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Nonlinear thermal buckling of axially functionally graded micro and nanobeams
Accepted Manuscript Nonlinear thermal buckling of axially functionally graded micro and nanobeams Navvab Shafiei, Seyed Sajad Mirjavadi, Behzad Mohasel Afshari, Samira Rabby, A.M.S. Hamouda PII: DOI: Reference:
S0263-8223(16)32441-2 http://dx.doi.org/10.1016/j.compstruct.2017.02.048 COST 8269
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
6 November 2016 23 December 2016 10 February 2017
Please cite this article as: Shafiei, N., Mirjavadi, S.S., Afshari, B.M., Rabby, S., Hamouda, A.M.S., Nonlinear thermal buckling of axially functionally graded micro and nanobeams, Composite Structures (2017), doi: http:// dx.doi.org/10.1016/j.compstruct.2017.02.048
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Nonlinear thermal buckling of axially functionally graded micro and nanobeams Navvab Shafieia*, Seyed Sajad Mirjavadib, Behzad Mohasel Afsharic, Samira Rabbyb, A.M.S. Hamoudab a
Department of Mechanical Engineering, Payame Noor University (PNU), P. O. Box, 19395-3697, Tehran, Iran. b Department of Mechanical and Industrial Engineering, Qatar University, P.O.Box 2713, Doha, Qatar. c School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran.
* Corresponding author: Navvab Shafiei* Tel: + 98 84 33368848 Email: [email protected] 0000-0003-0992-9660.
Abstract In this study, the nonlinear thermal buckling of axially functionally graded (AFG) EulerBernoulli micro/nanobeams is analyzed. The Eringen’s nonlocal elasticity theory is used to develop the governing equations of nanobeam and the modified couple stress theory is used to study the microbeam. The micro- and nanobeams are made of pure metal, pure ceramic and axially functionally graded material which is the composition of metal and ceramic. Boundary conditions are considered as clamped (CC) and simply supported (SS). The generalized differential quadrature method (GDQM) is used along with the iteration technique to solve the nonlinear equations. The parametric studies are served to examine the effects of the small scale parameters, length to height ratio (L/h), nonlinear amplitude and AFG power index on the buckling temperature of the micro- and nanobeams.
Keyword: Nonlinear buckling; Thermal buckling; Axially Functionally Graded; Eringen nonlocal theory; modified couple stress.
1
1. Introduction The emerging nano and micro-sciences have made so many nano- and micro-structures with many different functions. Among them, there are a lot of nano- and micro-systems that work in thermal areas and under thermal stresses. Thus, it is needed to study these small-scale systems in thermal environments. Accordingly, many researchers have studied the thermal buckling of microbeams. Based on the modified couple stress theory, Ke et. al [1] studied the free vibration and buckling of microbeams based on Timoshenko beam theory. Chiao and Lin [2] studied the self-buckling behavior of microbeams under resistive heating. Nateghi and Salamat-talab [3] studied the thermal effect on buckling and free vibration behavior of functionally graded (FG) microbeams on the basis of the modified couple stress theory. Ansari et. al [4] studied the thermal postbuckling of FG microbeams under thermal loads. Wang et. al [5] performed analysis on the elastic buckling of micro- and nano-rods/tubes based on Eringen's nonlocal elasticity theory and the Timoshenko beam theory. Mohammadabadi et. al [6] studied the thermal effect on size-dependent buckling behavior of micro composite laminated beams. Mohammadi and Mahzoon [7] studied the thermal effects on postbuckling of microbeams. Fang and Wickert [8] studied the static deformation of microbeams considering prescribed in-plane compressive stress. Şimşek and Reddy [9] introduced a unified higher order beam theory for buckling of a functionally graded (FG) microbeam embedded in elastic Pasternak medium. Malekzadeh et. al [10] studied the small scale effect on the thermal buckling of nanoplates embedded in an elastic medium. Ke et. al [11] studied the bending, buckling and free vibration of annular FG microplates. Akgöz and Civalek [12] investigated the buckling of size-dependent FG microbeams. Besides, many researchers have carried out investigations on the buckling behavior of nanostructures such as nanobeams, nanoplates, etc. Ansari et. al [13] studied the thermal buckling of single-walled carbon nanotubes. Tounsi et. al [14] presented a higher-order nonlocal beam theory for the thermal buckling of nanobeams. Berrabah et. al [15] presented a unified nonlocal shear deformation theory to study bending, buckling and free vibration of nanobeams. Malekzadeh et. al [10] studied the small scale effect on the thermal buckling of nanoplates embedded in an elastic medium. Emam [16] investigated the buckling and postbuckling of the nonlocal nanobeams. Lim et. al [17] studied the buckling of nanostructures including nanobeam, nanorod, and nanotube in a temperature field. The surface effects waere studied by Yan and Jiang [18] on the vibrational and buckling of 2
piezoelectric Euler-Bernoulli nanobeams. Thai [19] presented a nonlocal shear deformation beam theory for bending, buckling, and vibration of nanobeams using the nonlocal differential constitutive relations of Eringen. Zenkour and Sobhy [20] studied the thermal buckling of single-layered graphene sheets lying on an elastic medium. Şimşek and Yurtcu [21] studied the static bending and buckling of a FG nanobeam. Nami et. al [22] presented analysis on the thermal buckling analysis of functionally graded rectangular nanoplates. Chaht et. al [23] studied the bending and buckling behaviors of size-dependent FG nanobeams including the thickness stretching effect. Ansari et. al [24] studied the buckling and vibration responses of FG nanoplates subjected to thermal loading considering surface stress effects. Zhang et. al [25] investigated the hybrid nonlocal Euler-Bernoulli beam model for the bending, buckling, and vibration of micro/nanobeams. Rafiee et. al [26] studied the nonlinear thermal bifurcation buckling behavior of carbon nanotube reinforced composite beams with surface-bonded piezoelectric layers. Wang et. al [27] analyzed the thermal buckling properties of a nanoplate with small-scale effects. Barati et. al [28] studied the thermal buckling behavior of size-dependent FG nanoplates. Shen [29] performed analysis on the thermal postbuckling of nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes considering a uniform temperature rise. The lack of a comprehensive study on the thermal behavior of axially functionally graded (AFG) micro- and nano-beams is noticeable. Thus, the authors decided to put their efforts on this subject in this paper. Here, we studied the nonlinear thermal buckling of micro-/nanobeams considering the boundary conditions as clamped-clamped (CC) and simply supportedsimply supported (SS). The microbeam is modeled using the modified couple stress theory and the governing equations of the nanobeam are derived according to the Eringen’s nonlocal elasticity theory. The parametric results are presented using the generalized differential quadrature method (GDQM).
Temperature of the upper surface,T U
Thermal load,
h
Pure Ceramic
Pure Metal
Length, x-direction Clamped
Temperature of the lower surface,TL
Fig. 1. Schematic of the AFG beam and applied thermal load.
3
Clamped
2. Problem and formulation Herein, axially functionally graded (AFG) nano- and micro-beams are investigated based on the Euler-Bernoulli beam theory. The nano- and microbeams have length of ‘L’, width ‘b’ and height ‘h’ along x, y and z directions, respectively, along with thermal load ( N T ) as shown in Fig. 1.
2.1.Axially functionally graded material The mechanical properties of the AFG nano and micro-beams are considered varying based on the changing material volume fraction through the axis (x). The power law function is the basis of this variation which is defined as below: n
x F (x ) = F1 + (F2 − F1 ) L
(1)
here n denotes the power-law exponent which is in the range of 0 to ∞, x is the position of the point from the left end and the subscripts ()1 and ()2 respectively denote the left (x = 0) and right (x = L) ends of the nanobeam. Accordingly, the density (ρ), Young's modulus (E), thermal distribution () and Poisson's ratio (ν) of the AFG nano- and micro-beams are obtained by Eq. (2) as:
x ρ ( x ) = ρ1 + ( ρ 2 − ρ1 )
n
(2).a
L
x E (x ) = E1 + ( E 2 − E1 ) L x α ( x ) = α1 + (α2 − α1 ) L x ν (x ) = ν1 + (ν 2 − ν1 )
n
(2).b
n
(2).c
n
(2).d
L
The mechanical properties are temperature-dependent. Therefore, the nonlinear thermoelasticity equation is used to evaluate the material properties at the desired temperature T as: P = P0 ( P−1T
−1
+ PT + P2T 2 + P3T 3 + 1) 1
(3)
here, , , , are the temperature-dependent coefficients of material properties which are shown in Table. 1. 4
Table. 1 Temperature dependent coefficients of Young’s modulus, thermal expansion coefficient and mass density [30]. Material SUS304
Al2O3
Properties E (Pa) α (K-1) ρ (Kg/m3) Ν
P0 2.0104e+11 1.23e-05 8166 0.3262
P-1 0 0 0 0
P1 0.000308 0.000809 0 -0.0002
P2 -6.53e-07 0 0 3.80e-07
P3 0 0 0 0
E (Pa) α (K-1) ρ (Kg/m3) Ν
3.4955e+11 6.8269e-06 3750 0.26
0 0 0 0
-0.0003853 0.0001838 0 0
4.027e-07 0 0 0
-1.673e-11 0 0 0
2.2.Mathematical modeling The governing equation and boundary conditions can be evaluated using Hamilton’s principle as: τ
∫ δ (T
− U +W ext ) dt = 0
(4)
0
Here, using Hamilton principle we derive the governing equation and boundary conditions. In Eq. (4), T, U and Wext, respectively denote the kinetic energy, strain energy and the external work which is due to the external force. The displacement field of an Euler-Bernoulli beam is defined as:
u x ( x , z , t ) = u ( x ,t ) − z
∂w ( x , t ) ∂x
(5)
uy = 0 uz = w ( x ,t )
where ux, uy and u z are the displacements in directions of x, y and z, respectively. Also, ∂w ∂x and w(x) respectively are the angle of rotation and the transverse deflection of the AFG beams. Considering the kinetic energy of the AFG micro and nanobeams to be zero we have:
T =0
(6)
The micro and nanobeams are experiencing large transverse displacements, moderate rotations and small strains. Hence, the nonlinear von-Kármán’s strain-displacement relation is stated as: 2
εxx
∂u 1 ∂w ∂ 2w = + − z ∂x 2 ∂x ∂x 2
(7)
The strains and stresses cause the strain energy in the AFG beam which is expressed as:
5
δU =
L
L
∫ ∫ σ ij δ εij dA dx =
∫ ∫σ
0 A
0 A
Ax L 1 = ∫ 2 0 Ax
xx
δ ε xx dA dx
dx 2 2 L L 2 2 ∂u 1 ∂w ∂u ∂ w 1 ∂w ∂w + dx − B x dx 2 ∫ ∫ ∂x 2 L 0 ∂x 2 L 0 ∂x ∂x 2 ∂x ∂x
∂u 2 1 + 2 L ∂ x
2 2 ∂w ∫0 ∂x dx + C x L
∂ 2w 2 ∂x
2
(8)
where h
2
b
2
( Ax , B x ,C x ) = ∫ ∫ − h −b 2
E ( x ,T
) (1, z , z 2 ) dydz
(9)
2
When an AFG nano- or micro-beam is faced with in a high temperature environment for a considerable time, the temperature distribution along the thickness becomes uniform which is why we also considered the uniform temperature rise. The initial temperature of the nanoand microbeams (T0=300 K) in which the stress is zero is changed to the final temperature by ∆T. Here, the external work is attributed to the stress which is caused by the temperature change. Accordingly, the first variation of the work corresponding to the temperature change can be is obtained as L
δW ex t = ∫ N 0
T
∂w ∂ w dx δ ∂x ∂x
(10)
is the thermal resultant and is defined as: where N
T
=
∫ E ( x ,T ) α ( x ,T ) (T
− T 0 ) dA
(11)
A
With the temperature of the upper and lower surfaces of the nanobeam which are respectively shown as TU and TL (Fig. 1), the nonlinear temperature gradient along the thickness can be stated using the power function through z direction as [31]: αT
1 z T = T 0 + ∆Τ + 2 h
(12)
where αT denotes the non-negative power index of temperature variation function and ΔΤ=TU-TL. Considering αT=1, the temperature variation along the thickness becomes linear.
