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Size-dependent torsion of functionally graded bars
Accepted Manuscript Size-dependent torsion of functionally graded bars M. Rahaeifard PII:
S1359-8368(15)00439-4
DOI:
10.1016/j.compositesb.2015.08.011
Reference:
JCOMB 3689
To appear in:
Composites Part B
Received Date: 7 January 2015 Revised Date:
10 May 2015
Accepted Date: 7 August 2015
Please cite this article as: Rahaeifard M, Size-dependent torsion of functionally graded bars, Composites Part B (2015), doi: 10.1016/j.compositesb.2015.08.011. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Size-dependent torsion of functionally graded bars M. Rahaeifard*
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Department of Mechanical Engineering, Golpayegan University of Technology, Golpayegan, Iran.
Abstract
In this paper, size-dependent static and dynamic behavior of functionally graded microbars is investigated on the basis of the modified couple stress theory. The equation of motion and corresponding boundary conditions are
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derived using Hamilton's principle and presented in the dimensionless form. Equivalent mechanical properties (i.e. shear modulus, density and length scale) are extracted for the functionally graded microbar based on the mechanical
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properties of the material constituents. In this work, it is shown that without any simplifying assumption, two equivalent length scale parameters can be defined for functionally graded bars and the size-dependent mechanical behavior of these components can be explained using these parameters. As an example, static and dynamic behavior of a functionally graded microbar with fixed-free boundary conditions is analyzed and the effect of size-dependency on mechanical behavior of this structure is discussed.
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Keywords: B. Mechanical properties; B. Microstructures; C. Analytical modelling; C. Micro-mechanics; Modified couple stress theory.
1. Introduction
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Functionally graded materials (FGMs) are inhomogeneous composites which are made of two different materials typically a metal and a ceramic. The volume fractions of material constituents in FGMs vary continuously in a
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certain direction of the structure. This continuous variation provides a smooth change in the mechanical properties and eliminates the high magnitude shear stresses which appear in laminated composites. FGMs can offer the benefits of both of material constituents and hence, these materials are widely used in various fields of engineering including aerospace, bioengineering and mechanical engineering. Many researchers have studied the mechanical behavior of structures made of functionally graded materials. Here some of these works are reviewed. Fallah et al. [1] investigated buckling and vibration of functionally graded Euler-
*
E-mail: [email protected] Tel: +983157243163
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Bernoulli beams under thermal loading. Nonlinear vibration of functionally graded beams is investigated by Ke et al. [2]. They considered the material properties to be varying through the thickness of the beam and took into account the nonlinearities caused by the mid-plane stretching. Sarkar and Ganguli [3] presented a closed form
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solution for natural frequency of functionally graded Timoshenko beams. The nonlinear mechanical response of functionally graded piezoelectric beams with properties varying through the thickness of the beam is analyzed by Lin and Muliana [4]. Su and Banerjee [5] utilized the dynamic stiffness method to investigate the dynamic behavior and natural frequency of functionally graded Timoshenko beams. Torsional responses of functionally graded bars
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and cylinders are investigated by Horgan and Chan [6] and Dung and Hoa [7, 8] respectively. Rahaeifard et al. showed that the sensitivity of atomic force microscopes can be enhanced using functionally graded materials [9, 10].
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A closed form solution is proposed for static response of functionally graded Kirchhoff plates by Apuzzo et al. [11]. Hosseini-Hashemia et al. investigated the in-plane and out of plane vibration of functionally graded rectangular plates with simply supported boundary conditions [12]. Furthermore, mechanical behavior of functionally graded shells has been also widely investigated by researchers (see for example [13-17]).
Nowadays, FGMs are also used in microsystems such as thin films in the form of shape memory alloys [18, 19] and micro-electro-mechanical systems [18, 20]. In these systems, the size of structures is of order of microns and sub-
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microns. Many experimental researches performed on the mechanical response of micro scale structures (see for example [21-24]) proved that the mechanical behavior of these components is different from that predicted by the classical continuum theory. According to these researches, the normalized stiffness and normalized deflection of
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micro scale structures, which according to the classical theory should be independent of the structure size, is significantly size-dependent. Furthermore, the stiffness of micro scale structures measured in experiment, is much
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more than the stiffness predicted by the classical theory. These experiments clearly showed that the classical continuum theory can neither evaluate the mechanical response of micro scale structures nor justify the size-dependency observed in mechanical behavior of these components. In contrast to the classical theory, some new continuum theories such as couple stress theory and modified couple stress theory can cover this gap and accurately model the mechanical behavior of micro scale structures. The modified couple stress theory was developed by Yang et al. [25] by performing a modification on the couple stress theory previously proposed by Toupin [26] and also Mindlin and Tiersten [27]. In the modified couple stress theory, beside the classical stresses (i.e. normal and shear stresses), a non-classical stress known as couple stress is
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also acting on an element of material. This higher order stress is related to the displacement field by a new material constant called length scale parameter. Furthermore, in this theory, in addition to the classical equilibrium equations, a non-classic equilibrium equation (i.e. the equilibrium equation of moment of couples) is satisfied.
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Many researchers utilized this theory to investigate the mechanical behavior of micro scale structures as well as to justify the size-dependency observed in experiment. Park and Gao [28] utilized this theory to derive the governing equations of Euler-Bernoulli microbeams and detected the size-dependency in the static behavior of these structures. Tsiatas [29] utilized this theory to investigate the mechanical behavior of Kirchhoff microplates with arbitrary
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shapes. Size-dependent static and dynamic behavior of electrostatically actuated microcantilevers is analyzed by Rahaeifard et al. [30, 31]. They showed that utilizing the modified couple stress theory can remove the gap between
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the experimental and theoretical results of the static pull-in of microcantilevers [30]. They also proposed a nonclassical yield criterion based on the modified couple stress theory to justify the size-dependent yielding of micro scale structures. The predictions of their criterion were in very good agreement with experimental observations while the yielding load predicted by the classical theory was up to 50% less than the yielding load measured in experiment [32].
This theory is also widely used by researchers to model the static and dynamic behavior of micro scale structures
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made of functionally graded materials. The size-dependent formulations of functionally graded Euler-Bernoulli and Timoshenko microbeams are developed by Asghari et al. [33, 34]. They utilized this formulation to model the sizedependent deflection and natural frequency of these components. The size-dependent mechanical behavior of
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functionally graded Mindlin microplates is investigated by Ke et al. [35]. Thai and Kim [36] derived the governing equations of motion for functionally graded Reddy plates based on the modified couple stress theory and
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investigated the static and dynamic behavior of these structures. In this paper based on the modified couple stress theory, a formulation is developed to model the static and dynamic torsion of functionally graded microbars. The volume fraction of material constituents is assumed to vary through the radial direction. On this basis, two equivalent length scale parameters are defined for functionally graded microbars based on the length scale parameters and volume fractions of material constituents. As a case study, static deformation and free vibration of a functionally graded fixed-free bar is analyzed. It is shown that the modified couple stress theory can successfully model the size-dependency of mechanical behavior of functionally graded bars.
