-
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:
[email protected] (K.B. Mustapha).
International Journal of Engineering Science 53 (2012) 4657
Contents lists available at SciVerse ScienceDirect
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
journal homepage: www.elsevier .com/locate / i jengsci
-
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
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vibration of pretwisted beams. Journal of Sound and Vibration, 321,
115136.Ma, H. M., Gao, X. L., & Reddy, J. N. (2008). A
microstructure-dependent Timoshenko beam model based on a modied
couple stress theory. Journal of the
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Farshidianfar, A., Tahani, M., Mahdavi, S., & Dalir, H. (2008).
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(1991). Continuum mechanics for engineers. Boca Raton: CRC
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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