Accordingly, Fig. 2 represents the temperature distribution types along thickness for different values αT.
6
Upper surface, h/2
Thic knes s
T
α = 0.0 T
α = 0.25 T
α = 0.5 T
α = 1.0 T
α = 2.0 T
α = 4.0
Lower surface, -h/2 T0
T
α = 8.0
∆T
T0+∆T
Fig. 2. Schematic of the temperature gradient along the thickness.
The first variation of total energy is obtained by substitution of Eq. (6), (8) and (10) into Eq. (4) and using the fundamental lemma of calculus. Then the governing equations and boundary conditions of the AFG Euler-Bernoulli beams are obtained as: ∂2M ∂ + (N + N 2 ∂x ∂x
T
) ∂∂wx = 0
(13)
where ∂w ( x , t ) 1 N = ∫ σ xx dA = A x 2 A ∂x
2
(14)
Using the following boundary conditions: ∂M =0 ∂x
or
w =0
at
x = 0 and x = L
(15)
M =0
or
∂w =0 ∂x
at
x = 0 and x = L
(16)
where M is the bending moment, the stress resultants are defined as: M = ∫ σ xx z dA
(17)
A
7
2.2.1. Eringen nonlocal theory
By considering the state of a point as a function of the strain of that point and neglecting the relation of the state of the whole body with the state of each point, classical continuum theories, do not yield exact results for the nano-size structures. On the other hand, non-local continuum mechanic considers the stress at a point related to the neighborhood of that point and considers the long distance interactions in the body. Besides, the forces between the atoms and internal length scale are also taken into consideration in the non-local theories. First, the nonlocal model was proposed by Eringen [32-34]. Eringen considered the nonlocal parameter to put the effects of the small size according to the atomic theory of lattice dynamics and also the experimental observations on phonon dispersion. In the nonlocal theory, Eringen [33] defined the nonlocal stress tensor at point as:
σ ij ( x ) = ∫ α ( x ' − x , τ ) t ij ( x ' ) d Γ ( x ' )
(18)
Γ
where the kernel function | − |, indicates the nonlocal modulus in which | − | is the distance (in the Euclidean norm) and τ is a material constant that is dependent on the internal and external characteristic lengths (such as the lattice spacing and wavelength, respectively). Also,
!" ′
is the classical macroscopic second Piola-Kirchhoff stress at point
x which is connected to the linear strain tensor ($%& ) components as: t ij = C ijkl ε kl
(19)
The generalized Hooke’s law is used to relate the macroscopic stress at point to the strain ε at the point x of a Hookean solid and τ is a constant which is obtained by:
τ=
e 0a l
(20)
here and ( represent the internal and external characteristic lengths, respectively and ) is a material constant. The integral constitutive relations given by Eq. (18) can be written in form of an equivalent differential for a class of physically admissible kernel as:
(1 − (e a ) ∇ ) σ 2
2
0
kl
= t kl
(21)
here, ∇2 defines the Laplacian operator and e0a is the length scale which puts the size effect into effect. The one-dimension nonlocal constitutive relation of an elastic material can be expressed as:
8
σ xx − (e 0a )
2
∂ 2σ xx = E ( x ,T ) εxx ∂x 2
(22)
where σ and * are the nonlocal stress and strain, respectively. For defining the force-strain and moment-strain relations of the Euler-Bernoulli AFG nanobeam according to the nonlocal theory, we integrate Eq. (22) over the beam’s cross-section area. Then, we have: M − (e 0a )
2
∂ 2w ( x , t ) ∂2M = − C x ∂x 2 ∂x 2
(23)
Also substitution of the second derivative of M from Eq. (23) into (13) yields the explicit relation of the nonlocal bending moment as below: M = −C x
∂ 2w ∂w 2 ∂ − (e 0a ) N +NT ) ( 2 ∂x ∂x ∂x
(24)
Substituting of M from Eq. (24) into (13) yields the nonlocal governing equations of EulerBernoulli AFG nanobeam in terms of the displacement as below: d 2C x ∂ 2w dC x ∂ 3w ∂ 4w + 2 + C x dx 2 ∂ x 2 dx ∂ x 3 ∂x 4 2 2 1 ∂ 2 1 ∂ ∂ w ∂w − + e a A x ( ) 0 2 ∂ x 2 ∂x 2 ∂x ∂ x
= ∆ΤCr
dN T ∂ w T dx ∂ x + N 3 T 2 d N − e a ( ) 0 3 dx
∂ ∂ w 2 ∂w A x ∂ x ∂ x ∂ x
∂ 2w ∂x 2 ∂w d 2 N T ∂ 2w dN T ∂ 3w +3 + 3 +N ∂x dx 2 ∂ x 2 dx ∂x 3
T
∂ 4w ∂x 4
(25)
where ΔΤ+, and e 0a respectively are the nonlinear critical buckling temperature change and the nonlocal parameter which is related to the nano-size effect. Also, can be defined as: αT
N
1 z = ∫ E ( x ,T ) α ( x ,T ) + dA 2 h A
T
(26)
Using the following boundary conditions: ∂ ∂x
∂ 2w ∂w 2 ∂ T − C x ∂x 2 − (e 0a ) ∂x ( N + N ) ∂x = 0
−C x
∂ 2w ∂w 2 ∂ N +NT ) = 0 or − (e 0a ) ( 2 ∂x ∂x ∂x
or w = 0 at
∂w = 0 at ∂x
x = 0 and x = L
x = 0 and x = L
(27).a
(27).b
2.2.2. The modified couple stress theory of AFG nanobeam
Similar to the Eringen’s nonlocal elasticity theory, the modified couple stress theory 9
considers the small scale parameter. The modified couple stress theory which is a useful and sound theory for analysis of the micro-scale structures has been used a lot in studies on micro-elements [35-38]. The strain energy (U) is defined using the modified couple stress theory as below:
U =
1 ∫∫∫ (σ : ε + m : χ ) d υ 2 Λ
(28)
where Λ defines the integration region and the definition of the other parameters are as below: Cauchy stress σ:
σ = λ tr (ε ) I + 2µε
(29)
where trϵ denotes the trace tensor of *00 strain and the classical strain is written as: ε=
1 T ∇u + ( ∇u ) 2
(
)
(30)
here, u is the displacement field and the symmetric curvature m is:
m = 2l 2 µ χ
(31)
here, l denotes the length scale parameter of the material which is attributed to the couple stress at a certain point. Deviation part of couple stress 1 is: χ =
1 T ∇θ + ( ∇ θ ) 2
(
)
(32)
where θ denotes and rotation vector which is expressed as below: 1 θ = curl u 2
(33)
λ and μ which are in Eq. (29) and (31) are the Láme’s constants and are defined as below: µ (x ) = λ (x ) =
E ( x ,T
) 2 (1 + ν ( x ) ) E ( x ,T )ν ( x ,T
(34)
)
(35)
(1 + ν ( x ,T ) ) (1 − 2ν ( x ,T ) )
Considering the stresses and strains of the microbeam, the strain energy of the microbeam is written as:
10
2 2 2 2 2 L A ∂u ( x , t ) + 1 ∂w ( x , t ) dx + C ∂ w ( x , t ) ∫ ∂x x ∂x 2 x ∂x 2 L 0 2 2 2 L L 1 ∂u ( x , t ) 1 ∂w ( x , t ) 2 1 ∂ w (x ,t ) U = ∫ Ax dx dx + 2 D x l 2 ∫ 2 0 ∂x 2L 0 ∂x ∂x 2 2 2 2 L ∂u ( x , t ) ∂ w ( x , t ) ∂ w ( x ,t ) 1 ∂w ( x , t ) + dx 2 − B x ∂x ∂x 2 2 L ∫0 ∂x ∂ x
(36)
Ax, Bx, Cx are previously defined in Eq. (9), and h
Dx =
2
b
2
∫ ∫ µ ( x ,T ) dy dz −h
2
−b
(37)
2
Substituting Eqs. (6), (10) and (37) into Eq. (4) and using the fundamental lemma of calculus yield the first variation of total energy. The governing equations and boundary conditions of the AFG microbeams are then defined as: d 2 (C x + l 2 D x ) ∂ 2w d (C x + l 2 D x ) ∂ 3w ∂ 4w 2 + 2 + C + l D ( ) x x dx 2 ∂x 2 dx ∂x 3 ∂x 4 2 2 dN T ∂w 1 ∂ ∂w ∂w T ∂ w − = ∆Τ + N A x Cr 2 ∂x ∂x 2 ∂x ∂x dx ∂x
(38)
Using the following boundary conditions: ∂ ∂ 2w 2 C + l D =0 x x ∂x ∂x 2
(
(C
)
or
w =0
at
x = 0 and x = L
(39).a
or
∂w =0 ∂x
at
x = 0 and x = L
(39).b
2
x
+ l 2 Dx
) ∂∂xw
2
=0
3. Soloution methodology
The generalized differential quadrature method (GDQM) is employed for obtaining the results through solving the nonlinear equations. The r-th order derivative of function f ( xi ) in GDQM is: k ∂r f ( x ) C ij( r )f ( x i ) = ∑ ∂x r x = x j =1
(40)
P
k is the grid points number along x direction and C ij is:
11
C ij(1) =
M (x i )
(x
− x j ) M (x j )
i
n
∑
C ij( ) = − 1
;
i , j = 1, 2,..., n
and
i≠ j (41)
C ij( ) ; 1
i = j
j =1,i ≠ j
where M(x) is: k
∏ (x
M (x i ) =
i
−x j )
(42)
j =1, j ≠i
The weighting coefficient C(r), along x direction is derived as: (r )
C ij
( r −1) (1) C ij( r −1) = r C ij C ij − ; ( x i − x j ) n
∑
(r )
C ii = −
i , j = 1, 2,..., k , i ≠ j
and
2 ≤ r ≤ k −1 (43)
(r )
C ij ;
i , j = 1, 2,..., k
and
1≤ r ≤ k −1
j =1,i ≠ j
Chebyshev-Gauss-Lobatto approach is employed to obtain the distribution of the mesh points as:
xi =
( i − 1) L π 1 − cos 2 N − 1 ( )
i = 1,2,3, ... , k
(44)
Each of the nonlinear motion equations and boundary conditions of the AFG nanobeam (Eq. (25) and Eqs. (27)) and AFG microbeam (Eq. (38) Eqs. (39)) are considered to be the combination of three matrixes. Then the linear and nonlinear stiffness can be obtained as
{[ K ]
Linear
+ [ K ]non −Linear − ∆ΤCr [ M
]}{λ} = 0
(45)
The linear motion equation is first solved by GDQM and then, by employing the weight coefficients (Eq. (43)) to the linear motion equation we have: for nanobeam: d 2Cx dx 2