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The results of the modified couple stress theory are compared with those evaluated based on the classical theory and the importance of size effect in mechanical behavior of micro scale functionally graded bars is discussed.
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2. Preliminaries According to the modified couple stress theory presented by Yang et al. [18], the strain energy of an elastic body can be written as follows.
in which
(
i, j = 1, 2,3) ,
(1)
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1 (σ ijε ij + mij χij ) dV 2 V∫
σ ij and ε ij , denote the components of stress and strain tensors and mij and χij refer to the deviatoric part
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U=
of couple stress and the symmetric curvature tensors respectively. These parameters can be expressed as
χij =
νE
(1 + ν ) (1 − 2ν )
ε kk δ ij + 2 µε ij ,
1 ( θi, j + θ j,i ) , 2
mij = 2l 2 µχ ij ,
(2)
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σ ij =
1 ( ui , j + u j , i ) , 2
(3)
(4)
(5)
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ε ij =
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where ui denotes the components of displacement field, l is the length scale parameter, E and
µ
denote the Yong
modulus and shear modulus respectively and θi refers to the components of the rotation vector related to the components of the displacement field as follows.
θi =
1 (curl ( u ) )i . 2
(6)
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3. Modelling of functionally graded bar
A. The microbar is under distributed torque (torque per unit length) denoted by
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Figure 1 displays a functionally graded microbar of length L having a circular cross section with radius R and area
Td .
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For this microbar the displacement field can be written as [37, 38]
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Figure 1. A functionally graded circular microbar.
u1 = − yϕ ( z , t ) , u2 = xϕ ( z , t ) , u3 ( z , t ) = 0 ,
where u1 , u2 and u3 refer to displacements along x, y, and z axes respectively and
(7)
ϕ
denotes the rotation angle of
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the cross section about z axis. According to the abovementioned displacement field, the components of the strain and symmetric curvature tensors can be calculated using equations (2) and (4) as follows.
1 ∂ϕ ∂ϕ 1 ∂ 2ϕ 1 ∂ 2ϕ , χ33 = , χ13 = χ31 = − x 2 , χ32 = χ 23 = − y 2 , χ12 = χ 21 = 0. 2 ∂z ∂z 4 ∂z 4 ∂z
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χ11 = χ 22 = −
1 ∂ϕ 1 ∂ϕ , x , ε13 = ε 31 = − y 2 ∂z 2 ∂z
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ε11 = ε 22 = ε 33 = ε12 = ε 21 = 0 , ε 23 = ε 32 =
(8)
(9)
It is assumed that the microbar is made of two materials (one metal and one ceramic) in a way that the volume fractions of material constituents vary through the radial direction. According to Mori-Tanaka relation, the equivalent shear modulus of a functionally graded material (i.e. 41].
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µeq ) can found from the following equation [39-
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µeq − µm = µc − µm
fc µ − µm 1 + (1 − f c ) c µm + f1
(10)
µ m ( 9 K m + 8µ m ) 6 ( K m + 2µm )
(11)
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f1 =
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where subscripts m and c refer to metal and ceramic respectively, f c denotes the volume fraction of ceramic and
in which K m denotes the bulk modulus of metal. Using the equivalent shear modulus, the classical shear stresses can
∂ϕ , ∂z ∂ϕ σ 13 = σ 13 = 2 µeq ε13 = − µeq y , ∂z
σ 23 = σ 32 = 2 µeq ε 23 = µeq x
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be obtained as
(12)
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Furthermore, the higher order stresses (couple stresses) in metal and ceramic phases are evaluated as follows
∂ϕ ∂ϕ , ( m33 )m / c = 2 µm / c lm2 / c , ∂z ∂z ∂ 2ϕ ∂ 2ϕ 1 1 = − µm / c lm2 / c x 2 , ( m23 )m / c = ( m32 )m / c = − µm / c lm2 / c y 2 , 2 2 ∂z ∂z
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( m13 )m / c = ( m31 )m/ c
(13)
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( m11 )m/ c = ( m22 )m/ c = −µm/ c lm2 / c
where the subscript m/c (i.e. m or c) means “metal or ceramic”. Substituting equations (8)-(13) into equation (1), yields the strain energy of the functionally graded microbar as
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L
U=
1 {σ ijε ij + (mij )m χij fm (r ) + (mij )c χij fc (r )} dA dz 2 ∫0 ∫A
2 L 1 2 ∂ϕ 2 µ x + y eq 2 ∫0 ∫A ∂z
(
=
)
2 2 ∂ϕ 2 1 2 2 ∂ ϕ + µm lm2 f m (r ) + µclc2 f c (r ) 3 + x + y 2 dA dz ∂z 4 ∂z
)
(
)
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(
(14)
2 L R ∂ϕ 2 1 ∂ 2ϕ 2 ∂ϕ 3 2 2 µ µ = π ∫ ∫ r 3 µeq + l f ( r ) + l f ( r ) c c c mm m 3r ∂z + 4 r ∂z 2 dr dz , z ∂ 0 0
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where r denotes the distance of an arbitrary point from the center of the cross section ( r =
x 2 + y 2 ) and f m and
f c refer to the volume fraction of metal and ceramic respectively noted that f m + f c = 1 . Defining new equivalent
lˆA =
A
A
=
0
R2
r 2 µeq dA J
∫ (µ A
m
,
R
=
2π ∫ µeq r 3dr 0
J
)
f m lm2 + µc f c lc2 dA
µA A
∫ (µ A
m
)
,
=
f m lm2 + µc f c lc2 r 2 dA
µJ J
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lˆJ =
A
R
2 ∫ µeq rdr
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µJ
∫ =
µeq dA
2π ∫
R
0
=
(µ
m
)
f m lm2 + µc f c lc2 r dr
µA A
2π ∫
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µA
∫ =
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parameters for the functionally graded bar as follows
R
0
(µ
m
)
µJ J
(16)
(17)
,
f m lm2 + µc f c lc2 r 3 dr
(15)
,
(18)
∫
where J = r 2 dA is polar area moment of inertia, the strain energy can be written as A
2 2 L 2 1 1 2 ∂ϕ 2 ∂ ϕ ˆ ˆ U = ∫ µ J J + 3µ A Al A + µ J J lJ 2 20 ∂z 4 ∂z
(
)
(
)
dz .
The total kinetic energy of the microbar (K) can be expressed as
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(19)
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L L ∂u1 2 ∂u2 2 ∂u3 2 1 1 ∂ϕ f + f + + dA dz = J ρJ ρ ρ ( ) m m c c dz , ∫ 2 ∫0 ∫A ∂ t ∂ t ∂ t 2 ∂t 0 2
ρJ
∫ =
A
is the equivalent density of the functionally graded microbar defined as
r 2 ( ρ m f m + ρ c f c ) dA J
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ρJ
where
(20)
(21)
.
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K=
The variation of the external work done by the concentrated and distributed couples on the microbar can be
∂ϕ , ∂z z =0, L
L
δ W = ∫ T d δϕ ( z ) dz + T c δϕ z = 0, L + T h δ 0
(22)
T c and T h stand for the classical and higher order torques applied on the end sections of the bar. In order
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in which
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mentioned as
to derive the governing equation of motion, Hamilton's principle is utilized as follows.
t2
∫ (δ K − δ U + δ W ) dt = 0 .
