Mechanics of Advanced Composite Structures 3 (2016) 99-112
Semnan University
Mechanics of Advanced Composite Structures
journal homepage: http://MACS.journals.semnan.ac.ir Size-dependent Effects on the Vibration Behavior of a Ti-
moshenko Microbeam subjected to Pre-stress Loading based on DQM
M. Mohammadimehr*, H. Mohammadi Hooyeh, H. Afshari, M.R. Salarkia
Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
P A P E R I N F O
A B S T R A C T
Pa per hist ory: Received: 2016-04-05 Revised: 2016-05-19 Accepted: 2016-06-25
In this paper, size-dependent effects on the vibration behavior of Timoshenko microbeams under pre-stress loading embedded in an elastic foundation, using modified strain gradient theory (MSGT) and surface stress effects, were studied. To consider the surface stress effects, the Gurtin–Murdoch continuum mechanical approach was employed. Using Hamilton’s principle, the govern-ing equations of motion and boundary conditions were obtained and solved numerically using the differential quadrature method (DQM). The effects of pre-stress loading, surface residual stress, surface mass density, Young’s modulus applied to the surface layer, three material length scale parameters, and the elastic foundation coefficients were investigated. For higher aspect ratios, this study found that the effect of the pre-stress loading was negligible for higher modes. Considering size-dependent effects led to increase the stiffness of the matrix and enhance the dimensionless natural frequencies of the Timoshenko microbeam. The MSGT results were higher than those found using other theories. In addition, this research discovered that there were negli-gible surface stress effects with each of the three material length scale parameters.
Keyw ord s: Size dependent effect Pre-stress loading DQM Vibration behavior of Timoshen-ko microbeam
Nano technology is one of the most powerful technologies that can produce many materials and devices across a range of applications, such as elec-tronics, biomaterials, medicine, and energy produc-tion [1-3]. Vibration analysis of composite beams has been a research topic in many engineering fields because vibration plays an important role in the design of turbine blades, helicopter blades, propel-ler blades, drill bits, and fluted cutters. In practice, these structures are typically modeled as either Eu-ler or Timoshenko beams. The design of micro- and nano-electro-mechanical systems (MEMS/NEMS) requires widespread use of micro-rods and mi-crobeams with different complex behaviors. Cur-rently, micro-composite beams are employed in mi-cro-turbo machines, ultrasonic piezoelectric micro-motor designs, and medical micro devices.
Recently, many researchers have investigated the mechanical behaviors of micro- and nano-scale materials using beam, plate, and shell theories. Ghorbanpour Arana et al. [4] analyzed the pulsating fluid-induced dynamic instability of double-walled carbon nano-tubes (DWCNTs), based on a sinusoidal strain gradient theory using the differential quadra-ture method (DQM) and the Bolotin method. Their results depicted that the imposed magnetic field was an effective controlling parameter for dynamic instability of visco-DWCNTs. In another work, Ghor-banpour Arani et al. [5] presented the nonlinear vibration of coupled nano- and microstructures conveying fluid flow based on a Timoshenko beam model under a two-dimensional magnetic field. They expressed that the magnetic field played an important role in the stability of the carbon nano-tubes (CNTs) and controls the stability of the nanosystem.
100 M. Mohammadimehr et al. / Mechanics of Advanced Composite Structures 3 (2016) 99-112
Simsek [6] studied the free vibration analysis of nanobeams with various boundary conditions, based on the nonlocal elasticity theory for large am-plitude. In that research, the effect of nonlocal pa-rameters on the nonlinear frequency ratio was ex-amined. Their results showed that the nonlocal ef-fects should be considered in the analysis of the me-chanical behavior of nanostructures. Sahmani and Bahrami [7] analyzed the dynamic stability of mi-crobeams subjected to piezoelectric voltage, using the strain gradient theory (SGT). In their results, for a special value of applied piezoelectric voltage, in-creasing the dimensionless length scale parameter decreased the difference between stability respons-es predicted by the classical and non-classical beam models. In addition, Mohammadimehr and Golzari [8] investigated the elliptic phenomenon effect of cross-sections on the torsional buckling of a nano-composite beam reinforced by a single-walled car-bon nanotube (SWCNT). With an increase in the ma-trix thickness, the tangential and longitudinal strains of SWCNT decreased, and the opposite effect occurred for the interface stress and the dimension-less stress of the outer surface.