n
∑C rs ( )W s + 2 2
s =1
dCx dx
n
n
∑C rs ( )W s + Cx ∑C rs ( )W s 3
s =1
4
s =1
dN (1) ( 2) T dx ∑C rs W s + N ∑C rs W s s =1 s =1 n d 3N T n (1) ( 4) T = ∆ΤCr C W s + N ∑C rs W 3 ∑ rs s =1 − (e a )2 dx s =1 0 T n d 2N T n dN C ( 2)W s + 3 C rs ( 3)W s s +3 ∑ 2 ∑ rs dx s =1 dx s =1 T
n
n
for microbeam:
12
(46)
d 2 (Cx + Dxl 2 ) dx 2 dN = ∆ΤCr dx
n
∑C rs W s + 2 ( 2)
d (Cx + Dxl 2 ) dx
s =1 T
n
∑C
n
W s + N T ∑C rs
(1) rs
s =1
n
n
∑C rs ( )W s + (Cx + Dxl 2 ) ∑C rs ( )W s 3
s =1
4
s =1
Ws
(47)
(2)
s =1
Using the boundary conditions Eqs. (27) and (39), respectively for nano and microbeams, and assembling the related matrixes to the boundary conditions and governing equations, the linear fundamental thermal buckling of nano and microbeams can be calculated as below: [ K dd [ K bd
[ M dd ] [ M db ] {λd } ] [ K db ] {λd } = ∆ΤCr Linear ] [ K bb ] {λb } [ M bd ] [ M bb ] {λb }
(48)
where b and d indexes denote the boundary and domain, respectively and 5 is the mode shape. To solve the nonlinear equations, we need the linear mode shapes and then the mode shape of W can be obtained using Eq. (48). Then, the nonlinear mode shapes can be calculated by substitution of the calculated linear mode shapes into the nonlinear stiffness matrix. Afterwards, using Eqs. (25) and (38), the nonlinear temperature buckling and mode shapes are calculated by coupling the linear and nonlinear stiffness matrixes with the mass matrix. Afterwards, the iteration method is used to obtain the convergence of nonlinear results using the recalculation process. for nanobean: d 2Cx dx 2
n
∑C rs ( )W s + 2 2
s =1
dCx dx
n
+ (e 0a )
n
2
n
∑C
( 2) rs
s =1
3
s =1
1 1 1 −∑C rs ( ) A x ∑C rs ( )W s s =1 s =1 2 n
n
∑C rs ( )W s + Cx ∑C rs ( )W s 4
s =1 2
n
∑C s =1
Ws
(1) rs
2 n n n (1) 1 (1) (1) C A C W C W ∑ rs x ∑ rs s ∑ rs s s =1 s =1 2 s =1
dN (1) ( 2) T dx ∑C rs W s + N ∑C rs W s s =1 s =1 n d 3N T n (1) T ( 4) = ∆ΤCr C W s + N ∑C rs W 3 ∑ rs s =1 − (e a )2 dx s =1 0 T n d 2N T n dN C ( 2)W s + 3 C rs ( 3)W s s +3 ∑ 2 ∑ rs dx s =1 dx s =1 T
n
n
for microbeam:
13
(49)
d 2 (Cx + Dxl 2 ) dx 2 n
−∑C rs
(1)
s =1
n
∑C rs W s + 2 ( 2)
d (Cx + Dxl 2 ) dx
s =1
1 n (1) Ax ∑C rs W s s =1 2
2
n
∑C s =1
n
3
s =1
dN W s = ∆ΤCr dx
(1) rs
n
∑C rs ( )W s + (Cx + Dxl 2 ) ∑C rs ( )W s 4
s =1
T
n
∑C s =1
(1) rs
Ws + N
n
T
∑C s =1
Ws
(50)
(2 ) rs
4. Numerical results
The explained solution procedure is applied to pure metal, pure ceramic and AFG EulerBernoulli nano- and microbeams with different AFG power indexes (n). The results are depicted for low and high temperatures, different L/h values and power indexes of temperature variation function (α), under clamped-clamped (CC) and simply supportedsimply supported (SS) boundary conditions. The non-dimensional parameters are defined as below to yield better interpretation of the results: µ=
e0 a L
(51).a
l0 =
l h
(51).b
A mp = a
h2 12
(51).c
here µ, l0 and Amp are the non-dimensional nonlocal parameter, non-dimensional micro scale parameter (small scale parameter), non-dimensional frequency and amplitude of nonlinearity, respectively. The normalized critical buckling temperature (∆ΤCr) is also defined as below: Normalized ∆Τ Cr =
Nonlinear critical buckling temperature of micro or nanobeam Linear critical buckling temperature of local pure ceramic beam
(52)
The explained solving process is carried out on nano- and micro-beams for clamped (CC) and simply-supported (SS) boundary conditions. The problem is solved to find the linear and nonlinear critical buckling temperature change (∆Tcr) for various values of nonlinearity, aspect ratio (L/h), nonlocal value ‘μ’ (for nanobeams), small scale parameter ‘l0’ (for microbeams), and power index of temperature variation function (αT). To illustrate the effects of different parameters on buckling temperature of the micro- and nanobeams, several figures and two tables are presented. It should be noted that by neglecting the small scale effects (µ and l0), the nonlinear buckling temperatures of micro- and nanobeams are equal and the same as that of the classic theory. To prove the correctness of the provided results, the answers are compared with the other 14
studies. The comparison of the uniform and linear temperature changes with the results of Ebrahimi and Salari [39] are shown in Table. 2 for different values of the nonlocal parameter. The buckling temperature of Euler-Bernoulli beam is higher than the referred answers which are for the Timoshenko beam. It is because the degree of freedom of Timoshenko beam is more than that of Euler-Bernoulli beams. Table. 3 shows the mechanical properties of the ceramic and metal which are used in this paper. Table. 2. Comparison of the uniform and linear temperature changes with the results of Ebrahimi and Salari for different values of nonlocal parameter. (e0 a)2 =0 (nm)
(e0 a)=1 (nm)
(e0a)=2 (nm)
(e0a)=3 (nm)
Present, Euler-Bernoulli
[39], Timoshenko
Present, Euler-Bernoulli
[39], Timoshenko
Present, Euler-Bernoulli
[39], Timoshenko
Present, Euler-Bernoulli
[39], Timoshenko
L/h=40
UTC1 LTC2
71.36165 142.7233
68.6671 127.334
64.95122 129.9024
62.4988 114.998
59.59756 119.1951
57.3473 104.695
55.05926 110.1185
52.9803 95.9606
L/h=50
UTC LTC
45.67146 91.34292
43.9712 77.9423
42.958 85.91599
40.0212 70.0424
40.54888 81.09776
36.7224 63.4449
38.39562 76.79125
33.9261 57.8521
L/h=50
UTC LTC
31.71629 63.43258
30.5447 51.0893
30.38352 60.76704
27.8008 45.6016
29.15824 58.31649
25.5093 41.0186
28.02796 56.05592
23.5668 37.1336
UTC1: Uniform temperature change. LTC2: Linear temperature change.
Table. 3. The mechanical properties (Young’s modulus, mass density and Poisson’s ratio) of ceramic (Al2O3) and metal (SUS304), [30]. Material SUS304
Al2O3
Properties E (Pa) α (K-1) ρ (Kg/m3) Ν
X0 2.0104e+11 1.23e-05 8166 0.3262
X-1 0 0 0 0
X1 0.000308 0.000809 0 -0.0002
X2 -6.53e-07 0 0 3.80e-07
X3 0 0 0 0
E (Pa) α (K-1) ρ (Kg/m3) ν
3.4955e+11 6.8269e-06 3750 0.26
0 0 0 0
-0.0003853 0.0001838 0 0
4.027e-07 0 0 0
-1.673e-11 0 0 0
The normalized critical buckling temperature (∆Tcr) of SS and CC nanobeams are depicted respectively in Fig. 3 and Fig. 4 versus the amplitude of nonlinearity (Amp) for different values of power indexes of temperature variation function (αT). It can be seen that the normalized critical buckling temperature increases with the nonlinear amplitude as the nonlinear effect increases the stiffness of the beams. Also, the increment of power index of temperature variation function (αT) increases the buckling temperature too. Besides, it can be observed that as αT increases, the effect of the nonlinear amplitude on the critical buckling temperature increases too. Moreover, Fig. 3 and Fig. 4 show that the ∆Tcr of the AFG nanobeam is lower than ceramic and higher than metal. It is because the ∆Tcr is proportional to the stiffness, and as the stiffness increases, the value of ∆Tcr increases too.
15
T
(b): α = 0.25
Pure Metal n=1 n=2 n=5 n=10 Pure Ceramic
2.5 Normalize ∆TCr
T
(a): α = 0, Uniform temperature change
2
1.5
2 Normalize ∆TCr
3
1.5
1
1
0.5 0 -2
-1.5
-1
-0.5
0 Amp
0.5
1
1.5
0.5 -2
2
-1
T
2
1
2
(d): α = 2
5.5
3.5
5 4.5
Normalize ∆TCr
3 Normalize ∆TCr
1
T
(c): α = 1, Linear temperature change
2.5
4
3.5
2
3
2.5
1.5
2 1.5
1 -2
0 Amp
-1.5
-1
-0.5
0 Amp
0.5
1
1.5
2
-2
-1
0 Amp
Fig. 3. Normalized critical buckling temperature (∆Tcr) of SS nanobeam versus the nonlinear amplitude for different values of αT and AFG power indexes when L/h=40.
Comparison between Fig. 3 and Fig. 4 shows that the normalized critical buckling temperature (∆Tcr) of clamped nanobeam is lower than that of the simply supported nanobeam. The reason is that the stiffness of the CC nanobeam is lower than SS.
16
T
1.5
(b): α = 0.25
1.6
Pure Metal n=1 n=2 n=5 n=10 Pure Ceramic
2 Normalize ∆TCr
T
(a): α = 0, Uniform temperature change
1.4 Normalize ∆TCr
2.5
1
1.2 1 0.8
0.5 0.6 0 -2
-1.5
-1
-0.5
0 Amp
0.5
1
1.5
2
-2
-1
T
2.5
0 Amp
1
2
1
2
T
(c): α = 1, Linear temperature change
(d): α = 2
2
Normalize ∆TCr
Normalize ∆TCr
3.5 3
2.5
1.5
1 -2
2
1.5 -1.5
-1
-0.5
0 Amp
0.5
1
1.5
2
-2
-1
0 Amp
Fig. 4. Normalized critical buckling temperature (∆Tcr) of CC nanobeam versus the nonlinear amplitude for different values of αT and AFG power indexes when L/h=40.