(23)
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t1
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Substituting equations (19)-(22) into equation (23) and following some straightforward mathematical manipulations gives the equation of motion and boundary conditions of the functionally graded microbar as
−
2 2 ∂ ˆ 2 ∂ϕ + ∂ 1 µ J lˆ 2 ∂ ϕ + ∂ ( ρ J ) ∂ϕ = T d ( z ) , J + 3 Al µ µ J A A J J 2 J ∂z ∂z ∂z 2 4 ∂t ∂z ∂t
(
)
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(24)
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2 ∂ 1 2∂ ϕ 2 ∂ϕ +Tc =0 µ J J lˆJ 2 − µ J J + 3µ A AlˆA ∂z 4 ∂z ∂z z = 0, L
)
2 1 h ˆ2 ∂ ϕ =0 µ J J lJ 2 − T ∂z 4 z =0, L
or
or
∂ϕ =0. ∂z z =0, L
δ
(25)
(26)
Considering the same material properties for metal and ceramic ( µ m =
µc = µ , ρ m = ρ c = ρ , lm = lc = l ),
µ A = µ J = µ , ρ J = ρ , lˆA = lˆJ = l , and the formulation presented here, is reduced to the governing
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results in:
δϕ z =0, L = 0 ,
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(
equations of motion and boundary conditions of homogenous microbars derived based on the modified couple stress
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theory [42]. Moreover, letting l=0 and considering the same material properties for metal and ceramic, the equation of motion and boundary conditions are reduced to those of a homogenous bar derived based on the classical theory [38]. Considering uniform cross section for the microbar and utilizing the following dimensionless parameters
z r µ J , d T d L2 , c T c L , h T h , , r = , τ =t T = T = T = L R µJ J µJ J µJ J ρ J L2
(27)
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ζ=
equations (24)-(26) can be expressed in the dimensionless form as follows.
∂ 4ϕ ∂ 2ϕ ∂ 2ϕ − β + = T d (ζ ) 1 ∂ζ 4 ∂ζ 2 ∂τ 2
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β2
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∂ 3ϕ ∂ϕ − β1 +T c =0 β2 3 ζ ζ ∂ ∂ ζ = 0,1
∂ 2ϕ −T h =0 β2 2 ∂ζ ζ =0,1
or
or
(28)
δϕ ζ =0,1 = 0
(29)
∂ϕ =0 ∂ζ z =0,1
δ
(30)
where 2
β1 = 1 + 6
µ A lˆA , µ J R
2
1l β 2 = J . 4 L ˆ
(31)
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From equations (28)-(31) it can be concluded that considering the radius and the length of the bar much larger than the equivalent length scale parameters yields
β1 = 1
and
β2 = 0 .
Hence for macro scale bars, the equation of
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motion and boundary conditions are reduced to those derived based on the classical theory.
4. Results and discussion
To have numerical results, it is assumed that the microbar is made of Aluminum and Alumina as metal and ceramic, with the following material properties: µ m = 26 GPa, Km = 72 GPa, ρm = 2700 kg/m3 , µ c = 157 GPa, Kc = 228 GPa,
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ρm = 3960 kg/m3 . Furthermore, the variation of the volume fraction of material constituents along the radial direction is considered as n
n
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n r r f m (r ) = 1 − ( r ) = 1 − , f c ( r ) = r n = R R
(32)
which n is a positive real constant noting that n=0 and n=∞ are two extremes which means that the microbar is made
4.1. Static deformation
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of pure ceramic and pure metal respectively.
In order to examine size-dependent static behavior of a functionally graded microbar, a numerical example is presented here. Consider a fixed-free functionally graded microbar under dimensionless constant distributed torque
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T d (see figure 1). Assuming that the fixed support is not capable to apply higher order torque, the governing
β2
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equation of the static deformation and corresponding boundary conditions for this microbar can be written as
d 4ϕ d 2ϕ β − =Td 1 dζ 4 dζ 2
ϕ (0) = 0,
∂ 2ϕ ∂ζ 2
(33)
∂ 3ϕ ∂ϕ − β1 =0, = 0 , β2 3 ∂ ∂ ζ ζ ζ = 0,1 ζ =1
The solution of equation (33) can be written in the following form
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(34)
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ϕ (ζ ) = c1 + c2 ζ + c3 cosh(γ ζ ) + c4 sinh(γ ζ ) −
Tdζ2 . 2 β1
(35)
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where γ = β1 / β 2 and c1 , c2 , c3 , c4 are real constants that can be determined from boundary conditions. By
follows.
T d sinh ( γ ζ ) + sinh ( γ − γ ζ ) − sinh( γ ) ζ 2 − +ζ . 2 2 β1 γ sinh ( γ )
It is noted that setting
ζ 2 . 2
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(36)
ˆl = ˆl = 0 (which yields β = 1 and β = 0 ) ,the torsion angle of the microbar is derived A J 1 2
based on the classical theory as follows.
ϕC ( ζ ) = T d ζ −
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ϕ( ζ ) =
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applying the boundary conditions mentioned in equation (34), the torsion angle of the microbar is obtained as
Figure 2 shows the torsion angle of the microbar under constant distributed torque
(37)
T d = 1 , considering n=1 and
L / lˆJ = 20 . As shown in this figure, the modified couple stress theory evaluates the microbar stiffer than the
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classical theory and hence, the torsion angle predicted by the modified couple stress theory is smaller than that predicted by the classical theory. This figure indicates that, when the radius of the microbar is in order of the
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equivalent length scale, the difference between the torsion angle predicted by the classical theory and that evaluated based on the modified couple stress theory is considerable.
Figure 2. Torsion angle of the functionally graded bar.
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The torsion angle of the free end of the microbar with L / lˆJ = 20 is depicted in figure 3. The size-dependency of the static behavior of functionally graded microbar can be clearly observed in this figure. According to this figure, the modified couple stress theory predicts size-dependent static behavior for the microbar while according to the
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classical theory, the normalized static behavior of the microbar is independent of its size. As the bar radius increases, the results of the modified couple stress theory converge to those of the classical theory.
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Figure 3. Torsion angle of the free end of the bar versus parameter R / lˆA .
4.2. Free vibration
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In this section, free vibration of the considered fixed-free functionally graded microbar is investigated. The equation governing the free vibration of the microbar can be expressed as
∂ 4ϕ ∂ 2ϕ ∂ 2ϕ − + =0 β 1 ∂ζ 4 ∂ζ 2 ∂τ 2
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β2
(38)
To calculate natural frequencies, the mode shapes of torsional vibration of the bar are considered as
λ = 1, 2, 3,... ,
(39)
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1 ϒ λ (ζ ) = sin λ − πζ 2
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and the dynamic torsion of the microbar is expressed as
∞
∞
λ =1
λ =1
ϕ (ζ ,τ ) = ∑ψ (τ ) ϒ λ (ζ ) = ∑ψ (τ ) sin λ − πζ . 2 1
(40)
The mode shapes presented in equation (39), satisfy the essential and classical natural boundary conditions at both ends, as well as the non-classical natural boundary condition at the fixed end. Hence, only the non-classical natural boundary condition at ζ = 1 is not satisfied. However, in approximation of mode shapes of structures, it is common that some natural boundary conditions are not satisfied. Substituting equation (40) into equation (38), yields
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(41)
2 4 1 2 1 4 λ − π β1 + λ − π β 2 ϕ + ϕ&& = 0 2 2
τ . From equation (41) the dimensionless
natural frequencies of the functionally graded bar can be calculated as follows.