Alternately, Mohammadimehr and Rahmati [9] considered the small-scale effects on electro-thermo-mechanical vibration analysis of a single-walled boron nitride nanorod under electric excita-tion. They represented that the natural frequency decreased with an increase in the small-scale effects or aspect ratios. On the other hand, the small-scale effects were significant for lower aspect ratios and higher natural frequencies. Atabakhshian et al. [10] employed vibration of a smart coupled electro-thermal nanobeam system with an internal flow, based on nonlocal elasticity theory, while Ansari et al. [11] derived free vibration analysis from the evaluation of single and double-walled carbon nano-tubes based on nonlocal elastic shell models. They concluded that the small-scale effects in the non-local model made nanotubes more flexible. Akgoz and Civalek [12] studied higher-order shear defor-mation in microbeam models, based on the strain gradient elasticity theory. Their results showed that microbeams derived from the non-classical theories, specifically modified strain gradient theory (MSGT), were stiffer than those based on the classical theory (CT).
Asgharifard Sharabiani and Haeri Yazdi [13] il-lustrated the nonlinear free vibrations for function-ally graded (FG) nanobeams, including their surface effects. The results showed that the surface effects at higher volume fraction indices were either less or more dominant, in small and large amplitude ratios, respectively. Ke et al. [14] investigated the nonline-ar vibrations of piezoelectric nanobeams based on the nonlocal and the Timoshenko beam theories.
Their results demonstrated that a change in the ex-ternal electric voltage from a positive value to a negative value led to a decrease in the nonlinear frequency ratio. Ansari et al. [15] analyzed the bend-ing, buckling, and free vibration responses of FG Timoshenko microbeams, and they observed that the critical buckling loads and natural frequencies predicted by the beam models, based on MSGT and CT, provided the maximum and minimum values, respectively. Tounsi et al. [16] illustrated size-dependent bending and vibration analysis of FG mi-crobeams, based on MCST and neutral surface posi-tions. They represented that the inclusion of the couple stress effect makes a microbeam stiffer and decreased the vertical displacement and increased the natural frequency.
Alternately, Nazemnezhad et al. [17] employed an analytical study on the nonlinear free vibration of nanoscale beams incorporating surface density ef-fects. They observed that the effect of the surface density on the variation of the natural frequency of the nanobeam versus the thickness ratio decreases consistently with the increase of the mode number. Nejat Pishkenari et al. [18] examined the surface elasticity and size effects on the vibrational behavior of silicon nanoresonators. They developed a contin-uum model for nanobeam vibrations that was capa-ble of predicting the results of molecular dynamics (MD) simulations with considerably lower computa-tional effort. Yue et al. [19] proposed a microscale Timoshenko beam model for piezoelectricity using flexoelectricity and surface effects. Their results observed that the change of surface properties not only directly affected the static bending but also significantly changed the natural frequency of the beam. Preethi et al. [20] presented surface and non-local effects of the nonlinear analysis of Timoshenko beams using Eringen’s nonlocal theory and the Gur-tin-Murdoch approach, where the nonlocal parame-ters and the positive surface parameters’ values de-creased the stiffness of the beam and resulted in larger deflections and lower frequencies.
In this research, a Timoshenko microbeam mod-el, based on the modified strain gradient theory (MSGT) and surface stress effects subjected to pre-stress loading, is presented. The MSGT and surface stress effects were considered together in this study because both of them affect the structure at the mi-croscale. Despite the fact that the surface and small scale effects have been investigated individually in some papers, the novelty of this study lies in the evaluation of size-dependent effects, including three material length scale parameters, and the surface residual stress based on strain gradient, and the surface stress elasticity effects on the dimensionless natural frequency of Timoshenko microbeams, sub-jected to pre-stress loading and considered simulta-
M. Mohammadimehr et al. / Mechanics of Advanced Composite Structures 3 (2016) 99-112 101
neously at a microscale. Moreover, the size-dependent effects increased the dimensionless nat-ural frequency due to increasing flexural rigidity, which then enhanced the stability of the microstruc-tures. The governing equations of motion were ob-tained using Hamilton’s principle and energy meth-od. The equations were solved using the differential quadrature method (DQM).
2. The Governing Equations of Timo-shenko microbeams
A schematic view of a straight Timoshenko mi-crobeam model based on surface layers, an elastic medium, and pre-stress load is shown in Figure 1. The displacement fields for this model can be stated as [21]
0 0( ,z, ) ( , ) ( , )u x t u x t z x t (1)
v( ,z,t) 0x (2)
0( ,z,t) ( , )w x w x t (3)
where0
u and0
w are axial and transverse displace-
ments for the neutral axis, respectively, and0
is the
rotational transverse normal angle about the x-axis.
The components of normal ( xx ) and shear ( xz )
strains, using Eqs. (1), (2), and (3), are considered as follows:
0 0
xx
uz
x x
, (4)
0
0xz
w
x
.
(5)
Figure 1. A schematic view of a Timoshenko microbeam model
with a surface layer, elastic medium, and pre-stress load.
