speeds

The speed of propagation of a wave pulse in a medium is given by its phase velocity. Besides, when a wave packet or awave pulse comprising harmonic waves of different wavenumbers travels through such a dispersive medium as an elasticmicro beam, the speed of the wave packet or wave pulse is bound to be determined by the group velocity of the waves. Addi-tionally, the magnitude of the frequency of a wave pulse traveling within an elastic dispersive medium depends on the mag-nitude of the cut-off frequency. It is therefore of great interest to investigate the degree to which the material and thegeometric parameters of the twisted micro scale beam contributes to its cut-off frequency and the wave speeds (phaseand group velocity).

The cut-off frequency of the twisted micro scale beam is obtained by setting k = 0 in the dispersion relation dened by Eq.(31) which leads to:

Fig. 3. Effect of material length scale on the dispersion curve of a micro scale beam with zero twist.

Fig. 4. The spectrum curves a micro scale beam at a rate of twist of p/4 radian and different material length scales.

Fromthe raintrodof theof twiratio (This issectio

Figphase

54 K.B. Mustapha, Z.W. Zhong / International Journal of Engineering Science 53 (2012) 4657xcf

Eh2c4#r Eh2a2c4#r Ehc4#r

48a2h2qA2h2a2qAh2a4qA

p qA

pr

26

p 39

Eq. (39), it is obvious that the cut-off frequency is independent of the material length scale. However, it is a function ofte of twist and the second moment of inertia in the zx and zy planes. In Eq. (39), we have divided through by IXX anduced a2 = (IYY/IXX)0.5, where a2 now represents the thickness-to-width ratio of the micro scale beam. The rate of changecut-off frequency with the rate of twist is shown in Fig. 6 for different values of the thickness-to-width ratio. The ratest considered is in the range 0 6 c 6 2p radian. It is interesting to note that when the value of the thickness-to-widtha) of the twisted micro scale beam is 1, the relationship between the cut-off frequency and the rate of twist disappears.because when the thickness-to-width ratio is 1, the rectangular cross section is essentially reduced to a square crossn. Furthermore, the cut-off frequency increases with a higher value of the thickness-to-width ratio.s. 7 and 8 show the trend of the group velocity and the phase speed for different values of the rate of twist. Both thespeed and the group velocity of the bending wave in the twisted micro scale beam are found to be a function of the

Fig. 5. Real and imaginary components of the dispersion relation at a rate of twist of p/4 radian and different material length scales.

Fig. 6. Variation of the rate of twist and the cut-off frequency of the twisted micro scale beam.

K.B. Mustapha, Z.W. Zhong / International Journal of Engineering Science 53 (2012) 4657 55material length scale and other geometric parameters. By denition, the phase speed (cp) is a ratio of the propagating fre-quency and the wavenumbers (that is, cp =x/k), while the group velocity is (cg) is ox/ok. The two expressions for the phasevelocity and the group speed were numerically obtained from Eq. (38). An important observation in the analysis of the wavespeed for the twisted micro scale beam is that, unlike in the classical EB beam, the group velocity is more than twice thephase speed for all values of the wavenumbers obtained. Furthermore, it is noticed that both the phase speed and the groupvelocity show high propagating frequency at lower values of the rate of twist. Increasing the rate of twist therefore leads to adecrease in the magnitude of the propagating frequency.

5. Conclusion

The coupled elastodynamics governing equation of a twisted micro scale beam based on the modied couple stress theoryis presented. Based on the derived equation, the propagating characteristics of a monochromatic bending elastic wave arestudied. The following points are deduced from the analyses carried out:

Fig. 7. Variation of the phase speed and the propagating frequency of the wave prole at h = 2 l.

Fig. 8. Variation of the group velocity and the propagating frequency of the wave prole at h = 2 l.

Sciences, 37, 423439.

56 K.B. Mustapha, Z.W. Zhong / International Journal of Engineering Science 53 (2012) 4657Lin, S.-M., Wang, W.-R., & Lee, S.-Y. (2001). The dynamic analysis of nonuniformly pretwisted Timoshenko beams with elastic boundary conditions.International Journal of Mechanical Sciences, 43, 23852405.

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parameter elastic medium. Computational Materials Science, 50, 742751.Mustapha, K. B., & Zhong, Z. W. (2012). Stability of single-walled carbon nanotubes and single-walled carbon nanocones under self-weight and an axial tip

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Solids and Structures, 40, 385400.(i) The rate of twist of the micro scale beam is observed to cause the bifurcation of the spectrum curve of the wave prole.(ii) The wave motion in the micro scale beam of the modied couple stress theory is found to travel at a set of faster fre-

quency values than that of the classical theory.(iii) It is conrmed that the evanescent components of the wave prole decay exponentially in the spatial framework and

can thus not be used for effective energy transport.(iv) Both the phase speed and the group velocity of the bending wave in the twisted micro scale beam are found to be a

function of the material length scale.(v) The cut-off frequency is found to be independent of the material length scale but greatly affected by the rate of twist.(vi) The analysis of the wave speed for the geometry also reveals that, unlike in the classical EB beam, the group velocity is

more than twice the phase speed for all values of the wavenumbers determined.

The model developed in this study is easily extended to study the transverse vibration of a twisted micro scale beam. It isexpected that the study will also provide useful insight into the dynamics of twisted micro scale structures.

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Wave propagation characteristics of a twisted micro scale beam1 Introduction2 Fundamental relation of the modified couple stress theory3 Governing equation of a twisted micro scale beam4 Bending wave analysis and numerical discussion4.1 Wave dispersion relation4.2 Cut-off frequency and wave speeds

5 ConclusionReferences