2
4
1 1 Ω λ = λ − π 2 β1 + λ − π 4 β 2 , 2 2
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where a dot denotes the derivative with respect to dimensionless time
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(42)
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and the ratio of the natural frequency given by the modified couple stress theory to that predicted based on the classical theory is obtained as
CS ΩΜ 1 λ = β1 + λ − π 2 β 2 Classical 2 Ωλ 2
(43)
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Figure 4 shows the first and second dimensionless natural frequencies of the microbar versus the ratio of microbar radius to the equivalent length scale ( R / lˆA ) considering L / lˆJ = 20 . As can be seen in this figure, the dimensionless natural frequency predicted by the modified couple stress theory is size-dependent. Also, the
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modified couple stress theory predicts the natural frequency of the microbar higher than the classical theory.
Figure 4. Natural frequency of the bar versus parameter R / lˆA .
Figures 5 and 6 show the ratio of first and forth natural frequencies predicted by the modified couple stress theory to those given by the classical theory considering n=1.
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Figure 5. The ratio of the first natural frequency of the bar given by the modified couple stress theory to that given by classical theory.
by classical theory.
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Figure 6. The ratio of the forth natural frequency of the bar given by the modified couple stress theory to that given
As shown in these figures and can be understood from equation (43), the difference between the results of modified
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couple stress and classical theories is increased for higher frequencies. From these figures, it is also concluded that for higher frequencies, the effect of L / lˆJ on natural frequency of the microbar is more tangible. According to these
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figures, for the microbars with length and radius much more than the equivalent length scales, the natural frequency calculate based on the modified couple stress theory, are in agreement with those evaluated based on the classical theory.
5. Conclusion
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In this paper, based on the modified couple stress theory, a formulation is derived for size-dependent static and dynamic behavior of functionally graded microbars. The governing equations of motion and corresponding boundary conditions are derived using Hamilton's principle. It is assumed that the volume fraction of material constituents is varying in the radial direction of the microbar. On this basis, equivalent material properties including
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length scale parameters are defined for the functionally graded bar and the static deformation and free vibration of this structures are modeled using these equivalent parameters. Results indicate that the modified couple stress theory
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is capable to model the size-dependent static and dynamic response of small structures. According to the findings of this paper, when the radius of the bar is of order of the equivalent length scale, the size-effect plays an important role in mechanical behavior of the structure. In this situation, the classical theory underestimates the stiffness of the structure and gives a rough estimation of the static and dynamic response of the microbar.
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functionally graded materials using higher-order equivalent single layer theories. Composites Part B 2014;67(0):490-509.
[18] Fu Y, Du H, Huang W, Zhang S, Hu M. TiNi-based thin films in MEMS applications: a review. Sens Actuators A 2004;112(2):395-408.
[19] Fu Y, Du H, Zhang S. Functionally graded TiN/TiNi shape memory alloy films. Mater Lett 2003;57(20):29959.
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[20] Witvrouw A, Mehta A. The use of functionally graded poly-SiGe layers for MEMS applications. Materials Science Forum: Trans Tech Publ; 2005. p. 255-260.
[21] Fleck N, Muller G, Ashby M, Hutchinson J. Strain gradient plasticity: theory and experiment. Acta Metal Mater
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1994;42(2):475-487.
[22] McFarland AW, Colton JS. Role of material microstructure in plate stiffness with relevance to microcantilever
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sensors. J Micromech Microeng 2005;15(5):1060. [23] Lam D, Yang F, Chong A, Wang J, Tong P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids. 2003;51(8):1477-1508. [24] Stölken J, Evans A. A microbend test method for measuring the plasticity length scale. Acta Mater 1998;46(14):5109-5115.
[25] Yang F, Chong A, Lam D, Tong P. Couple stress based strain gradient theory for elasticity. Int J Solids Struct 2002;39(10):2731-2743. [26] Toupin RA. Elastic materials with couple-stresses. Arch Ration Mech Anal 1962;11(1):385-414.
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[27] Mindlin R, Tiersten H. Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 1962;11(1):415448. [28] Park S, Gao X. Bernoulli–Euler beam model based on a modified couple stress theory. J Micromech Microeng
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2006;16(11):2355. [29] Tsiatas GC. A new Kirchhoff plate model based on a modified couple stress theory. Int J Solids Struct 2009;46(13):2757-64.
[30] Rahaeifard M, Kahrobaiyan M, Asghari M, Ahmadian M. Static pull-in analysis of microcantilevers based on
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the modified couple stress theory. Sens Actuators A 2011;171(2):370-374.
[31] Rahaeifard M, Ahmadian MT, Firoozbakhsh K. Size-dependent dynamic behavior of microcantilevers under
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suddenly applied DC voltage. IMECE, Part C 2013, doi: 10.1177/0954406213490376.
[32] Kahrobaiyan M, Rahaeifard M, Ahmadian M. A size-dependent yield criterion. Int J Eng Sci 2014;74:151-161. [33] Asghari M, Ahmadian M, Kahrobaiyan M, Rahaeifard M. On the size-dependent behavior of functionally graded micro-beams. Mater Des 2010;31(5):2324-2329.
[34] Asghari M, Rahaeifard M, Kahrobaiyan M, Ahmadian M. The modified couple stress functionally graded Timoshenko beam formulation. Mater Des 2011;32(3):1435-1443.
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[35] Ke LL, Yang J, Kitipornchai S, Bradford MA, Wang YS. Axisymmetric nonlinear free vibration of sizedependent functionally graded annular microplates. Composites Part B 2013;53(0):207-217. [36] Thai H-T, Kim S-E. A size-dependent functionally graded Reddy plate model based on a modified couple stress
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theory. Composites Part B 2013;45(1):1636-1645.
[37] Kahrobaiyan MH, Tajalli S, Movahhedy M, Akbari J, Ahmadian M. Torsion of strain gradient bars. Int J Eng
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Sci 2011;49(9):856-866.
[38] Rao SS. Vibration of continuous systems: John Wiley & Sons; 2007. [39] Cheng ZQ, Batra RC. Three-dimensional thermoelastic deformations of a functionally graded elliptic plate. Composites Part B 2000;31(2):97-106. [40] Belabed Z, Houari MSA, Tounsi A, Mahmoud S, Bég OA. An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates. Composites Part B 2014;60:274-283.
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[41] Tounsi A, Al-Basyouni K, Mahmoud S. Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position. Composite Struct 2015;125:621630.
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[42] Gheshlaghi B, Hasheminejad SM. Size dependent torsional vibration of nanotubes. Physica E 2010;43(1):45-
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48.