The strain energy for the linear isotropic elastic material, based on MSGT, is considered as follows [21,22]:
(1) (1)1( )
2 ij ij ijk ijki ii j ijU m dVp
(6)
whereij
andij are the Cauchy stress tensor and
the strain tensor, respectively. The expressions i
andij
denote the dilatation gradient tensor, the
deviatoric stretch gradient tensor, and the symmet-ric rotation gradient tensor, respectively, which are defined as the following forms [23,24]
,
ii j
j
uu
x
(7)
,i mm i (8)
, , ,
, , , ,
, ,
(1) )
( )
1(
31 1
2 215 151
21
(
5
)
( )
jk i ki j ij k
ij mm k mk
i
m ki mm j mj m
jk mm mi m
k
i
j
(9)
, ,( )
1
2ij i j j iu u (10)
,
1
2ij jkl l kie u (11)
where mmand
iu are the dilatation strain and the
displacement vector, respectively, according to
Akgöz and Civalek [25]. The Knocker symbol is ij ,
and the permutation symbol is ijke :
1
1
0
fora forwared permutation of ijk
fora backwared permutation of ijk
if i,j,k is equalijk
e
(12)
The constitutive equations for linear, elastic, and isotropic materials are given by the following forms [25] 2
ij ij mm ij (13)
2
02
i ip l (14)
(1) (1)2
12
ijk ijkl (15)
2
22
ij ijlm (16)
102 M. Mohammadimehr et al. / Mechanics of Advanced Composite Structures 3 (2016) 99-112
where ij is the deviatoric strain, which can be writ-
ten as follows [25]:
1
3ij ij mm ij (17)
where0l ,
1l , and 2l denote three additional inde-
pendent material length scale parameters associat-ed with the dilatation gradient tensor, deviatoric stretch gradient tensor, and symmetric rotation gradient tensor, respectively. In addition, the pa-rameters λ and μ are the Lame coefficients which are given as [26,27]
(1 )(1 2 )
E
,
2(1 )
E
(18)
where E and denote Young’s modulus and Pois-
son’s ratio, respectively. Using Eqs. (3), (4), and (5), the following equa-
tions are given by 2 2
0 02 2xu z
x x
(19)
0z x
(20)
Using Eqs. (3), (4), and (5), the nonzero compo-
nents of the deviatoric stretch gradient tensor(1)ijk
can be derived as follows:
(1)
111
(1)
333
(1) (1) (1)
113 131 311
(1) (1) (1)
313
2 2
0 0
2 22
0 0
22
0
331 133
(1) (1) (1)
122 212 221
(1) (
32
0
22 2
0 0
2 22 2
0
2 23
2
2
0
2
8 4
15 151 1
5 51 1
5
2 2
5 52
5
5
1
5
uz
x xw
x xw
x xu
zx xu
zx x
2
0 01) (1)
23 22
2 1
15 15
w
x x
(21)
Substituting Eqs. (19)–(21) into Eq. (11) yields the following form:
2
0 0
2
1)(
4xy yx
w
xx
(22)
Using Eqs. (15) and (21), the higher order stresses for MSGT are obtained as the following forms:
(1) 2 2
111 1 1
(1) (1) (1) 2 2
313 331 133 1 1
(1) 2 2
333 1 1
(1) (1) (1) 2 2
113 131 311 1 1
(1) (1) (1)
122
2 2
0 0
2 2
21
2 2
0 0
2 22
0 0
22
2
0
1
0
22
2
2 2
5 5
16 8
15 15
4 4
5 5
4 2
5 5
l l
l l
l
uz
x xu
zx
l
l l
xw
x xw
x x
2 2
1 1
(1) (1) (1)
322 232
2 2
0 0
2 22
2
2 20
2
0
13 1 2
2 2
5 54 2
15 15
uz
x xw
l l
l lx x
(23)
Using Eqs. (14), (16), (19), (20) and (22), we have 2
2 0 0
2 2
2 2
2 0 0
0 2 2
2 0
0
)1
(2
2
2
xy yx
x
z
wm m l
xxu
p l zx x
p lx
(24)
The non-zero stressesij are obtained as follows:
0 0
0
0
( 2 )
)( )(
xx
x sz
uz
x xw
xk
(25)
In Eq. (25),sk denotes the shear correction factor
which depends on the shape of microbeam cross-section.