The normalized critical buckling temperature (∆Tcr) of SS and CC nanobeams are respectively shown in Fig. 5 and Fig. 6 versus the nonlinear amplitude for different values of AFG index and nonlocal parameter. It is observed that the normalized critical buckling temperatures (∆Tcr) of SS and CC nanobeams decrease by increasing the nonlocal value. It is because the nonlocal effect decreases the stiffness of the nanobeam and decreasing the stiffness of the nanobeam decreases the critical buckling temperature.
17
(a): Pure Ceramic
3
µ = 0.0 µ = 0.1 µ = 0.2 µ = 0.3
2
1.5
0.5 0
0.5
1
1.5 Amp
2
2.5
0
3
0
0.5
1
(c): n=1 µ = 0.0 µ = 0.1 µ = 0.2 µ = 0.3
1.5
2
2.5
3
2
2.5
3
µ = 0.0 µ = 0.1
2.5 Normalize ∆TCr
2
1.5 Amp (d): n=10
3
2.5 Normalize ∆TCr
1
0.5
3
2
1.5
1
0.5 0
2
1.5
1
0
µ = 0.0 µ = 0.1 µ = 0.2 µ = 0.3
2.5 Normalize ∆TCr
Normalize ∆TCr
2.5
(b): Pure Metal
3
1
0.5
0
0.5
1
1.5 Amp
2
2.5
3
0
0
0.5
1
1.5 Amp
Fig. 5. Normalized critical buckling temperature (∆Tcr) of SS nanobeam versus the nonlinear amplitude for different nonlocal values and AFG power indexes when αT=0.1 and L/h=40.
18
(a): Pure Ceramic
2.5
µ = 0.0 µ = 0.1 µ = 0.2 µ = 0.3
1.5
1.5
1
0.5 0
µ = 0.0 µ = 0.1 µ = 0.2 µ = 0.3
2 Normalize ∆TCr
Normalize ∆TCr
2
(b): Pure Metal
2.5
1
0.5
0
1
2
3
4
0
5
0
1
2
Amp (c): n=1
2.5 µ = 0.0 µ = 0.1 µ = 0.2 µ = 0.3
1
0.5 0
5
3
4
5
µ = 0.0 µ = 0.1
2 Normalize ∆TCr
Normalize ∆TCr
1.5
4
(d): n=10
2.5
2
3 Amp
1.5 1 0.5
0
1
2
3
4
5
0
0
1
2 Amp
Amp
Fig. 6. Normalized critical buckling temperature (∆Tcr) of CC nanobeam versus the nonlinear amplitude for different nonlocal values and AFG power indexes when αT=0.1 and L/h=40.
Fig. 7 and Fig. 8 show the (∆Tcr) of SS and CC microbeams, respectively, for different values of small scale parameter (l0) versus the nonlinear amplitude. It is seen that as the nonlocal parameter increases, the value of (∆Tcr) increases too. The reason is that the nonlocal parameter increases the stiffness of the microbeam, which leads to a higher value of normalized critical buckling temperature. Fig. 8 shows that the normalized critical buckling temperature (∆Tcr) of CC microbeam is lower than that of SS microbeam.
19
(a): Pure Ceramic 7
(b): Pure Metal 7
l = 0.0
l 0 = 0.0
6
l 0 = 0.2
5
l = 0.4
4
l = 0.8
Normalize ∆TCr
Normalize ∆TCr
0
0 0
3 2 1 0
1
Normalize ∆TCr
7
2
3
4
5
l 0 = 0.4
4
l 0 = 0.8
3 2
0
5
l 0 = 0.2
5
l 0 = 0.4
4
l 0 = 0.8
0
1
2
Amp
Amp
(c): n=1
(d): n=10 7
l 0 = 0.0
6
3
4
5
3
4
5
l = 0.0 0
6
l = 0.2
5
l 0 = 0.4
4
l = 0.8
0
3 2
0
3 2 1
1 0
l 0 = 0.2
1
Normalize ∆TCr
0
6
0
1
2
3
4
5
0
0
1
2 Amp
Amp
Fig. 7. Normalized critical buckling temperature (∆Tcr) of SS microbeam versus the nonlinear amplitude for different nonlocal small scale parameters and AFG power indexes when αT=0.1 when L/h=40.
20
(a): Pure Ceramic
4
(b): Pure Metal
4
3
Normalize ∆TCr
Normalize ∆TCr
l 0 = 0.0
2
1
0
0
1
2
3
4
l 0 = 0.4 l 0 = 0.8
2
1
0
5
l 0 = 0.2
3
0
1
2
(c): n=1
4
3
4
5
3
4
5
Amp
Amp
(d): n=10
4 l 0 = 0.2
3
Normalize ∆TCr
Normalize ∆TCr
l 0 = 0.0 l 0 = 0.4 l 0 = 0.8
2
1
0
0
1
2
3
4
5
3
2
1
0
0
1
2 Amp
Amp
Fig. 8. Normalized critical buckling temperature (∆Tcr) of CC microbeam versus the nonlinear amplitude for different nonlocal small scale parameters and AFG power indexes when αT=0.1 when
Table. 4 shows the value of nonlinear critical buckling temperature (in Kelvin) of SS nanobeam for different values of nonlinearity, nonlocal value, L/h and the AFG power index when the power index of temperature variation function is αT=0.2. It is seen that increasing the L/h decreases the ∆Tcr. This is because as the L/h increases, the stiffness of the nanobeam decreases which leads to a lower value of ∆Tcr. Similar to Fig. 3, Fig. 4, Fig. 5 and Fig. 6, Table. 4 shows that increasing the nonlinear amplitude increases the critical buckling temperature and increasing the nonlocal value decreases the ∆Tcr.
21
Table. 4. The nonlinear critical buckling temperature (in Kelvin) of SS nanobeam in differen values of Amp, n, μ, and L/h when αT=0.2. Amp=1 Amp=2 Amp=3 μ=0 μ=0.1 μ=0 μ=0.1 μ=0 μ=0.1 Pure Metal 191.0444 176.5612 279.6634 265.0778 425.619 410.897 n=1 266.7629 243.3819 390.4709 365.4423 594.218 566.5207 n=2 302.3162 280.854 438.2844 417.1361 662.2083 641.62 L/h=20 n=5 342.4768 325.6039 493.1526 483.4594 741.2869 743.4784 n=10 365.2835 205.8098 528.0418 389.1458 796.0806 692.2056 Pure Ceramic 406.3386 375.5339 594.8252 563.8026 905.263 873.9504
L/h=40
Pure Metal n=1 n=2 n=5 n=10 Pure Ceramic
47.76109 66.69071 75.57906 85.61919 91.32087 101.5846
44.1403 60.84549 70.21351 81.40097 51.45246 93.88346
69.91584 97.61772 109.5711 123.2881 132.0105 148.7063
66.26944 91.36057 104.284 120.8648 97.28645 140.9507
106.4047 148.5545 165.5521 185.3217 199.0202 226.3158
102.7243 141.6302 160.405 185.8696 173.0514 218.4876
L/h=80
Pure Metal n=1 n=2 n=5 n=10 Pure Ceramic
11.94027 16.67268 18.89476 21.4048 22.83022 25.39616
11.03508 15.21137 17.55338 20.35024 12.86312 23.47087
17.47896 24.40443 27.39278 30.82204 33.00261 37.17658
16.56736 22.84014 26.071 30.21621 24.32161 35.23766
26.60119 37.13863 41.38802 46.33043 49.75504 56.57894
25.68106 35.40754 40.10125 46.4674 43.26285 54.6219
The value of the nonlinear critical buckling temperature of SS microbeam is shown in Table. 5 for different values of L/h, small scale parameter, AFG power index, and nonlinear amplitude. Comparison between Table. 4 and Table. 5 shows that, when the small scale effect is neglected in the Eringen’s nonlocal elasticity and the modified couple stress, we obtain the results of the classic theory for nano- and micro-beams. Besides, this comparison shows that increasing the nonlocal effect in the Eringen’s theory decreases the buckling temperature, while increasing the small scale parameter in the modified couple stress theory increases the buckling temperature.
22
Table. 5. The nonlinear critical buckling temperature (in Kelvin) of SS microbeam in differen values of Amp, n, l0 , and L/h when αT=0.2. Amp=1 Amp=2 Amp=3 l0=0 l0=0.1 l0 =0 l0=0.1 l0=0 l0=0.1 Pure Metal 191.0444 238.3069 279.6634 327.2004 425.619 473.5643 n=1 266.7629 332.7608 390.4709 456.852 594.218 661.1692 n=2 302.3162 377.5296 438.2844 513.9163 662.2083 738.4769 L/h=20 n=5 342.4768 428.0145 493.1526 579.1518 741.2869 827.9997 n=10 365.2835 456.3142 528.0418 619.5725 796.0806 888.3772 Pure Ceramic 406.3386 506.8628 594.8252 695.9334 905.263 1007.239
L/h=40
Pure Metal n=1 n=2 n=5 n=10 Pure Ceramic
47.76109 66.69071 75.57906 85.61919 91.32087 101.5846
59.57672 83.19019 94.38239 107.0036 114.0786 126.7157
69.91584 97.61772 109.5711 123.2881 132.0105 148.7063
81.8001 114.213 128.4791 144.788 154.8931 173.9833
106.4047 148.5545 165.5521 185.3217 199.0202 226.3158
118.3911 165.2923 184.6192 206.9999 222.0943 251.8099
L/h=80
Pure Metal n=1 n=2 n=5 n=10 Pure Ceramic
11.94027 16.67268 18.89476 21.4048 22.83022 25.39616
14.89418 20.79755 23.5956 26.75091 28.51964 31.67892
17.47896 24.40443 27.39278 30.82204 33.00261 37.17658
20.45003 28.55325 32.11977 36.19699 38.72328 43.49583
26.60119 37.13863 41.38802 46.33043 49.75504 56.57894
29.59777 41.32307 46.15481 51.74998 55.52357 62.95247
5. Conclusion
The studied problem was the nonlinear thermal buckling of axially functionally grade (AFG) micro- and nanobeams. The modified couple stress theory is the basis of the governing equations of the microbeam and Eringen’s nonlocal elasticity theory is used to study the nanobeam. The material of the micro- and nanobeams are pure metal, pure ceramic and AFG materials and boundary conditions are clamped (CC) and simply supported (SS). The nonlinear equations are solved using the generalized differential quadrature method (GDQM) along with the iteration technique. The results are depicted to illustrate the effects of the small scale parameters, length to height ratio (L/h), nonlinear amplitude and AFG power index on the buckling temperature of the micro- and nano-beams. The main results of this paper are briefly represented below: -
Increment of αT increases, the dependency of the critical buckling temperature on the nonlinear amplitude.
-
Increasing the small scale effect decreases the buckling temperature of nanobeams, while it increases the buckling temperature of microbeams. 23
-
The buckling temperature of AFG micro- and nano-beams is lower than that of ceramic and higher than metal.