Figure Captions:
Figure 1. A functionally graded circular microbar.
Figure 2. Torsion angle of the functionally graded bar.
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Figure 3. Torsion angle of the free end of the bar versus parameter R / lˆA . Figure 4. Natural frequency of the bar versus parameter R / lˆA .
Figure 5. The ratio of the first natural frequency of the bar given by the modified couple stress theory to that given by classical theory.
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by classical theory.
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Figure 6. The ratio of the forth natural frequency of the bar given by the modified couple stress theory to that given
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x T d ( z, t )
x y
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z
2R
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SC
Figure 1. A functionally graded circular microbar.
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0.5 0.45
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0.4
0.3 0.25 0.2
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Torsion angle: φ
0.35
0.15
0.05 0
0
0.1
0.2
0.3
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0.1
0.4
0.5
0.6
0.7
0.8
0.9
1
z/L
R / lˆA = 3
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R / lˆA = 5
R / lˆA = 10
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Figure 2. Torsion angle of the functionally graded bar.
Classical ( R >> lˆA )
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0.55
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0.45
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0.4
0.35
0.3
0.25
0.2
2
4
6
8
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Torsion angle of the tip: φ(1)
0.5
10
12
14
16
18
20
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R / lˆA
Modified couple stress (n=0.1)
Classical
Modified couple stress (n=10)
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Figure 3. Torsion angle of the free end of the bar versus parameter R / lˆA .
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2.5
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2.3 2.2 2.1
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2 1.9 1.8 1.7 1.6 1.5
2
4
6
8
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Dimensionless natural frequency
2.4
10
12
14
16
18
20
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R / lˆA
Modified couple stress (n=0.1) Classical
Modified couple stress (n=10)
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Figure 4. First natural frequency of the bar versus parameter R / lˆA .
L / lˆJ
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R / lˆA
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Ω1MCS/ Ω1Classical
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Figure 5. The ratio of the first natural frequency of the bar given by the modified couple stress theory to that given
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by classical theory.
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Ω4MCS/ Ω4Classical
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R / lˆA
L / lˆJ
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Figure 6. The ratio of the forth natural frequency of the bar given by the modified couple stress theory to
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that given by classical theory.
S1359-8368(15)00439-4
DOI:
10.1016/j.compositesb.2015.08.011
Reference:
JCOMB 3689
To appear in:
Composites Part B
Received Date: 7 January 2015 Revised Date:
10 May 2015
Accepted Date: 7 August 2015
Please cite this article as: Rahaeifard M, Size-dependent torsion of functionally graded bars, Composites Part B (2015), doi: 10.1016/j.compositesb.2015.08.011. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Size-dependent torsion of functionally graded bars M. Rahaeifard*
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Department of Mechanical Engineering, Golpayegan University of Technology, Golpayegan, Iran.
Abstract
In this paper, size-dependent static and dynamic behavior of functionally graded microbars is investigated on the basis of the modified couple stress theory. The equation of motion and corresponding boundary conditions are
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derived using Hamilton's principle and presented in the dimensionless form. Equivalent mechanical properties (i.e. shear modulus, density and length scale) are extracted for the functionally graded microbar based on the mechanical
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properties of the material constituents. In this work, it is shown that without any simplifying assumption, two equivalent length scale parameters can be defined for functionally graded bars and the size-dependent mechanical behavior of these components can be explained using these parameters. As an example, static and dynamic behavior of a functionally graded microbar with fixed-free boundary conditions is analyzed and the effect of size-dependency on mechanical behavior of this structure is discussed.
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Keywords: B. Mechanical properties; B. Microstructures; C. Analytical modelling; C. Micro-mechanics; Modified couple stress theory.
1. Introduction
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Functionally graded materials (FGMs) are inhomogeneous composites which are made of two different materials typically a metal and a ceramic. The volume fractions of material constituents in FGMs vary continuously in a
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certain direction of the structure. This continuous variation provides a smooth change in the mechanical properties and eliminates the high magnitude shear stresses which appear in laminated composites. FGMs can offer the benefits of both of material constituents and hence, these materials are widely used in various fields of engineering including aerospace, bioengineering and mechanical engineering. Many researchers have studied the mechanical behavior of structures made of functionally graded materials. Here some of these works are reviewed. Fallah et al. [1] investigated buckling and vibration of functionally graded Euler-
*
E-mail: [email protected] Tel: +983157243163
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Bernoulli beams under thermal loading. Nonlinear vibration of functionally graded beams is investigated by Ke et al. [2]. They considered the material properties to be varying through the thickness of the beam and took into account the nonlinearities caused by the mid-plane stretching. Sarkar and Ganguli [3] presented a closed form
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solution for natural frequency of functionally graded Timoshenko beams. The nonlinear mechanical response of functionally graded piezoelectric beams with properties varying through the thickness of the beam is analyzed by Lin and Muliana [4]. Su and Banerjee [5] utilized the dynamic stiffness method to investigate the dynamic behavior and natural frequency of functionally graded Timoshenko beams. Torsional responses of functionally graded bars
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and cylinders are investigated by Horgan and Chan [6] and Dung and Hoa [7, 8] respectively. Rahaeifard et al. showed that the sensitivity of atomic force microscopes can be enhanced using functionally graded materials [9, 10].
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A closed form solution is proposed for static response of functionally graded Kirchhoff plates by Apuzzo et al. [11]. Hosseini-Hashemia et al. investigated the in-plane and out of plane vibration of functionally graded rectangular plates with simply supported boundary conditions [12]. Furthermore, mechanical behavior of functionally graded shells has been also widely investigated by researchers (see for example [13-17]).
Nowadays, FGMs are also used in microsystems such as thin films in the form of shape memory alloys [18, 19] and micro-electro-mechanical systems [18, 20]. In these systems, the size of structures is of order of microns and sub-
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microns. Many experimental researches performed on the mechanical response of micro scale structures (see for example [21-24]) proved that the mechanical behavior of these components is different from that predicted by the classical continuum theory. According to these researches, the normalized stiffness and normalized deflection of
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micro scale structures, which according to the classical theory should be independent of the structure size, is significantly size-dependent. Furthermore, the stiffness of micro scale structures measured in experiment, is much
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more than the stiffness predicted by the classical theory. These experiments clearly showed that the classical continuum theory can neither evaluate the mechanical response of micro scale structures nor justify the size-dependency observed in mechanical behavior of these components. In contrast to the classical theory, some new continuum theories such as couple stress theory and modified couple stress theory can cover this gap and accurately model the mechanical behavior of micro scale structures. The modified couple stress theory was developed by Yang et al. [25] by performing a modification on the couple stress theory previously proposed by Toupin [26] and also Mindlin and Tiersten [27]. In the modified couple stress theory, beside the classical stresses (i.e. normal and shear stresses), a non-classical stress known as couple stress is
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also acting on an element of material. This higher order stress is related to the displacement field by a new material constant called length scale parameter. Furthermore, in this theory, in addition to the classical equilibrium equations, a non-classic equilibrium equation (i.e. the equilibrium equation of moment of couples) is satisfied.