Because of a high surface-to-volume ratio, the surface stress effect plays an important role with micro- and nanoscale materials. For this purpose, the constitutive equation of the Gurtin–Murdoch continuum mechanics approach is considered as follows [28]:
,
,
( )
2( )
s
s s s rr
s
s s s
s s
z s z
u
u
(26)
where sis the residual surface stress under un-
strained condition, and sand s
are the surface
Lame constants. The components of normal and shear surface stress can be written as follows:
0 0
0
(2 )s
xx s s s
s
xz s
uz
x xw
x
(27)
M. Mohammadimehr et al. / Mechanics of Advanced Composite Structures 3 (2016) 99-112 103
The classical beam theory does not satisfy our model. For solving this problem, it is assumed that the stress component zz
varies linearly through the
beam thickness and satisfies the balance conditions on the surfaces. Therefore, zz
can be written as
[28] 2 2
0 0
2 2
2 2
0 0
2 2
( ) ( )
2
( ) ( )
s ss sxz xz
zz
s ss sxz xz
w w
x xt t
w w
x xt t zh
(28)
Using Eq. (27), zz can be written as
2 2
0 0
2 2
2 τ ρ
zz s s
w wz
h x t
(29)
The normal and shear components, considering bulk and surface effects and using Eq. (29), can be written as
0 0
2 2
0 0
2 2
0
0
( 2 )( )
2 (τ ρ
(
)(1
)
)
xx
s s
xz s
uz
x xw wz
h x tw
kx
(30)
is the total potential energy that includes the strain energy, kinetic energy, and work done by the external loads, which can written as [4]
(K )tot tot tot
U (31)
where s
tot
s
tot
elastic
tot
U U U
K K K
V
(32)
and where U , sU , K and sK are the strain and
kinematic energies for bulk and surface effects,
respectively. Moreover, and elasticV are the work
done by the external forces, including the pre-stress load and the elastic foundation, respectively.
Using Hamilton’s principle and a variational method for Timoshenko microbeam model, based on strain gradient theory and the surface stress ef-fects embedded in an elastic medium subjected to pre-stress loading , yields the following equation [4]:
0
[ ] 0t
s s elasticU U K K V dt (33)
The surface strain energy is obtained as
0
1
2
L
s
s s
ij ijU dSdx
(34)
where
2
2
( , ) 1,
( , ) 1,
A I z dA
S J z dS
Using the presented equations, the strain energy for bulk and surface effects are explained in Appen-dix A with details. Using Eqs. (14)–(16), the kinetic energy of the Timoshenko microbeam model for bulk and surface effects can be written as
2 2
0 2 2 2
0 0 0
2 2
0 2 2 2
0 0 0
1( )
2 1
[2
1( )
2
]
]1
[2
L
A
Ls
ss
s
I A A
u wk dAdx
t tw u
t
J S S
t tu w
k dSdxt t
w u
t t t
(35)
The work done by external forces, including pre-stress load and elastic foundation, can be written as
0
2
0 0
2
0 0 0
0
2
0 00
0
1
1( )
2
( )2
x
Elastic Mediu
L
x
m w P
x
L
w P
A
F k w G w
V k w G w w dx
N w dxx
N
(36)
where 0 x is the pre-stress load, andwk and
PG are
Winkler’s spring and Pasternak’s shear modulli of elastic foundation, respectively.
104 M. Mohammadimehr et al. / Mechanics of Advanced Composite Structures 3 (2016) 99-112
By substituting Eqs. (A-1), (A-2), (35) and (36) into Eq. (33), one can obtain the governing equa-tions of motion and boundary conditions as follows:
0
0
2
2
024
2 2
0 1 042
02
3 42 2
0 1 04
0
22 2 2
0 1 0 02
2 2
1 2
:
)
)
:
( ( ))( )
) bh
( ( 2 ) 2( )( 2 ))(
4(2 ) (
5
2 (b h) ( ) 0
4(2 ) ( )
5 12
32 1( 2
15 416
( )
b15 4
)1
(
s
s s
s
s
s
bh b h ux
l l bh ux
bh ut
bhl l
xk w
x
k l l lx
u
bh b h
b
l
h
l
2 3
2 3
2 2
3
033 2
02
2
2
3 2
023 3
0 03
0
2
02
0
2 2