-
Increasing the nonlinear amplitude, AFG power index and temperature variation function (αT) increase the buckling temperature and increasing L/h decreases the buckling temperature.
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26
S0263-8223(16)32441-2 http://dx.doi.org/10.1016/j.compstruct.2017.02.048 COST 8269
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
6 November 2016 23 December 2016 10 February 2017
Please cite this article as: Shafiei, N., Mirjavadi, S.S., Afshari, B.M., Rabby, S., Hamouda, A.M.S., Nonlinear thermal buckling of axially functionally graded micro and nanobeams, Composite Structures (2017), doi: http:// dx.doi.org/10.1016/j.compstruct.2017.02.048
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Nonlinear thermal buckling of axially functionally graded micro and nanobeams Navvab Shafieia*, Seyed Sajad Mirjavadib, Behzad Mohasel Afsharic, Samira Rabbyb, A.M.S. Hamoudab a
Department of Mechanical Engineering, Payame Noor University (PNU), P. O. Box, 19395-3697, Tehran, Iran. b Department of Mechanical and Industrial Engineering, Qatar University, P.O.Box 2713, Doha, Qatar. c School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran.
* Corresponding author: Navvab Shafiei* Tel: + 98 84 33368848 Email: [email protected] 0000-0003-0992-9660.
Abstract In this study, the nonlinear thermal buckling of axially functionally graded (AFG) EulerBernoulli micro/nanobeams is analyzed. The Eringen’s nonlocal elasticity theory is used to develop the governing equations of nanobeam and the modified couple stress theory is used to study the microbeam. The micro- and nanobeams are made of pure metal, pure ceramic and axially functionally graded material which is the composition of metal and ceramic. Boundary conditions are considered as clamped (CC) and simply supported (SS). The generalized differential quadrature method (GDQM) is used along with the iteration technique to solve the nonlinear equations. The parametric studies are served to examine the effects of the small scale parameters, length to height ratio (L/h), nonlinear amplitude and AFG power index on the buckling temperature of the micro- and nanobeams.
Keyword: Nonlinear buckling; Thermal buckling; Axially Functionally Graded; Eringen nonlocal theory; modified couple stress.
1
1. Introduction The emerging nano and micro-sciences have made so many nano- and micro-structures with many different functions. Among them, there are a lot of nano- and micro-systems that work in thermal areas and under thermal stresses. Thus, it is needed to study these small-scale systems in thermal environments. Accordingly, many researchers have studied the thermal buckling of microbeams. Based on the modified couple stress theory, Ke et. al [1] studied the free vibration and buckling of microbeams based on Timoshenko beam theory. Chiao and Lin [2] studied the self-buckling behavior of microbeams under resistive heating. Nateghi and Salamat-talab [3] studied the thermal effect on buckling and free vibration behavior of functionally graded (FG) microbeams on the basis of the modified couple stress theory. Ansari et. al [4] studied the thermal postbuckling of FG microbeams under thermal loads. Wang et. al [5] performed analysis on the elastic buckling of micro- and nano-rods/tubes based on Eringen's nonlocal elasticity theory and the Timoshenko beam theory. Mohammadabadi et. al [6] studied the thermal effect on size-dependent buckling behavior of micro composite laminated beams. Mohammadi and Mahzoon [7] studied the thermal effects on postbuckling of microbeams. Fang and Wickert [8] studied the static deformation of microbeams considering prescribed in-plane compressive stress. Şimşek and Reddy [9] introduced a unified higher order beam theory for buckling of a functionally graded (FG) microbeam embedded in elastic Pasternak medium. Malekzadeh et. al [10] studied the small scale effect on the thermal buckling of nanoplates embedded in an elastic medium. Ke et. al [11] studied the bending, buckling and free vibration of annular FG microplates. Akgöz and Civalek [12] investigated the buckling of size-dependent FG microbeams. Besides, many researchers have carried out investigations on the buckling behavior of nanostructures such as nanobeams, nanoplates, etc. Ansari et. al [13] studied the thermal buckling of single-walled carbon nanotubes. Tounsi et. al [14] presented a higher-order nonlocal beam theory for the thermal buckling of nanobeams. Berrabah et. al [15] presented a unified nonlocal shear deformation theory to study bending, buckling and free vibration of nanobeams. Malekzadeh et. al [10] studied the small scale effect on the thermal buckling of nanoplates embedded in an elastic medium. Emam [16] investigated the buckling and postbuckling of the nonlocal nanobeams. Lim et. al [17] studied the buckling of nanostructures including nanobeam, nanorod, and nanotube in a temperature field. The surface effects waere studied by Yan and Jiang [18] on the vibrational and buckling of 2
piezoelectric Euler-Bernoulli nanobeams. Thai [19] presented a nonlocal shear deformation beam theory for bending, buckling, and vibration of nanobeams using the nonlocal differential constitutive relations of Eringen. Zenkour and Sobhy [20] studied the thermal buckling of single-layered graphene sheets lying on an elastic medium. Şimşek and Yurtcu [21] studied the static bending and buckling of a FG nanobeam. Nami et. al [22] presented analysis on the thermal buckling analysis of functionally graded rectangular nanoplates. Chaht et. al [23] studied the bending and buckling behaviors of size-dependent FG nanobeams including the thickness stretching effect. Ansari et. al [24] studied the buckling and vibration responses of FG nanoplates subjected to thermal loading considering surface stress effects. Zhang et. al [25] investigated the hybrid nonlocal Euler-Bernoulli beam model for the bending, buckling, and vibration of micro/nanobeams. Rafiee et. al [26] studied the nonlinear thermal bifurcation buckling behavior of carbon nanotube reinforced composite beams with surface-bonded piezoelectric layers. Wang et. al [27] analyzed the thermal buckling properties of a nanoplate with small-scale effects. Barati et. al [28] studied the thermal buckling behavior of size-dependent FG nanoplates. Shen [29] performed analysis on the thermal postbuckling of nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes considering a uniform temperature rise. The lack of a comprehensive study on the thermal behavior of axially functionally graded (AFG) micro- and nano-beams is noticeable. Thus, the authors decided to put their efforts on this subject in this paper. Here, we studied the nonlinear thermal buckling of micro-/nanobeams considering the boundary conditions as clamped-clamped (CC) and simply supportedsimply supported (SS). The microbeam is modeled using the modified couple stress theory and the governing equations of the nanobeam are derived according to the Eringen’s nonlocal elasticity theory. The parametric results are presented using the generalized differential quadrature method (GDQM).
Temperature of the upper surface,T U
Thermal load,
h
Pure Ceramic
Pure Metal
Length, x-direction Clamped
Temperature of the lower surface,TL
Fig. 1. Schematic of the AFG beam and applied thermal load.
3
Clamped
2. Problem and formulation Herein, axially functionally graded (AFG) nano- and micro-beams are investigated based on the Euler-Bernoulli beam theory. The nano- and microbeams have length of ‘L’, width ‘b’ and height ‘h’ along x, y and z directions, respectively, along with thermal load ( N T ) as shown in Fig. 1.
2.1.Axially functionally graded material The mechanical properties of the AFG nano and micro-beams are considered varying based on the changing material volume fraction through the axis (x). The power law function is the basis of this variation which is defined as below: n
x F (x ) = F1 + (F2 − F1 ) L
(1)
here n denotes the power-law exponent which is in the range of 0 to ∞, x is the position of the point from the left end and the subscripts ()1 and ()2 respectively denote the left (x = 0) and right (x = L) ends of the nanobeam. Accordingly, the density (ρ), Young's modulus (E), thermal distribution () and Poisson's ratio (ν) of the AFG nano- and micro-beams are obtained by Eq. (2) as:
x ρ ( x ) = ρ1 + ( ρ 2 − ρ1 )
n
(2).a
L
x E (x ) = E1 + ( E 2 − E1 ) L x α ( x ) = α1 + (α2 − α1 ) L x ν (x ) = ν1 + (ν 2 − ν1 )
n
(2).b
n
(2).c
n
(2).d
L
The mechanical properties are temperature-dependent. Therefore, the nonlinear thermoelasticity equation is used to evaluate the material properties at the desired temperature T as: P = P0 ( P−1T
−1
+ PT + P2T 2 + P3T 3 + 1) 1
(3)
here, , , , are the temperature-dependent coefficients of material properties which are shown in Table. 1. 4
Table. 1 Temperature dependent coefficients of Young’s modulus, thermal expansion coefficient and mass density [30]. Material SUS304
Al2O3
Properties E (Pa) α (K-1) ρ (Kg/m3) Ν
P0 2.0104e+11 1.23e-05 8166 0.3262
P-1 0 0 0 0
P1 0.000308 0.000809 0 -0.0002
P2 -6.53e-07 0 0 3.80e-07
P3 0 0 0 0
E (Pa) α (K-1) ρ (Kg/m3) Ν
3.4955e+11 6.8269e-06 3750 0.26
0 0 0 0
-0.0003853 0.0001838 0 0
4.027e-07 0 0 0
-1.673e-11 0 0 0
2.2.Mathematical modeling The governing equation and boundary conditions can be evaluated using Hamilton’s principle as: τ
∫ δ (T
− U +W ext ) dt = 0
(4)
0
Here, using Hamilton principle we derive the governing equation and boundary conditions. In Eq. (4), T, U and Wext, respectively denote the kinetic energy, strain energy and the external work which is due to the external force. The displacement field of an Euler-Bernoulli beam is defined as:
u x ( x , z , t ) = u ( x ,t ) − z
∂w ( x , t ) ∂x
(5)
uy = 0 uz = w ( x ,t )
where ux, uy and u z are the displacements in directions of x, y and z, respectively. Also, ∂w ∂x and w(x) respectively are the angle of rotation and the transverse deflection of the AFG beams. Considering the kinetic energy of the AFG micro and nanobeams to be zero we have:
T =0
(6)
The micro and nanobeams are experiencing large transverse displacements, moderate rotations and small strains. Hence, the nonlinear von-Kármán’s strain-displacement relation is stated as: 2
εxx
∂u 1 ∂w ∂ 2w = + − z ∂x 2 ∂x ∂x 2
(7)
The strains and stresses cause the strain energy in the AFG beam which is expressed as:
5
δU =
L
L
∫ ∫ σ ij δ εij dA dx =
∫ ∫σ
0 A
0 A
Ax L 1 = ∫ 2 0 Ax
xx
δ ε xx dA dx
dx 2 2 L L 2 2 ∂u 1 ∂w ∂u ∂ w 1 ∂w ∂w + dx − B x dx 2 ∫ ∫ ∂x 2 L 0 ∂x 2 L 0 ∂x ∂x 2 ∂x ∂x
∂u 2 1 + 2 L ∂ x
2 2 ∂w ∫0 ∂x dx + C x L
∂ 2w 2 ∂x
2
(8)
where h
2
b
2
( Ax , B x ,C x ) = ∫ ∫ − h −b 2
E ( x ,T
) (1, z , z 2 ) dydz
(9)
2
When an AFG nano- or micro-beam is faced with in a high temperature environment for a considerable time, the temperature distribution along the thickness becomes uniform which is why we also considered the uniform temperature rise. The initial temperature of the nanoand microbeams (T0=300 K) in which the stress is zero is changed to the final temperature by ∆T. Here, the external work is attributed to the stress which is caused by the temperature change. Accordingly, the first variation of the work corresponding to the temperature change can be is obtained as L
δW ex t = ∫ N 0
T
∂w ∂ w dx δ ∂x ∂x
(10)
is the thermal resultant and is defined as: where N
T
=
∫ E ( x ,T ) α ( x ,T ) (T
− T 0 ) dA
(11)
A
With the temperature of the upper and lower surfaces of the nanobeam which are respectively shown as TU and TL (Fig. 1), the nonlinear temperature gradient along the thickness can be stated using the power function through z direction as [31]: αT
1 z T = T 0 + ∆Τ + 2 h
(12)
where αT denotes the non-negative power index of temperature variation function and ΔΤ=TU-TL. Considering αT=1, the temperature variation along the thickness becomes linear.