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Many researchers utilized this theory to investigate the mechanical behavior of micro scale structures as well as to justify the size-dependency observed in experiment. Park and Gao [28] utilized this theory to derive the governing equations of Euler-Bernoulli microbeams and detected the size-dependency in the static behavior of these structures. Tsiatas [29] utilized this theory to investigate the mechanical behavior of Kirchhoff microplates with arbitrary
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shapes. Size-dependent static and dynamic behavior of electrostatically actuated microcantilevers is analyzed by Rahaeifard et al. [30, 31]. They showed that utilizing the modified couple stress theory can remove the gap between
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the experimental and theoretical results of the static pull-in of microcantilevers [30]. They also proposed a nonclassical yield criterion based on the modified couple stress theory to justify the size-dependent yielding of micro scale structures. The predictions of their criterion were in very good agreement with experimental observations while the yielding load predicted by the classical theory was up to 50% less than the yielding load measured in experiment [32].
This theory is also widely used by researchers to model the static and dynamic behavior of micro scale structures
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made of functionally graded materials. The size-dependent formulations of functionally graded Euler-Bernoulli and Timoshenko microbeams are developed by Asghari et al. [33, 34]. They utilized this formulation to model the sizedependent deflection and natural frequency of these components. The size-dependent mechanical behavior of
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functionally graded Mindlin microplates is investigated by Ke et al. [35]. Thai and Kim [36] derived the governing equations of motion for functionally graded Reddy plates based on the modified couple stress theory and
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investigated the static and dynamic behavior of these structures. In this paper based on the modified couple stress theory, a formulation is developed to model the static and dynamic torsion of functionally graded microbars. The volume fraction of material constituents is assumed to vary through the radial direction. On this basis, two equivalent length scale parameters are defined for functionally graded microbars based on the length scale parameters and volume fractions of material constituents. As a case study, static deformation and free vibration of a functionally graded fixed-free bar is analyzed. It is shown that the modified couple stress theory can successfully model the size-dependency of mechanical behavior of functionally graded bars.
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The results of the modified couple stress theory are compared with those evaluated based on the classical theory and the importance of size effect in mechanical behavior of micro scale functionally graded bars is discussed.
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2. Preliminaries According to the modified couple stress theory presented by Yang et al. [18], the strain energy of an elastic body can be written as follows.
in which
(
i, j = 1, 2,3) ,
(1)
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1 (σ ijε ij + mij χij ) dV 2 V∫
σ ij and ε ij , denote the components of stress and strain tensors and mij and χij refer to the deviatoric part
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U=
of couple stress and the symmetric curvature tensors respectively. These parameters can be expressed as
χij =
νE
(1 + ν ) (1 − 2ν )
ε kk δ ij + 2 µε ij ,
1 ( θi, j + θ j,i ) , 2
mij = 2l 2 µχ ij ,
(2)
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σ ij =
1 ( ui , j + u j , i ) , 2
(3)
(4)
(5)
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ε ij =
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where ui denotes the components of displacement field, l is the length scale parameter, E and
µ
denote the Yong
modulus and shear modulus respectively and θi refers to the components of the rotation vector related to the components of the displacement field as follows.
θi =
1 (curl ( u ) )i . 2
(6)
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3. Modelling of functionally graded bar
A. The microbar is under distributed torque (torque per unit length) denoted by
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Figure 1 displays a functionally graded microbar of length L having a circular cross section with radius R and area
Td .
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For this microbar the displacement field can be written as [37, 38]
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Figure 1. A functionally graded circular microbar.
u1 = − yϕ ( z , t ) , u2 = xϕ ( z , t ) , u3 ( z , t ) = 0 ,
where u1 , u2 and u3 refer to displacements along x, y, and z axes respectively and
(7)
ϕ
denotes the rotation angle of
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the cross section about z axis. According to the abovementioned displacement field, the components of the strain and symmetric curvature tensors can be calculated using equations (2) and (4) as follows.
1 ∂ϕ ∂ϕ 1 ∂ 2ϕ 1 ∂ 2ϕ , χ33 = , χ13 = χ31 = − x 2 , χ32 = χ 23 = − y 2 , χ12 = χ 21 = 0. 2 ∂z ∂z 4 ∂z 4 ∂z
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χ11 = χ 22 = −
1 ∂ϕ 1 ∂ϕ , x , ε13 = ε 31 = − y 2 ∂z 2 ∂z
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ε11 = ε 22 = ε 33 = ε12 = ε 21 = 0 , ε 23 = ε 32 =
(8)
(9)
It is assumed that the microbar is made of two materials (one metal and one ceramic) in a way that the volume fractions of material constituents vary through the radial direction. According to Mori-Tanaka relation, the equivalent shear modulus of a functionally graded material (i.e. 41].
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µeq ) can found from the following equation [39-
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µeq − µm = µc − µm
fc µ − µm 1 + (1 − f c ) c µm + f1
(10)
µ m ( 9 K m + 8µ m ) 6 ( K m + 2µm )
(11)
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f1 =
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where subscripts m and c refer to metal and ceramic respectively, f c denotes the volume fraction of ceramic and
in which K m denotes the bulk modulus of metal. Using the equivalent shear modulus, the classical shear stresses can
∂ϕ , ∂z ∂ϕ σ 13 = σ 13 = 2 µeq ε13 = − µeq y , ∂z
σ 23 = σ 32 = 2 µeq ε 23 = µeq x
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be obtained as
(12)
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Furthermore, the higher order stresses (couple stresses) in metal and ceramic phases are evaluated as follows
∂ϕ ∂ϕ , ( m33 )m / c = 2 µm / c lm2 / c , ∂z ∂z ∂ 2ϕ ∂ 2ϕ 1 1 = − µm / c lm2 / c x 2 , ( m23 )m / c = ( m32 )m / c = − µm / c lm2 / c y 2 , 2 2 ∂z ∂z
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( m13 )m / c = ( m31 )m/ c
(13)
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( m11 )m/ c = ( m22 )m/ c = −µm/ c lm2 / c
where the subscript m/c (i.e. m or c) means “metal or ceramic”. Substituting equations (8)-(13) into equation (1), yields the strain energy of the functionally graded microbar as
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L
U=
1 {σ ijε ij + (mij )m χij fm (r ) + (mij )c χij fc (r )} dA dz 2 ∫0 ∫A
2 L 1 2 ∂ϕ 2 µ x + y eq 2 ∫0 ∫A ∂z
(
=
)
2 2 ∂ϕ 2 1 2 2 ∂ ϕ + µm lm2 f m (r ) + µclc2 f c (r ) 3 + x + y 2 dA dz ∂z 4 ∂z
)
(
)
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(
(14)
2 L R ∂ϕ 2 1 ∂ 2ϕ 2 ∂ϕ 3 2 2 µ µ = π ∫ ∫ r 3 µeq + l f ( r ) + l f ( r ) c c c mm m 3r ∂z + 4 r ∂z 2 dr dz , z ∂ 0 0
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where r denotes the distance of an arbitrary point from the center of the cross section ( r =
x 2 + y 2 ) and f m and
f c refer to the volume fraction of metal and ceramic respectively noted that f m + f c = 1 . Defining new equivalent
lˆA =
A
A
=
0
R2
r 2 µeq dA J
∫ (µ A
m
,
R
=
2π ∫ µeq r 3dr 0
J
)
f m lm2 + µc f c lc2 dA
µA A
∫ (µ A
m
)
,
=
f m lm2 + µc f c lc2 r 2 dA
µJ J
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lˆJ =
A
R
2 ∫ µeq rdr
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µJ
∫ =
µeq dA
2π ∫
R
0
=
(µ
m
)
f m lm2 + µc f c lc2 r dr
µA A
2π ∫
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µA
∫ =
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parameters for the functionally graded bar as follows
R
0
(µ
m
)
µJ J
(16)
(17)
,
f m lm2 + µc f c lc2 r 3 dr
(15)
,
(18)
∫
where J = r 2 dA is polar area moment of inertia, the strain energy can be written as A
2 2 L 2 1 1 2 ∂ϕ 2 ∂ ϕ ˆ ˆ U = ∫ µ J J + 3µ A Al A + µ J J lJ 2 20 ∂z 4 ∂z
(
)
(
)
dz .