1
(( 2 ) ( 2 )(12
( ( )( )12
06 1
h( )
))( )2
6 1
:
(
16 1(
1
6
)2 6
( ) ( )
( 2 ( ))( )
( ))
5 4
( )
) (
s s
s
s s
s s s
s
bh h
bh h
bh bh
wx
bh
x
b
bh
t
w wx x t
w
k w kx
x
h b h bh
b h
bhl l
3
034
2 2
1 2 042
022 2
0 0 0 02 2
)
1
4
2 ( )
( ) ( )
8( ) ( )
15
( )( )
0w
s
P
bh b h
k G
x
l l bh wx
wt
N w w wx x
(37)
For boundary conditions
0
3
32
2 2
0
2
2
2 2
1 2 0
2 2
1 0
0
0
2 2
1 2 0
2 2
0
0
21
0
2
8( ) ( )
1516 1
(15 4
(
8( ) ( )
1516 1
(15 4
:
1
4
) ( )
( 2 ( ))( )
( ))
:
1
4
) ( )
( 2 2( )( 2 ))(( )
2
:
)
(
s s
s s
s s
l l bh wx
l lx
k wx
w
bh
bh b h
k
l l bh wx
l
bh b h
w
lx
bh b h u
l
x
h
u
x
b
32
1 0
2 2
0 1 0
3
0
2 2 2
1 0 0
2 2
1 0
32 2
0 1 0
2
3
2
2
2 3
2
2
2 2
0
3
32
2
4(
5
4(2 ) (
5
(( 2 ) ( 2 )(12
32 1( 2
15 416 1
(15 4
4(2 ) ( )
5
) ) ( )
:
)
:
))( )2 6
) bh( )
)bh( )
12
6
( )
s s
s
s
l bh ux
l l bh ux
bh
x
l l lx
l
b h
u
l wx
bhl
x
lx
b
h h
x
b
h w
32 2
1 0
2
20
1
4(2 )
5 12
:x
bhl l
x
(38)
M. Mohammadimehr et al. / Mechanics of Advanced Composite Structures 3 (2016) 99-112 105
where
1
3 13
2 2 2
7 1 2 0
2 2 2 2
2 1 0 5 1 2
32 2 2 2
6 1 2 1 0
3 2 3
2
( 2 ) 2( )( 2 )
2( )
32 1( 2 )
15 44 16 1
( 2 ) ( )5 15 4
8 1 4( ) M ( 2 )
15 4 5 12
( 2 ) ( 2 )( )12 2 6
12(1 )
s s
s s
s s
s
A bh b h
A b h A bh
A l l l bh
A l l bh A l l bh
bhA l l bh l l
bh bh hD
bhE G
2
3 2 3
1 3
12(1 )
I 2( ) I ( )12 2 6
s
s s
bh
bh bh hbh b h
(39)
The dimensionless geometric, mechanical, and surface residual stress, surface mass density, Young’s modulus of surface layer, and three material length scale parameters can be defined as follows [28]:
* *110 31
1 32 210 10110 10
2 2 4
10 110 110
0 1 3 13
1 3 13
110 110
2 5 6 7
2 5 6 7 2
110
2
* *
110
110 110
,W ,X , ,
,d ,I ,I
,e ,m
( , , ),N ,( , , )
( , , , )( , , , ) ,
K , , ( 2 ),w PW P
u w x hU
L h L L
A IIt D
L I IA h I h
G E Mg
I h A h A h
N A A Aa a a
A A
A A A Aa a a a
A h
k L GG A bh
A A
10
I h
(40)
To use the differential quadrature (DQ) method, first we should convert Eqs. (37) and (38) into di-mensionless equations. Thus, substituting Eq. (40) into Eqs. (37) and (38) yields the Eqs. (41a) and (41b).
2 4 22 *
1 2 12 4 2
3 2 43 2 4
5 133 2 42 3
2 3
7 132 2
3 23 * 23
33 2
3 2 42 3
5 13 63 2 42
13 3 2
3
:
0
:
02
:
2
U
U U Ua a I
X X
Wa d m a
X X XW W
a a gXX X
aW We I
XXW
W Wa a a
X X XW
a aX X
a
X
2*
22 2
* *
12 20
W
P
WN K W
XW W
G IX
(41a)
3 23 2 3
6 5 13 3 133 2
22
6 52
32 3
1 2 3
2
2
2 32 3
7 5 2 3
2
2
0 ( ) ( ) 02
0 ( ) 0
0 or 02
0 or 0
0 or (d ) ( ) 0
0 0
aW WW or a a a a a
X X X
W Wor a a e
X X X
aU UU a a
X X
U U
X X
Wa a e m
X X X
orX X
(41b)
The dimensionless simply supported (SS) boundary conditions for the microbeam model are considered as follows:
22
7 5 23
3
3
(d ) ( )
0
WU W a a e
X X
mX
(42)
106 M. Mohammadimehr et al. / Mechanics of Advanced Composite Structures 3 (2016) 99-112
3. Using the DQ method to solve the Timo-shenko microbeam model
The (DQ) method was used to solve Eq. (41) and the associated boundary conditions in Eq. (42) to determine the free vibration frequencies of the beam. The basic concept of the DQM was defined as the derivative of a function at a given point that can be approximated as a linear sum of a weighted func-tion at all sample points [29,30]. Using this approx-imation, the differential equations are then reduced to a set of algebraic equations. This approach is con-venient for solving problems governed by fourth- or higher-order differential equations.