Accordingly, Fig. 2 represents the temperature distribution types along thickness for different values αT.
6
Upper surface, h/2
Thic knes s
T
α = 0.0 T
α = 0.25 T
α = 0.5 T
α = 1.0 T
α = 2.0 T
α = 4.0
Lower surface, -h/2 T0
T
α = 8.0
∆T
T0+∆T
Fig. 2. Schematic of the temperature gradient along the thickness.
The first variation of total energy is obtained by substitution of Eq. (6), (8) and (10) into Eq. (4) and using the fundamental lemma of calculus. Then the governing equations and boundary conditions of the AFG Euler-Bernoulli beams are obtained as: ∂2M ∂ + (N + N 2 ∂x ∂x
T
) ∂∂wx = 0
(13)
where ∂w ( x , t ) 1 N = ∫ σ xx dA = A x 2 A ∂x
2
(14)
Using the following boundary conditions: ∂M =0 ∂x
or
w =0
at
x = 0 and x = L
(15)
M =0
or
∂w =0 ∂x
at
x = 0 and x = L
(16)
where M is the bending moment, the stress resultants are defined as: M = ∫ σ xx z dA
(17)
A
7
2.2.1. Eringen nonlocal theory
By considering the state of a point as a function of the strain of that point and neglecting the relation of the state of the whole body with the state of each point, classical continuum theories, do not yield exact results for the nano-size structures. On the other hand, non-local continuum mechanic considers the stress at a point related to the neighborhood of that point and considers the long distance interactions in the body. Besides, the forces between the atoms and internal length scale are also taken into consideration in the non-local theories. First, the nonlocal model was proposed by Eringen [32-34]. Eringen considered the nonlocal parameter to put the effects of the small size according to the atomic theory of lattice dynamics and also the experimental observations on phonon dispersion. In the nonlocal theory, Eringen [33] defined the nonlocal stress tensor at point as:
σ ij ( x ) = ∫ α ( x ' − x , τ ) t ij ( x ' ) d Γ ( x ' )
(18)
Γ
where the kernel function | − |, indicates the nonlocal modulus in which | − | is the distance (in the Euclidean norm) and τ is a material constant that is dependent on the internal and external characteristic lengths (such as the lattice spacing and wavelength, respectively). Also,
!" ′
is the classical macroscopic second Piola-Kirchhoff stress at point
x which is connected to the linear strain tensor ($%& ) components as: t ij = C ijkl ε kl
(19)
The generalized Hooke’s law is used to relate the macroscopic stress at point to the strain ε at the point x of a Hookean solid and τ is a constant which is obtained by:
τ=
e 0a l
(20)
here and ( represent the internal and external characteristic lengths, respectively and ) is a material constant. The integral constitutive relations given by Eq. (18) can be written in form of an equivalent differential for a class of physically admissible kernel as:
(1 − (e a ) ∇ ) σ 2
2
0
kl
= t kl
(21)
here, ∇2 defines the Laplacian operator and e0a is the length scale which puts the size effect into effect. The one-dimension nonlocal constitutive relation of an elastic material can be expressed as:
8
σ xx − (e 0a )
2
∂ 2σ xx = E ( x ,T ) εxx ∂x 2
(22)
where σ and * are the nonlocal stress and strain, respectively. For defining the force-strain and moment-strain relations of the Euler-Bernoulli AFG nanobeam according to the nonlocal theory, we integrate Eq. (22) over the beam’s cross-section area. Then, we have: M − (e 0a )
2
∂ 2w ( x , t ) ∂2M = − C x ∂x 2 ∂x 2
(23)
Also substitution of the second derivative of M from Eq. (23) into (13) yields the explicit relation of the nonlocal bending moment as below: M = −C x
∂ 2w ∂w 2 ∂ − (e 0a ) N +NT ) ( 2 ∂x ∂x ∂x
(24)
Substituting of M from Eq. (24) into (13) yields the nonlocal governing equations of EulerBernoulli AFG nanobeam in terms of the displacement as below: d 2C x ∂ 2w dC x ∂ 3w ∂ 4w + 2 + C x dx 2 ∂ x 2 dx ∂ x 3 ∂x 4 2 2 1 ∂ 2 1 ∂ ∂ w ∂w − + e a A x ( ) 0 2 ∂ x 2 ∂x 2 ∂x ∂ x
= ∆ΤCr
dN T ∂ w T dx ∂ x + N 3 T 2 d N − e a ( ) 0 3 dx
∂ ∂ w 2 ∂w A x ∂ x ∂ x ∂ x
∂ 2w ∂x 2 ∂w d 2 N T ∂ 2w dN T ∂ 3w +3 + 3 +N ∂x dx 2 ∂ x 2 dx ∂x 3
T
∂ 4w ∂x 4
(25)
where ΔΤ+, and e 0a respectively are the nonlinear critical buckling temperature change and the nonlocal parameter which is related to the nano-size effect. Also, can be defined as: αT
N
1 z = ∫ E ( x ,T ) α ( x ,T ) + dA 2 h A
T
(26)
Using the following boundary conditions: ∂ ∂x
∂ 2w ∂w 2 ∂ T − C x ∂x 2 − (e 0a ) ∂x ( N + N ) ∂x = 0
−C x
∂ 2w ∂w 2 ∂ N +NT ) = 0 or − (e 0a ) ( 2 ∂x ∂x ∂x
or w = 0 at
∂w = 0 at ∂x
x = 0 and x = L
x = 0 and x = L
(27).a
(27).b
2.2.2. The modified couple stress theory of AFG nanobeam
Similar to the Eringen’s nonlocal elasticity theory, the modified couple stress theory 9
considers the small scale parameter. The modified couple stress theory which is a useful and sound theory for analysis of the micro-scale structures has been used a lot in studies on micro-elements [35-38]. The strain energy (U) is defined using the modified couple stress theory as below:
U =
1 ∫∫∫ (σ : ε + m : χ ) d υ 2 Λ
(28)
where Λ defines the integration region and the definition of the other parameters are as below: Cauchy stress σ:
σ = λ tr (ε ) I + 2µε
(29)
where trϵ denotes the trace tensor of *00 strain and the classical strain is written as: ε=
1 T ∇u + ( ∇u ) 2
(
)
(30)
here, u is the displacement field and the symmetric curvature m is:
m = 2l 2 µ χ
(31)
here, l denotes the length scale parameter of the material which is attributed to the couple stress at a certain point. Deviation part of couple stress 1 is: χ =
1 T ∇θ + ( ∇ θ ) 2
(
)
(32)
where θ denotes and rotation vector which is expressed as below: 1 θ = curl u 2
(33)
λ and μ which are in Eq. (29) and (31) are the Láme’s constants and are defined as below: µ (x ) = λ (x ) =
E ( x ,T
) 2 (1 + ν ( x ) ) E ( x ,T )ν ( x ,T
(34)
)
(35)
(1 + ν ( x ,T ) ) (1 − 2ν ( x ,T ) )
Considering the stresses and strains of the microbeam, the strain energy of the microbeam is written as:
10
2 2 2 2 2 L A ∂u ( x , t ) + 1 ∂w ( x , t ) dx + C ∂ w ( x , t ) ∫ ∂x x ∂x 2 x ∂x 2 L 0 2 2 2 L L 1 ∂u ( x , t ) 1 ∂w ( x , t ) 2 1 ∂ w (x ,t ) U = ∫ Ax dx dx + 2 D x l 2 ∫ 2 0 ∂x 2L 0 ∂x ∂x 2 2 2 2 L ∂u ( x , t ) ∂ w ( x , t ) ∂ w ( x ,t ) 1 ∂w ( x , t ) + dx 2 − B x ∂x ∂x 2 2 L ∫0 ∂x ∂ x
(36)
Ax, Bx, Cx are previously defined in Eq. (9), and h
Dx =
2
b
2
∫ ∫ µ ( x ,T ) dy dz −h
2
−b
(37)
2
Substituting Eqs. (6), (10) and (37) into Eq. (4) and using the fundamental lemma of calculus yield the first variation of total energy. The governing equations and boundary conditions of the AFG microbeams are then defined as: d 2 (C x + l 2 D x ) ∂ 2w d (C x + l 2 D x ) ∂ 3w ∂ 4w 2 + 2 + C + l D ( ) x x dx 2 ∂x 2 dx ∂x 3 ∂x 4 2 2 dN T ∂w 1 ∂ ∂w ∂w T ∂ w − = ∆Τ + N A x Cr 2 ∂x ∂x 2 ∂x ∂x dx ∂x
(38)
Using the following boundary conditions: ∂ ∂ 2w 2 C + l D =0 x x ∂x ∂x 2
(
(C
)
or
w =0
at
x = 0 and x = L
(39).a
or
∂w =0 ∂x
at
x = 0 and x = L
(39).b
2
x
+ l 2 Dx
) ∂∂xw
2
=0
3. Soloution methodology
The generalized differential quadrature method (GDQM) is employed for obtaining the results through solving the nonlinear equations. The r-th order derivative of function f ( xi ) in GDQM is: k ∂r f ( x ) C ij( r )f ( x i ) = ∑ ∂x r x = x j =1
(40)
P
k is the grid points number along x direction and C ij is:
11
C ij(1) =
M (x i )
(x
− x j ) M (x j )
i
n
∑
C ij( ) = − 1
;
i , j = 1, 2,..., n
and
i≠ j (41)
C ij( ) ; 1
i = j
j =1,i ≠ j
where M(x) is: k
∏ (x
M (x i ) =
i
−x j )
(42)
j =1, j ≠i
The weighting coefficient C(r), along x direction is derived as: (r )
C ij
( r −1) (1) C ij( r −1) = r C ij C ij − ; ( x i − x j ) n
∑
(r )
C ii = −
i , j = 1, 2,..., k , i ≠ j
and
2 ≤ r ≤ k −1 (43)
(r )
C ij ;
i , j = 1, 2,..., k
and
1≤ r ≤ k −1
j =1,i ≠ j
Chebyshev-Gauss-Lobatto approach is employed to obtain the distribution of the mesh points as:
xi =
( i − 1) L π 1 − cos 2 N − 1 ( )
i = 1,2,3, ... , k
(44)
Each of the nonlinear motion equations and boundary conditions of the AFG nanobeam (Eq. (25) and Eqs. (27)) and AFG microbeam (Eq. (38) Eqs. (39)) are considered to be the combination of three matrixes. Then the linear and nonlinear stiffness can be obtained as
{[ K ]
Linear
+ [ K ]non −Linear − ∆ΤCr [ M
]}{λ} = 0
(45)
The linear motion equation is first solved by GDQM and then, by employing the weight coefficients (Eq. (43)) to the linear motion equation we have: for nanobeam: d 2Cx dx 2