The total kinetic energy of the microbar (K) can be expressed as
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(19)
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L L ∂u1 2 ∂u2 2 ∂u3 2 1 1 ∂ϕ f + f + + dA dz = J ρJ ρ ρ ( ) m m c c dz , ∫ 2 ∫0 ∫A ∂ t ∂ t ∂ t 2 ∂t 0 2
ρJ
∫ =
A
is the equivalent density of the functionally graded microbar defined as
r 2 ( ρ m f m + ρ c f c ) dA J
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ρJ
where
(20)
(21)
.
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K=
The variation of the external work done by the concentrated and distributed couples on the microbar can be
∂ϕ , ∂z z =0, L
L
δ W = ∫ T d δϕ ( z ) dz + T c δϕ z = 0, L + T h δ 0
(22)
T c and T h stand for the classical and higher order torques applied on the end sections of the bar. In order
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in which
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mentioned as
to derive the governing equation of motion, Hamilton's principle is utilized as follows.
t2
∫ (δ K − δ U + δ W ) dt = 0 .
(23)
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t1
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Substituting equations (19)-(22) into equation (23) and following some straightforward mathematical manipulations gives the equation of motion and boundary conditions of the functionally graded microbar as
−
2 2 ∂ ˆ 2 ∂ϕ + ∂ 1 µ J lˆ 2 ∂ ϕ + ∂ ( ρ J ) ∂ϕ = T d ( z ) , J + 3 Al µ µ J A A J J 2 J ∂z ∂z ∂z 2 4 ∂t ∂z ∂t
(
)
8
(24)
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2 ∂ 1 2∂ ϕ 2 ∂ϕ +Tc =0 µ J J lˆJ 2 − µ J J + 3µ A AlˆA ∂z 4 ∂z ∂z z = 0, L
)
2 1 h ˆ2 ∂ ϕ =0 µ J J lJ 2 − T ∂z 4 z =0, L
or
or
∂ϕ =0. ∂z z =0, L
δ
(25)
(26)
Considering the same material properties for metal and ceramic ( µ m =
µc = µ , ρ m = ρ c = ρ , lm = lc = l ),
µ A = µ J = µ , ρ J = ρ , lˆA = lˆJ = l , and the formulation presented here, is reduced to the governing
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results in:
δϕ z =0, L = 0 ,
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(
equations of motion and boundary conditions of homogenous microbars derived based on the modified couple stress
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theory [42]. Moreover, letting l=0 and considering the same material properties for metal and ceramic, the equation of motion and boundary conditions are reduced to those of a homogenous bar derived based on the classical theory [38]. Considering uniform cross section for the microbar and utilizing the following dimensionless parameters
z r µ J , d T d L2 , c T c L , h T h , , r = , τ =t T = T = T = L R µJ J µJ J µJ J ρ J L2
(27)
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ζ=
equations (24)-(26) can be expressed in the dimensionless form as follows.
∂ 4ϕ ∂ 2ϕ ∂ 2ϕ − β + = T d (ζ ) 1 ∂ζ 4 ∂ζ 2 ∂τ 2
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β2
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∂ 3ϕ ∂ϕ − β1 +T c =0 β2 3 ζ ζ ∂ ∂ ζ = 0,1
∂ 2ϕ −T h =0 β2 2 ∂ζ ζ =0,1
or
or
(28)
δϕ ζ =0,1 = 0
(29)
∂ϕ =0 ∂ζ z =0,1
δ
(30)
where 2
β1 = 1 + 6
µ A lˆA , µ J R
2
1l β 2 = J . 4 L ˆ
(31)
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From equations (28)-(31) it can be concluded that considering the radius and the length of the bar much larger than the equivalent length scale parameters yields
β1 = 1
and
β2 = 0 .
Hence for macro scale bars, the equation of
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motion and boundary conditions are reduced to those derived based on the classical theory.
4. Results and discussion
To have numerical results, it is assumed that the microbar is made of Aluminum and Alumina as metal and ceramic, with the following material properties: µ m = 26 GPa, Km = 72 GPa, ρm = 2700 kg/m3 , µ c = 157 GPa, Kc = 228 GPa,
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ρm = 3960 kg/m3 . Furthermore, the variation of the volume fraction of material constituents along the radial direction is considered as n
n
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n r r f m (r ) = 1 − ( r ) = 1 − , f c ( r ) = r n = R R
(32)
which n is a positive real constant noting that n=0 and n=∞ are two extremes which means that the microbar is made
4.1. Static deformation
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of pure ceramic and pure metal respectively.
In order to examine size-dependent static behavior of a functionally graded microbar, a numerical example is presented here. Consider a fixed-free functionally graded microbar under dimensionless constant distributed torque
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T d (see figure 1). Assuming that the fixed support is not capable to apply higher order torque, the governing
β2
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equation of the static deformation and corresponding boundary conditions for this microbar can be written as
d 4ϕ d 2ϕ β − =Td 1 dζ 4 dζ 2
ϕ (0) = 0,
∂ 2ϕ ∂ζ 2
(33)
∂ 3ϕ ∂ϕ − β1 =0, = 0 , β2 3 ∂ ∂ ζ ζ ζ = 0,1 ζ =1
The solution of equation (33) can be written in the following form
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(34)
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ϕ (ζ ) = c1 + c2 ζ + c3 cosh(γ ζ ) + c4 sinh(γ ζ ) −
Tdζ2 . 2 β1
(35)
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where γ = β1 / β 2 and c1 , c2 , c3 , c4 are real constants that can be determined from boundary conditions. By
follows.
T d sinh ( γ ζ ) + sinh ( γ − γ ζ ) − sinh( γ ) ζ 2 − +ζ . 2 2 β1 γ sinh ( γ )
It is noted that setting
ζ 2 . 2
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(36)
ˆl = ˆl = 0 (which yields β = 1 and β = 0 ) ,the torsion angle of the microbar is derived A J 1 2
based on the classical theory as follows.