According to this method, the mth order deriva-tive of the function f(x) with respect to x at a grid point xi, is approximated by a linear sum of all the functional values in the whole domain as follows [22]:
( )
1
(x )
i
m Nm
ij jmjx x
d fC f
dx
(43)
where xi is the location of ith sample point in the
domain; N is the number of sampling points; (x )j
f
is the functional value at point xi, ( )m
ijC
is the
weighting coefficient of the mth order differentiation attached to these functional values. To avoid ill-conditioning, the Lagrange interpolation basis func-tions are used as the following form [22]:
1, , 1,(1)
1,
(x x )/ (x x ) (i j)
, 1,2,....,
1(i j)
(x x )
N N
i k j kk k i j k k j
ij
N
k k i i k
C i j N
(44)
To determine the unequally-spaced positions of the grid points, the Chebyshev–Gauss–Lobatto polyno-mials were employed as follows [22]:
2 11 cos
2 1i
L ix π
N
(45)
The first order weighting matrix can be obtained completely from Eq. (44). Higher-order coefficient matrices can be obtained from the first-order weighting matrix as follows [22]:
(2) (1) (1)
1
(3) (1) (2) (2) (1)
1 1
(4) (1) (3) (3) (1)
1 1
N
ij ik kjkN N
ij ik kj ik kjk kN N
ij ik kj ik kjk k
C C C
C C C C C
C C C C C
(46)
Then, substituting Eqs. (44) and (46) into Eqs. (41a) and (42) obtained the following equations of motion using the DQ method
2(2) 2 (4) *
1 2 1 21 1
3 (3) 2 (2)
5 71 1
4 (4) (1)3
13 131 12 2
3 (1) * 2
32 21
2 (3) * (2)
5 13 31
0
( ) ( ) Ψ
Ψ Ψ ( )2Ψ
( ) 0
( ) Ψ ( )
N N
ik k ik k
N N
ik k ik k
N N
ik k ik k
N
ik k
N
ik k P ik
Ua C U a ξ C U I
τa e ξ C W d a ξ C
amξ C a a ξ C W
gξ C W I ξτ τ
a ξ C a a N G ξ C
1
3 (4) (1)3
6 131 12
*
12
( ) Ψ2
( ) 0
N
k
N N
ik k ik k
W
W
aa ξ C W a C
K ξW I ξWτ
(47)
For (SS) boundary conditions, we have:
(1) 2 (2)
7 51 1
3 (3)
1
(d ) ( )
0
N N
ik k ik k
N
ik k
U W a C a e C W
m C
(48)
The general solutions of motion equations are considered as
U( , ) ( )e
W( , ) ( )e
( , ) ( )e
i
i
i
x t U X
x t W X
x t X
(49)
where 10
110
IL
A is the dimensionless natural fre-
quency. Ω is the fundamental natural frequency, and ρ denotes the density of microbeam.
The stiffness and mass matrices for the Timo-shenko microbeam, using strain gradient theory and surface stress effects under pre-stress loading, can be written as
2( K ){ , } 0T
b dM d d (50)
where K , M are the stiffness and mass matrices
and the subscripts b and d stand for the boundary and domain points, respectively. By solving Eq. (50), the dimensionless natural frequencies for and
their associated vibration mode shapes can be ex-tracted.
4. Numerical Results and Discussion The mechanical and geometric properties of a
Timoshenko microbeam is considered as [28,31]
3
2
4
2
210 , 2331 , 0.24,
4.488 , 2.774 , 0.605
3.17 7 ,k 5/6, 17.6
, 2 , 4 ,L 20h,k 2 10 ,G 10
s s s
s s
w P
kgE Gpa
m
N N N
m m m
kge l m
m
N Nb h h l
m m
(51)
M. Mohammadimehr et al. / Mechanics of Advanced Composite Structures 3 (2016) 99-112 107
The material length scale parameter is crucial for the successful application of the MSGT. Dehrouyeh-Semnani and Nikkhah-Bahrami [32] presented an in-depth discussion on how to determine this pa-rameter numerically, and they compared the nu-merical results obtained to those obtained by exper-imental testing [32]. They showed the bending rigid-ity of an epoxy micro-cantilever versus thickness for a modified couple stress model (MCST), using
17.6cl l m , and the experimental data reported
by Lam et al. and found that the results based on the constitutive beam model validated the experimental data, while the Euler-Bernoulli beam model overes-timated the bending rigidity of the micro-cantilever. In addition, they depicted that the material length scale parameter of epoxy-based materials on the Euler-Bernoulli beam model equals to
12.45EBl l m . According to the results of
Dehrouyeh-Semnani and Nikkhah-Bahrami, the Eu-ler-Bernoulli beam model validated the experi-mental data very well, but the constitutive beam model underestimated the bending rigidity of the epoxy micro-cantilever. Therefore, in this work, we used the material length scale parameter equal to 17.6 m .