n
∑C rs ( )W s + 2 2
s =1
dCx dx
n
n
∑C rs ( )W s + Cx ∑C rs ( )W s 3
s =1
4
s =1
dN (1) ( 2) T dx ∑C rs W s + N ∑C rs W s s =1 s =1 n d 3N T n (1) ( 4) T = ∆ΤCr C W s + N ∑C rs W 3 ∑ rs s =1 − (e a )2 dx s =1 0 T n d 2N T n dN C ( 2)W s + 3 C rs ( 3)W s s +3 ∑ 2 ∑ rs dx s =1 dx s =1 T
n
n
for microbeam:
12
(46)
d 2 (Cx + Dxl 2 ) dx 2 dN = ∆ΤCr dx
n
∑C rs W s + 2 ( 2)
d (Cx + Dxl 2 ) dx
s =1 T
n
∑C
n
W s + N T ∑C rs
(1) rs
s =1
n
n
∑C rs ( )W s + (Cx + Dxl 2 ) ∑C rs ( )W s 3
s =1
4
s =1
Ws
(47)
(2)
s =1
Using the boundary conditions Eqs. (27) and (39), respectively for nano and microbeams, and assembling the related matrixes to the boundary conditions and governing equations, the linear fundamental thermal buckling of nano and microbeams can be calculated as below: [ K dd [ K bd
[ M dd ] [ M db ] {λd } ] [ K db ] {λd } = ∆ΤCr Linear ] [ K bb ] {λb } [ M bd ] [ M bb ] {λb }
(48)
where b and d indexes denote the boundary and domain, respectively and 5 is the mode shape. To solve the nonlinear equations, we need the linear mode shapes and then the mode shape of W can be obtained using Eq. (48). Then, the nonlinear mode shapes can be calculated by substitution of the calculated linear mode shapes into the nonlinear stiffness matrix. Afterwards, using Eqs. (25) and (38), the nonlinear temperature buckling and mode shapes are calculated by coupling the linear and nonlinear stiffness matrixes with the mass matrix. Afterwards, the iteration method is used to obtain the convergence of nonlinear results using the recalculation process. for nanobean: d 2Cx dx 2
n
∑C rs ( )W s + 2 2
s =1
dCx dx
n
+ (e 0a )
n
2
n
∑C
( 2) rs
s =1
3
s =1
1 1 1 −∑C rs ( ) A x ∑C rs ( )W s s =1 s =1 2 n
n
∑C rs ( )W s + Cx ∑C rs ( )W s 4
s =1 2
n
∑C s =1
Ws
(1) rs
2 n n n (1) 1 (1) (1) C A C W C W ∑ rs x ∑ rs s ∑ rs s s =1 s =1 2 s =1
dN (1) ( 2) T dx ∑C rs W s + N ∑C rs W s s =1 s =1 n d 3N T n (1) T ( 4) = ∆ΤCr C W s + N ∑C rs W 3 ∑ rs s =1 − (e a )2 dx s =1 0 T n d 2N T n dN C ( 2)W s + 3 C rs ( 3)W s s +3 ∑ 2 ∑ rs dx s =1 dx s =1 T
n
n
for microbeam:
13
(49)
d 2 (Cx + Dxl 2 ) dx 2 n
−∑C rs
(1)
s =1
n
∑C rs W s + 2 ( 2)
d (Cx + Dxl 2 ) dx
s =1
1 n (1) Ax ∑C rs W s s =1 2
2
n
∑C s =1
n
3
s =1
dN W s = ∆ΤCr dx
(1) rs
n
∑C rs ( )W s + (Cx + Dxl 2 ) ∑C rs ( )W s 4
s =1
T
n
∑C s =1
(1) rs
Ws + N
n
T
∑C s =1
Ws
(50)
(2 ) rs
4. Numerical results
The explained solution procedure is applied to pure metal, pure ceramic and AFG EulerBernoulli nano- and microbeams with different AFG power indexes (n). The results are depicted for low and high temperatures, different L/h values and power indexes of temperature variation function (α), under clamped-clamped (CC) and simply supportedsimply supported (SS) boundary conditions. The non-dimensional parameters are defined as below to yield better interpretation of the results: µ=
e0 a L
(51).a
l0 =
l h
(51).b
A mp = a
h2 12
(51).c
here µ, l0 and Amp are the non-dimensional nonlocal parameter, non-dimensional micro scale parameter (small scale parameter), non-dimensional frequency and amplitude of nonlinearity, respectively. The normalized critical buckling temperature (∆ΤCr) is also defined as below: Normalized ∆Τ Cr =
Nonlinear critical buckling temperature of micro or nanobeam Linear critical buckling temperature of local pure ceramic beam
(52)
The explained solving process is carried out on nano- and micro-beams for clamped (CC) and simply-supported (SS) boundary conditions. The problem is solved to find the linear and nonlinear critical buckling temperature change (∆Tcr) for various values of nonlinearity, aspect ratio (L/h), nonlocal value ‘μ’ (for nanobeams), small scale parameter ‘l0’ (for microbeams), and power index of temperature variation function (αT). To illustrate the effects of different parameters on buckling temperature of the micro- and nanobeams, several figures and two tables are presented. It should be noted that by neglecting the small scale effects (µ and l0), the nonlinear buckling temperatures of micro- and nanobeams are equal and the same as that of the classic theory. To prove the correctness of the provided results, the answers are compared with the other 14
studies. The comparison of the uniform and linear temperature changes with the results of Ebrahimi and Salari [39] are shown in Table. 2 for different values of the nonlocal parameter. The buckling temperature of Euler-Bernoulli beam is higher than the referred answers which are for the Timoshenko beam. It is because the degree of freedom of Timoshenko beam is more than that of Euler-Bernoulli beams. Table. 3 shows the mechanical properties of the ceramic and metal which are used in this paper. Table. 2. Comparison of the uniform and linear temperature changes with the results of Ebrahimi and Salari for different values of nonlocal parameter. (e0 a)2 =0 (nm)
(e0 a)=1 (nm)
(e0a)=2 (nm)
(e0a)=3 (nm)
Present, Euler-Bernoulli
[39], Timoshenko
Present, Euler-Bernoulli
[39], Timoshenko
Present, Euler-Bernoulli
[39], Timoshenko
Present, Euler-Bernoulli
[39], Timoshenko
L/h=40
UTC1 LTC2
71.36165 142.7233
68.6671 127.334
64.95122 129.9024
62.4988 114.998
59.59756 119.1951
57.3473 104.695
55.05926 110.1185
52.9803 95.9606
L/h=50
UTC LTC
45.67146 91.34292
43.9712 77.9423
42.958 85.91599
40.0212 70.0424
40.54888 81.09776
36.7224 63.4449
38.39562 76.79125
33.9261 57.8521
L/h=50
UTC LTC
31.71629 63.43258
30.5447 51.0893
30.38352 60.76704
27.8008 45.6016
29.15824 58.31649
25.5093 41.0186
28.02796 56.05592
23.5668 37.1336
UTC1: Uniform temperature change. LTC2: Linear temperature change.
Table. 3. The mechanical properties (Young’s modulus, mass density and Poisson’s ratio) of ceramic (Al2O3) and metal (SUS304), [30]. Material SUS304
Al2O3
Properties E (Pa) α (K-1) ρ (Kg/m3) Ν
X0 2.0104e+11 1.23e-05 8166 0.3262
X-1 0 0 0 0
X1 0.000308 0.000809 0 -0.0002
X2 -6.53e-07 0 0 3.80e-07
X3 0 0 0 0
E (Pa) α (K-1) ρ (Kg/m3) ν
3.4955e+11 6.8269e-06 3750 0.26
0 0 0 0
-0.0003853 0.0001838 0 0
4.027e-07 0 0 0
-1.673e-11 0 0 0
The normalized critical buckling temperature (∆Tcr) of SS and CC nanobeams are depicted respectively in Fig. 3 and Fig. 4 versus the amplitude of nonlinearity (Amp) for different values of power indexes of temperature variation function (αT). It can be seen that the normalized critical buckling temperature increases with the nonlinear amplitude as the nonlinear effect increases the stiffness of the beams. Also, the increment of power index of temperature variation function (αT) increases the buckling temperature too. Besides, it can be observed that as αT increases, the effect of the nonlinear amplitude on the critical buckling temperature increases too. Moreover, Fig. 3 and Fig. 4 show that the ∆Tcr of the AFG nanobeam is lower than ceramic and higher than metal. It is because the ∆Tcr is proportional to the stiffness, and as the stiffness increases, the value of ∆Tcr increases too.
15
T
(b): α = 0.25
Pure Metal n=1 n=2 n=5 n=10 Pure Ceramic
2.5 Normalize ∆TCr
T
(a): α = 0, Uniform temperature change
2
1.5
2 Normalize ∆TCr
3
1.5
1
1
0.5 0 -2
-1.5
-1
-0.5
0 Amp
0.5
1
1.5
0.5 -2
2
-1
T
2
1
2
(d): α = 2
5.5
3.5
5 4.5
Normalize ∆TCr
3 Normalize ∆TCr
1
T
(c): α = 1, Linear temperature change
2.5
4
3.5
2
3
2.5
1.5
2 1.5
1 -2
0 Amp
-1.5
-1
-0.5
0 Amp
0.5
1
1.5
2
-2
-1
0 Amp
Fig. 3. Normalized critical buckling temperature (∆Tcr) of SS nanobeam versus the nonlinear amplitude for different values of αT and AFG power indexes when L/h=40.
Comparison between Fig. 3 and Fig. 4 shows that the normalized critical buckling temperature (∆Tcr) of clamped nanobeam is lower than that of the simply supported nanobeam. The reason is that the stiffness of the CC nanobeam is lower than SS.
16
T
1.5
(b): α = 0.25
1.6
Pure Metal n=1 n=2 n=5 n=10 Pure Ceramic
2 Normalize ∆TCr
T
(a): α = 0, Uniform temperature change
1.4 Normalize ∆TCr
2.5
1
1.2 1 0.8
0.5 0.6 0 -2
-1.5
-1
-0.5
0 Amp
0.5
1
1.5
2
-2
-1
T
2.5
0 Amp
1
2
1
2
T
(c): α = 1, Linear temperature change
(d): α = 2
2
Normalize ∆TCr
Normalize ∆TCr
3.5 3
2.5
1.5
1 -2
2
1.5 -1.5
-1
-0.5
0 Amp
0.5
1
1.5
2
-2
-1
0 Amp
Fig. 4. Normalized critical buckling temperature (∆Tcr) of CC nanobeam versus the nonlinear amplitude for different values of αT and AFG power indexes when L/h=40.