ϕC ( ζ ) = T d ζ −
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ϕ( ζ ) =
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applying the boundary conditions mentioned in equation (34), the torsion angle of the microbar is obtained as
Figure 2 shows the torsion angle of the microbar under constant distributed torque
(37)
T d = 1 , considering n=1 and
L / lˆJ = 20 . As shown in this figure, the modified couple stress theory evaluates the microbar stiffer than the
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classical theory and hence, the torsion angle predicted by the modified couple stress theory is smaller than that predicted by the classical theory. This figure indicates that, when the radius of the microbar is in order of the
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equivalent length scale, the difference between the torsion angle predicted by the classical theory and that evaluated based on the modified couple stress theory is considerable.
Figure 2. Torsion angle of the functionally graded bar.
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The torsion angle of the free end of the microbar with L / lˆJ = 20 is depicted in figure 3. The size-dependency of the static behavior of functionally graded microbar can be clearly observed in this figure. According to this figure, the modified couple stress theory predicts size-dependent static behavior for the microbar while according to the
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classical theory, the normalized static behavior of the microbar is independent of its size. As the bar radius increases, the results of the modified couple stress theory converge to those of the classical theory.
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Figure 3. Torsion angle of the free end of the bar versus parameter R / lˆA .
4.2. Free vibration
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In this section, free vibration of the considered fixed-free functionally graded microbar is investigated. The equation governing the free vibration of the microbar can be expressed as
∂ 4ϕ ∂ 2ϕ ∂ 2ϕ − + =0 β 1 ∂ζ 4 ∂ζ 2 ∂τ 2
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β2
(38)
To calculate natural frequencies, the mode shapes of torsional vibration of the bar are considered as
λ = 1, 2, 3,... ,
(39)
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1 ϒ λ (ζ ) = sin λ − πζ 2
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and the dynamic torsion of the microbar is expressed as
∞
∞
λ =1
λ =1
ϕ (ζ ,τ ) = ∑ψ (τ ) ϒ λ (ζ ) = ∑ψ (τ ) sin λ − πζ . 2 1
(40)
The mode shapes presented in equation (39), satisfy the essential and classical natural boundary conditions at both ends, as well as the non-classical natural boundary condition at the fixed end. Hence, only the non-classical natural boundary condition at ζ = 1 is not satisfied. However, in approximation of mode shapes of structures, it is common that some natural boundary conditions are not satisfied. Substituting equation (40) into equation (38), yields
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(41)
2 4 1 2 1 4 λ − π β1 + λ − π β 2 ϕ + ϕ&& = 0 2 2
τ . From equation (41) the dimensionless
natural frequencies of the functionally graded bar can be calculated as follows.
2
4
1 1 Ω λ = λ − π 2 β1 + λ − π 4 β 2 , 2 2
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where a dot denotes the derivative with respect to dimensionless time
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(42)
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and the ratio of the natural frequency given by the modified couple stress theory to that predicted based on the classical theory is obtained as
CS ΩΜ 1 λ = β1 + λ − π 2 β 2 Classical 2 Ωλ 2
(43)
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Figure 4 shows the first and second dimensionless natural frequencies of the microbar versus the ratio of microbar radius to the equivalent length scale ( R / lˆA ) considering L / lˆJ = 20 . As can be seen in this figure, the dimensionless natural frequency predicted by the modified couple stress theory is size-dependent. Also, the
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modified couple stress theory predicts the natural frequency of the microbar higher than the classical theory.
Figure 4. Natural frequency of the bar versus parameter R / lˆA .
Figures 5 and 6 show the ratio of first and forth natural frequencies predicted by the modified couple stress theory to those given by the classical theory considering n=1.
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Figure 5. The ratio of the first natural frequency of the bar given by the modified couple stress theory to that given by classical theory.
by classical theory.
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Figure 6. The ratio of the forth natural frequency of the bar given by the modified couple stress theory to that given
As shown in these figures and can be understood from equation (43), the difference between the results of modified
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couple stress and classical theories is increased for higher frequencies. From these figures, it is also concluded that for higher frequencies, the effect of L / lˆJ on natural frequency of the microbar is more tangible. According to these
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figures, for the microbars with length and radius much more than the equivalent length scales, the natural frequency calculate based on the modified couple stress theory, are in agreement with those evaluated based on the classical theory.
5. Conclusion
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In this paper, based on the modified couple stress theory, a formulation is derived for size-dependent static and dynamic behavior of functionally graded microbars. The governing equations of motion and corresponding boundary conditions are derived using Hamilton's principle. It is assumed that the volume fraction of material constituents is varying in the radial direction of the microbar. On this basis, equivalent material properties including
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length scale parameters are defined for the functionally graded bar and the static deformation and free vibration of this structures are modeled using these equivalent parameters. Results indicate that the modified couple stress theory
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is capable to model the size-dependent static and dynamic response of small structures. According to the findings of this paper, when the radius of the bar is of order of the equivalent length scale, the size-effect plays an important role in mechanical behavior of the structure. In this situation, the classical theory underestimates the stiffness of the structure and gives a rough estimation of the static and dynamic response of the microbar.
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48.
Figure Captions:
Figure 1. A functionally graded circular microbar.
Figure 2. Torsion angle of the functionally graded bar.
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Figure 3. Torsion angle of the free end of the bar versus parameter R / lˆA . Figure 4. Natural frequency of the bar versus parameter R / lˆA .
Figure 5. The ratio of the first natural frequency of the bar given by the modified couple stress theory to that given by classical theory.
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by classical theory.
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Figure 6. The ratio of the forth natural frequency of the bar given by the modified couple stress theory to that given
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x T d ( z, t )
x y
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z
2R
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Figure 1. A functionally graded circular microbar.
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0.5 0.45
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0.4
0.3 0.25 0.2
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Torsion angle: φ
0.35
0.15
0.05 0
0
0.1
0.2
0.3
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0.1
0.4
0.5
0.6
0.7
0.8
0.9
1
z/L
R / lˆA = 3
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R / lˆA = 5
R / lˆA = 10
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Figure 2. Torsion angle of the functionally graded bar.
Classical ( R >> lˆA )
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0.55
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0.45
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0.4
0.35
0.3
0.25
0.2
2
4
6
8
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Torsion angle of the tip: φ(1)
0.5
10
12
14
16
18
20
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R / lˆA
Modified couple stress (n=0.1)
Classical
Modified couple stress (n=10)
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Figure 3. Torsion angle of the free end of the bar versus parameter R / lˆA .
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2.5
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2.3 2.2 2.1
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2 1.9 1.8 1.7 1.6 1.5
2
4
6
8
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Dimensionless natural frequency
2.4
10
12
14
16
18
20
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R / lˆA
Modified couple stress (n=0.1) Classical
Modified couple stress (n=10)
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Figure 4. First natural frequency of the bar versus parameter R / lˆA .
L / lˆJ
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R / lˆA
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Ω1MCS/ Ω1Classical
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Figure 5. The ratio of the first natural frequency of the bar given by the modified couple stress theory to that given
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by classical theory.
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Ω4MCS/ Ω4Classical
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R / lˆA
L / lˆJ
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Figure 6. The ratio of the forth natural frequency of the bar given by the modified couple stress theory to
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that given by classical theory.