Table 1 gives the dimensionless natural frequen-cies for the Timoshenko microbeam under various boundary conditions. An excellent agreement was found between the present results and the analytical solutions.
The results, obtained by the present work, are compared with the reported results by Ansari et al. [33] in Figure 2, where they demonstrate good agreement each other. In addition, the trend of the results was the same. On the other hand, increasing the aspect ratio (L/h) reduced the dimensionless natural frequency. Moreover, the stiffness of the Timoshenko microbeam decreased with increasing the aspect ratio. Table 1. Comparison of dimensionless natural frequencies with
various thicknesses for different boundary conditions.
Thickness (nm)
S-S S-C C-C
Ansari et al. [28] h=1
0.1830 0.2148 0.2524 Present work 0.1863 0.2169 0.2553 Ansari et al. [28]
h=5 0.1255 0.1643 0.2117
Present work 0.1258 0.1652 0.2120
Figure 2. The dimensionless natural frequency versus aspect
ratio.
Table 2 shows the first three dimensionless nat-
ural frequencies of the Timoshenko microbeam model for the different values of aspect ratio
( 1
L
h), and surface residual stress ( s ). As shown
in Table 2, by increasing the aspect ratio, the value of the first three dimensionless natural frequencies decreases, and the opposite occurs for the surface residual stress.
The latter subject has been illustrated for dimen-sionless fundamental natural frequencies in Figure 3. Considering the surface residual stress, the Timo-shenko beam at a microscale becomes stiffer, but the effect of this parameter on the dimensionless natural frequency is not noticeable. Therefore, sur-face residual stress can be ignored in the results. Table 2. First, second, and third dimensionless natural frequen-cies of a Timoshenko microbeam model for the different values
of 1
L
h and
s for 0 1 2( 1 ) l l l m .
1
L
h
1 2 3
10 3.6763 7.3545 11.0340 90
s
N
m
15 3.6387 7.2933 10.9432
20 3.6208 7.2618 10.8893
10 3.6762 7.3543 11.0336 0
s
N
m 15 3.6386 7.2931 10.9429
20 3.6207 7.2616 10.8891
10 3.6761 7.3540 11.0333 90
s
N
m
15 3.6385 7.2929 10.9426
20 3.6207 7.2614 10.8888
108 M. Mohammadimehr et al. / Mechanics of Advanced Composite Structures 3 (2016) 99-112
Figure 3. The dimensionless fundamental natural frequency
versus aspect ratio for different values ofs 0 1 2( 1 ) l l l m
Tables 3 and 4 depict the first three dimension-
less natural frequencies of the Timoshenko mi-crobeam model for the values of the aspect ratio
( 1
L
h), surface mass density ( s ), and Young’s
modulus of surface layer ( 2 s s ), respectively. By
increasing of the 2 s s , the value of the first three
dimensionless natural frequencies increases and vice versa for surface mass density. A change in
2 s s and s led to increase stiffness and mass of
the micro structure, respectively. Moreover, the re-sults, shown in Figures 4 and 5, are similar to those shown in Tables 3 and 4. Furthermore, Figures 4
and 5 demonstrate that the effect of s on the di-
mensionless natural frequency is more than 2 s s . However, the effect of 2 s s on the di-
mensionless natural frequency is not noticeable, and it can be ignored in the results.
Table 3. First, second, and third dimensionless natural fre-quencies of the Timoshenko microbeam model for the differ-
M. Mohammadimehr et al. / Mechanics of Advanced Composite Structures 3 (2016) 99-112 109
Figures 6a and 6b show the influence of pre-stress load on the dimensionless first and third nat-ural frequencies versus aspect ratio, respectively. These results demonstrated that the effect of pre-stress load on the greater mode is negligible for higher aspect ratios, and this effect was similar to the lower aspect ratios for all modes. Clearly, the stiffness of microbeam increased at lower aspect ratios. In this figure, the effect of the positive pre-stress load on the natural frequency was higher than that of the negative pre-stress load. Consequently, positive and negative pre-stress loads led to in-crease and decrease stiffness of the Timoshenko microbeam, respectively. These results are the same for dimensionless natural frequencies.
(a)
(b)
Figure 6. The influence of pre-stress load on the dimensionless first (a) and third (b) natural frequencies versus aspect ratios
( 0.001, 0 , 0.001) N N N .
To consider the size-dependent effects (l denotes the material length scale parameter), the parameter at a microscale is taken into account, and it is non-zero for MSGT ( 0 1 2 l l l l ) or MCST ( 0 1 20, l l l l ),
while at a macro scale, it is zero for CT ( 0 1 2 0l l l l ).