The normalized critical buckling temperature (∆Tcr) of SS and CC nanobeams are respectively shown in Fig. 5 and Fig. 6 versus the nonlinear amplitude for different values of AFG index and nonlocal parameter. It is observed that the normalized critical buckling temperatures (∆Tcr) of SS and CC nanobeams decrease by increasing the nonlocal value. It is because the nonlocal effect decreases the stiffness of the nanobeam and decreasing the stiffness of the nanobeam decreases the critical buckling temperature.
17
(a): Pure Ceramic
3
µ = 0.0 µ = 0.1 µ = 0.2 µ = 0.3
2
1.5
0.5 0
0.5
1
1.5 Amp
2
2.5
0
3
0
0.5
1
(c): n=1 µ = 0.0 µ = 0.1 µ = 0.2 µ = 0.3
1.5
2
2.5
3
2
2.5
3
µ = 0.0 µ = 0.1
2.5 Normalize ∆TCr
2
1.5 Amp (d): n=10
3
2.5 Normalize ∆TCr
1
0.5
3
2
1.5
1
0.5 0
2
1.5
1
0
µ = 0.0 µ = 0.1 µ = 0.2 µ = 0.3
2.5 Normalize ∆TCr
Normalize ∆TCr
2.5
(b): Pure Metal
3
1
0.5
0
0.5
1
1.5 Amp
2
2.5
3
0
0
0.5
1
1.5 Amp
Fig. 5. Normalized critical buckling temperature (∆Tcr) of SS nanobeam versus the nonlinear amplitude for different nonlocal values and AFG power indexes when αT=0.1 and L/h=40.
18
(a): Pure Ceramic
2.5
µ = 0.0 µ = 0.1 µ = 0.2 µ = 0.3
1.5
1.5
1
0.5 0
µ = 0.0 µ = 0.1 µ = 0.2 µ = 0.3
2 Normalize ∆TCr
Normalize ∆TCr
2
(b): Pure Metal
2.5
1
0.5
0
1
2
3
4
0
5
0
1
2
Amp (c): n=1
2.5 µ = 0.0 µ = 0.1 µ = 0.2 µ = 0.3
1
0.5 0
5
3
4
5
µ = 0.0 µ = 0.1
2 Normalize ∆TCr
Normalize ∆TCr
1.5
4
(d): n=10
2.5
2
3 Amp
1.5 1 0.5
0
1
2
3
4
5
0
0
1
2 Amp
Amp
Fig. 6. Normalized critical buckling temperature (∆Tcr) of CC nanobeam versus the nonlinear amplitude for different nonlocal values and AFG power indexes when αT=0.1 and L/h=40.
Fig. 7 and Fig. 8 show the (∆Tcr) of SS and CC microbeams, respectively, for different values of small scale parameter (l0) versus the nonlinear amplitude. It is seen that as the nonlocal parameter increases, the value of (∆Tcr) increases too. The reason is that the nonlocal parameter increases the stiffness of the microbeam, which leads to a higher value of normalized critical buckling temperature. Fig. 8 shows that the normalized critical buckling temperature (∆Tcr) of CC microbeam is lower than that of SS microbeam.
19
(a): Pure Ceramic 7
(b): Pure Metal 7
l = 0.0
l 0 = 0.0
6
l 0 = 0.2
5
l = 0.4
4
l = 0.8
Normalize ∆TCr
Normalize ∆TCr
0
0 0
3 2 1 0
1
Normalize ∆TCr
7
2
3
4
5
l 0 = 0.4
4
l 0 = 0.8
3 2
0
5
l 0 = 0.2
5
l 0 = 0.4
4
l 0 = 0.8
0
1
2
Amp
Amp
(c): n=1
(d): n=10 7
l 0 = 0.0
6
3
4
5
3
4
5
l = 0.0 0
6
l = 0.2
5
l 0 = 0.4
4
l = 0.8
0
3 2
0
3 2 1
1 0
l 0 = 0.2
1
Normalize ∆TCr
0
6
0
1
2
3
4
5
0
0
1
2 Amp
Amp
Fig. 7. Normalized critical buckling temperature (∆Tcr) of SS microbeam versus the nonlinear amplitude for different nonlocal small scale parameters and AFG power indexes when αT=0.1 when L/h=40.
20
(a): Pure Ceramic
4
(b): Pure Metal
4
3
Normalize ∆TCr
Normalize ∆TCr
l 0 = 0.0
2
1
0
0
1
2
3
4
l 0 = 0.4 l 0 = 0.8
2
1
0
5
l 0 = 0.2
3
0
1
2
(c): n=1
4
3
4
5
3
4
5
Amp
Amp
(d): n=10
4 l 0 = 0.2
3
Normalize ∆TCr
Normalize ∆TCr
l 0 = 0.0 l 0 = 0.4 l 0 = 0.8
2
1
0
0
1
2
3
4
5
3
2
1
0
0
1
2 Amp
Amp
Fig. 8. Normalized critical buckling temperature (∆Tcr) of CC microbeam versus the nonlinear amplitude for different nonlocal small scale parameters and AFG power indexes when αT=0.1 when
Table. 4 shows the value of nonlinear critical buckling temperature (in Kelvin) of SS nanobeam for different values of nonlinearity, nonlocal value, L/h and the AFG power index when the power index of temperature variation function is αT=0.2. It is seen that increasing the L/h decreases the ∆Tcr. This is because as the L/h increases, the stiffness of the nanobeam decreases which leads to a lower value of ∆Tcr. Similar to Fig. 3, Fig. 4, Fig. 5 and Fig. 6, Table. 4 shows that increasing the nonlinear amplitude increases the critical buckling temperature and increasing the nonlocal value decreases the ∆Tcr.
21
Table. 4. The nonlinear critical buckling temperature (in Kelvin) of SS nanobeam in differen values of Amp, n, μ, and L/h when αT=0.2. Amp=1 Amp=2 Amp=3 μ=0 μ=0.1 μ=0 μ=0.1 μ=0 μ=0.1 Pure Metal 191.0444 176.5612 279.6634 265.0778 425.619 410.897 n=1 266.7629 243.3819 390.4709 365.4423 594.218 566.5207 n=2 302.3162 280.854 438.2844 417.1361 662.2083 641.62 L/h=20 n=5 342.4768 325.6039 493.1526 483.4594 741.2869 743.4784 n=10 365.2835 205.8098 528.0418 389.1458 796.0806 692.2056 Pure Ceramic 406.3386 375.5339 594.8252 563.8026 905.263 873.9504
L/h=40
Pure Metal n=1 n=2 n=5 n=10 Pure Ceramic
47.76109 66.69071 75.57906 85.61919 91.32087 101.5846
44.1403 60.84549 70.21351 81.40097 51.45246 93.88346
69.91584 97.61772 109.5711 123.2881 132.0105 148.7063
66.26944 91.36057 104.284 120.8648 97.28645 140.9507
106.4047 148.5545 165.5521 185.3217 199.0202 226.3158
102.7243 141.6302 160.405 185.8696 173.0514 218.4876
L/h=80
Pure Metal n=1 n=2 n=5 n=10 Pure Ceramic
11.94027 16.67268 18.89476 21.4048 22.83022 25.39616
11.03508 15.21137 17.55338 20.35024 12.86312 23.47087
17.47896 24.40443 27.39278 30.82204 33.00261 37.17658
16.56736 22.84014 26.071 30.21621 24.32161 35.23766
26.60119 37.13863 41.38802 46.33043 49.75504 56.57894
25.68106 35.40754 40.10125 46.4674 43.26285 54.6219
The value of the nonlinear critical buckling temperature of SS microbeam is shown in Table. 5 for different values of L/h, small scale parameter, AFG power index, and nonlinear amplitude. Comparison between Table. 4 and Table. 5 shows that, when the small scale effect is neglected in the Eringen’s nonlocal elasticity and the modified couple stress, we obtain the results of the classic theory for nano- and micro-beams. Besides, this comparison shows that increasing the nonlocal effect in the Eringen’s theory decreases the buckling temperature, while increasing the small scale parameter in the modified couple stress theory increases the buckling temperature.
22
Table. 5. The nonlinear critical buckling temperature (in Kelvin) of SS microbeam in differen values of Amp, n, l0 , and L/h when αT=0.2. Amp=1 Amp=2 Amp=3 l0=0 l0=0.1 l0 =0 l0=0.1 l0=0 l0=0.1 Pure Metal 191.0444 238.3069 279.6634 327.2004 425.619 473.5643 n=1 266.7629 332.7608 390.4709 456.852 594.218 661.1692 n=2 302.3162 377.5296 438.2844 513.9163 662.2083 738.4769 L/h=20 n=5 342.4768 428.0145 493.1526 579.1518 741.2869 827.9997 n=10 365.2835 456.3142 528.0418 619.5725 796.0806 888.3772 Pure Ceramic 406.3386 506.8628 594.8252 695.9334 905.263 1007.239
L/h=40
Pure Metal n=1 n=2 n=5 n=10 Pure Ceramic
47.76109 66.69071 75.57906 85.61919 91.32087 101.5846
59.57672 83.19019 94.38239 107.0036 114.0786 126.7157
69.91584 97.61772 109.5711 123.2881 132.0105 148.7063
81.8001 114.213 128.4791 144.788 154.8931 173.9833
106.4047 148.5545 165.5521 185.3217 199.0202 226.3158
118.3911 165.2923 184.6192 206.9999 222.0943 251.8099
L/h=80
Pure Metal n=1 n=2 n=5 n=10 Pure Ceramic
11.94027 16.67268 18.89476 21.4048 22.83022 25.39616
14.89418 20.79755 23.5956 26.75091 28.51964 31.67892
17.47896 24.40443 27.39278 30.82204 33.00261 37.17658
20.45003 28.55325 32.11977 36.19699 38.72328 43.49583
26.60119 37.13863 41.38802 46.33043 49.75504 56.57894
29.59777 41.32307 46.15481 51.74998 55.52357 62.95247
5. Conclusion
The studied problem was the nonlinear thermal buckling of axially functionally grade (AFG) micro- and nanobeams. The modified couple stress theory is the basis of the governing equations of the microbeam and Eringen’s nonlocal elasticity theory is used to study the nanobeam. The material of the micro- and nanobeams are pure metal, pure ceramic and AFG materials and boundary conditions are clamped (CC) and simply supported (SS). The nonlinear equations are solved using the generalized differential quadrature method (GDQM) along with the iteration technique. The results are depicted to illustrate the effects of the small scale parameters, length to height ratio (L/h), nonlinear amplitude and AFG power index on the buckling temperature of the micro- and nano-beams. The main results of this paper are briefly represented below: -
Increment of αT increases, the dependency of the critical buckling temperature on the nonlinear amplitude.
-
Increasing the small scale effect decreases the buckling temperature of nanobeams, while it increases the buckling temperature of microbeams. 23
-
The buckling temperature of AFG micro- and nano-beams is lower than that of ceramic and higher than metal.
-
Increasing the nonlinear amplitude, AFG power index and temperature variation function (αT) increase the buckling temperature and increasing L/h decreases the buckling temperature.
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26