Figures 7a and 7b are plotted to illustrate the in-fluence of various material length scale theories in-cluding modified strain gradient (MSGT) ( 0 1 2 l l l l ), modified couple stress (MCST)
( 0 1 20, l l l l ), and classical theories (CT)
( 0 1 2 0 l l l ) on the dimensionless first and third
natural frequencies versus h
l, respectively.
(a)
(b)
Figure 7. The influence of various material length scale theories on the dimensionless (a) first and (b) third natural frequencies
versus h/l.
10 15 20 25 30 35 40 45 500.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
L/h
1
Possitive pre-stress
No pre-stress
Negative pre-stress
10 15 20 25 30 35 40 45 501
1.5
2
2.5
3
3.5
4
L/h
3
Possitive pre-stress
No pre-stress
Negative pre-stress
2 4 6 8 10 120.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
h/l
1
CT
MCST
MSGT
2 4 6 8 10 121.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
h/l
3
CT
MCST
MSGT
110 M. Mohammadimehr et al. / Mechanics of Advanced Composite Structures 3 (2016) 99-112
The effect of the material length scale parame-ters on the dimensionless natural frequencies for MSGT was higher than that of the other states, such as MCST and CT. This indicates that considering three material length scale parameters
0 1 2( )l l l l led to increase stiffness of the Timo-
shenko microbeam model, and therefore the dimen-sionless natural frequencies for MSGT enhanced.
Figures 8 and 9 present the influence of trans-verse and shear constants of the elastic foundation on the dimensionless fundamental natural frequen-cies with different values of aspect ratios. The di-mensionless natural frequency increased with an increase in the transverse and shear constants of the elastic foundation, while the elastic foundation in-creased stiffness of the microstructure.
Figure 8. The influence of the transverse constant of the elastic foundation on the dimensionless fundamental natural frequen-
cies with different values of aspect ratios1
L
h, ( 0)PG .
Figure 9. The influence of shear constant of elastic foundation on the dimensionless fundamental natural frequencies with different
values of aspect ratios1
L
h, ( 0)wk .
Moreover, increasing the transverse and shear constants of the elastic foundation were directly related to the stiffness of the Timoshenko mi-crobeam and the dimensionless natural frequency.
5. Conclusions
Size-dependent effects on the free vibration analysis of the Timoshenko microbeam model, based on MSGT and surface stress effects subjected to pre-stress loading embedded in an elastic medi-um, were investigated. The Gurtin–Murdoch contin-uum mechanical approach was considered, and the set of governing equations were derived using a variational method and solved using DQM. Effects of pre-stress load, surface residual stress, surface mass density, Young’s modulus of surface layer, material length scale parameters, and elastic foundation coef-ficients were studied.
The results of this article can be listed as follows: By increasing the aspect ratio, the values of
natural frequencies decreased while the op-posite occurred for surface residual stress. In addition, when increasing the value of
2 s s , the value of the natural frequencies
increased, while the surface mass density decreased. Variations in 2 s s and s led
to increase stiffness and mass matrices for the micro structures, respectively. The nu-merical results showed that the effect of sur-face residual stress was more than the sur-face mass density or Young’s modulus of the surface layer.
The effect of pre-stress loading in higher modes was negligible for higher aspect rati-os, and this effect was similar to lower as-pect ratios across all modes.
The effect of the three material length scale parameters on the natural frequencies for MSGT was higher than that of the other the-ories. Application of each of the three mate-rial length scale parameters 0 1 2( , , 0)l l l , in-
creased the natural frequencies for MSGT, which was due to the increasing stiffness of the Timoshenko microbeam model.
Natural frequencies increased with an in-crease in the transverse and shear constants of the elastic foundation. Consequently, ap-plying the elastic foundation values led to increase stiffness of the Timoshenko mi-crobeam model.
Comparison between the material length scale parameters and the surface effect con-firmed that natural frequencies are more af-fected by the material length scale parame-ters than surface effects.
0 2 4 6 8 10
x 107
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
Spring constant of Winkler-type,Kw
1
L/h=15
L/h=20
L/h=22
L/h=24
0 10 20 30 40 50 60 70 800
0.5
1
1.5
Shear constant of Pasternak-type,GP
1
L/h=5
L/h=10
L/h=20
L/h=50
M. Mohammadimehr et al. / Mechanics of Advanced Composite Structures 3 (2016) 99-112 111
Acknowledgments
The authors would like to thank the reviewers for their reports that improved the clarity of this article. Moreover, the authors are grateful to the Iranian Nanotechnology Development Committee for their financial support. We are also grateful to the University of Kashan for supporting this work through Grant no. 463855/1.
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Appendix
The strain energies for bulk and surface effects are written as follows:
