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Effects of neutral surface deviation on nonlinear resonance of embedded temperature-dependent functionally graded nanobeams
Accepted Manuscript Effects of neutral surface deviation on nonlinear resonance of embedded temperature-dependent functionally graded nanobeams Yu-gang Tang, Ying Liu, Dong Zhao PII: DOI: Reference:
S0263-8223(17)30095-8 https://doi.org/10.1016/j.compstruct.2017.10.058 COST 9033
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
11 January 2017 13 October 2017 18 October 2017
Please cite this article as: Tang, Y-g., Liu, Y., Zhao, D., Effects of neutral surface deviation on nonlinear resonance of embedded temperature-dependent functionally graded nanobeams, Composite Structures (2017), doi: https:// doi.org/10.1016/j.compstruct.2017.10.058
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Effects of neutral surface deviation on nonlinear resonance of embedded temperature-dependent functionally graded nanobeams
Yu-gang Tang, Ying Liu*, Dong Zhao
Department of Mechanics, School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, PR China *Email: [email protected]; Tel: 86-10-51688763; Fax: 86-10-51682094
Abstract
In this paper, forced vibration of functionally graded (FG) nanobeams resting on the nonlinear elastic foundations are investigated based on the nonlocal strain gradient theory. The material parameters of FG nanobeams are assumed to be temperature-dependent and change continuously along the thickness direction according to the power-law function (PFGM) or sigmoid function (SFGM). Based on the Euler–Bernoulli beam theory and von-Kármán geometric nonlinearity, the governing equations of motion are derived by considering the deviation between the geometrical and physical neutral surfaces. Closed-form approximate solution for nonlinear forced vibration of a FG nanobeam is derived by using multiple time scale method. The results show that decrease of non-homogeneity index and material length scale parameter, or increase of temperature variation and nonlocal parameter will increase the resonance frequencies of FG nanobeams. The effect of the in-coincidence of physical and geometrical neutral surfaces on the nonlinear resonance of the nanobeams could not be ignored, especially for SFGM nanobeams with larger non-homogeneity index and stronger size effects, embedded in a softer medium
1
with obvious temperature variation.
Keywords: Functionally graded nanobeam; Nonlocal strain gradient theory; Physical neutral surface; Nonlinear forced vibration; Temperature dependent
1. Introduction
Composites have been widely used in different fields and their mechanical properties are always the attention of scientists [1-10]. As one kind of new improved composite materials that has smooth and continuous material parameter variation throughout certain dimension(s), functionally graded material (FGM) has attracted great attention due to their exceptional mechanical and thermal properties such as higher fracture toughness, enhanced thermal resistance and improved stress spreading, which promise prospective applications in micro/nanoelectromechanical systems (MEMS/NEMS) [11-16].
Both experiments and simulations have showed that size effects play a significant role on the performance of small-size structures [17-21]. Due to the limitation of classical elasticity theory (CET) in description size effects, several non-classical continuum theories such as nonlocal elasticity theory (NET) [22] and strain gradient theory (SGT) [23, 24] have been proposed to determine the size-dependent deformation of small-size structures. In contrast to the classical continuum theory, the nonlocal elasticity theory [22] considers the size effect by assuming that the stress at a reference point is a function of strains at all points in the body. Numerous works has been done based on NET to consider size effects of nanostructures [25-27]. Results show that although NET is suitable to capture softening stiffness, the capability in prediction the stiffness enhancement observed in the experimental or theoretical studies is weak. Strain gradient theory (SGT) considers the effects of stiffness enhancement. Based on SGT, Akgöz 2
and Civalek [28] investigated the static bending and free vibration behavior of simply supported microbeams. Ghorbanpour Arani et al. [29] analyzed the nonlinear vibration of a piezoelectric nanobeam. Ghayesh et al. [30] investigated the nonlinear forced vibration of a microbeam. More recently, Thai et al. [31] presented a review of continuum mechanics models for size effects on the mechanical response of beams and plates and also pointed out that NET could only predicted the softening stiffness, while SGT was good at prediction the stiffness enhancement. To address this issue, by combining NET and SGT together, Lim et al. [32] developed a nonlocal strain gradient theory (NSGT) by considering both of the nonlocal parameter and the material gradient length scale parameter, which can well predict the stiffness softening as well as stiffness enhancement simultaneously. By adopting NSGT, Tang et al. [33, 34], and Li et al. [35] investigated the size effects on the wave propagation in viscoelastic SWCNTs. Li et al. [36] developed a size-dependent model to study the buckling analysis of nonlinear Euler-Bernoulli beam. It is seen that more and more works have been done based on NSGT to reflect size effects of micro/nano-structures.
As basic elements in engineering application, mechanical performances of FG structures (beam, plate, etc.) are always the frontier [37-40]. For a FG beam, based on the classical continuum theory, Bourada et al. [41] performed an investigation on the bending and vibration of FG trigonometric higher-order beams including the shear deformation effect. Calim [42] discussed the free and forced vibrations of axially FG Timoshenko beams embedded in the viscoelastic foundation. Considering temperature variation, Ansari et al. [43] studied nonlinear forced vibration of FG nanobeams based on surface elasticity theory. Uymaz [44] investigated the forced vibration of FG beams based on NET. Niknam and Aghdam [45] investigated the large amplitude free vibration and buckling of a nonlocal FG beam resting on nonlinear elastic 3
foundation by adopting a semi analytical approach. El-Borgi et al. [46] explored the nonlocal effect on the free and forced vibrations of FG Euler–Bernoulli nanobeams resting on a nonlinear elastic foundation. Bouafia et al. [47] analyzed the bending and free flexural vibration behaviors of FG nanobeams based on a nonlocal quasi-3D theory. Thai et al. [48] investigated size effects on the static bending, buckling and free vibration behaviors of FG sandwich microbeams based on the modified couple stress theory. Based on NSGT, Şimşek [49] developed a novel size-dependent beam model to analyze the nonlinear free vibration of FG nanobeams. Trinh et al. [50] discussed the vibration and buckling behaviors of PFGM beams under mechanical and thermal loads based on a high-order beam theory. Ebrahimi and Salari [51] studied the thermal effects on the buckling and free vibration of PFGM nanobeams based on the nonlocal Timoshenko beam model. Ghadiri and Shafiei [52] explored thermal vibration of rotating PFGM microbeams based on modified couple stress theory. Referring to the modified SGT, Shenas et al. [53] investigated the vibration characteristics of rotating pre-twisted PFGM microbeams in the thermal environment. Ebrahimi and Barati [54] developed a quasi-3D beam model to study wave propagation in PFGM nanobeams in thermal environments based on NSGT.
It should be pointed out that in the above discussion, the geometrical symmetrical axis of the beams were taken as the neutral surface in the equation establishment. In fact, for a FG structure, the gradient distribution of materials in space definitely causes the deviation of the geometrical neutral axis from the physical one, which has attracted more and more attention in recent years. Bellifa et al. [55] proposed a simple shear deformation theory to study bending and dynamic behaviors of PFGM plates with the neutral surface position concept. Eltaher et al. [56] investigated the free vibration of FG macro/nanobeams based on the nonlocal continuum model considering the physical neutral axis. Barretta 4
et al. [57] developed a new Eringen gradient elasticity model to study the bending behaviors of FG Timoshenko nanobeams based on the physical neutral surface. Ahouel et al. [58] proposed a nonlocal trigonometric shear deformation beam theory to examine the bending, buckling, and vibration behaviors of PFGM nanobeams including neutral surface position concept. On the basis of the modified couple stress theory and the neutral surface, Al-Basyouni et al. [59] presented the size-dependent bending and vibration of PFGM microbeams. Ebrahimi and Barati [60] investigated the vibration characteristics of hydro-thermally affected PFGM viscoelastic nanobeams embedded in viscoelastic foundation based on NSGT. However, to per author’s best knowledge, the influence of the physical neutral surface derivation from geometrical one due to the temperature-dependent material gradient on the nonlinear resonance of FG nanobeams has not yet been fully addressed, especially when the nonlocal stress and higher order strain field are both considered.
The objective of the present manuscript is to study the forced vibration of temperature-dependent PFGM and SFGM nanobeams resting on the nonlinear elastic foundation by adopting NSGT. Firstly, referring to physical neutral surface, governing equations of motion are derived based on Euler–Bernoulli beam theory together with von-Kármán’s geometric nonlinearity. Then the closed-form approximate solution for the nonlinear forced vibration is presented based on the multiple time scale method. The influence of the neutral surface deviation caused by the temperature-dependent gradient, small scale parameters, as well as force amplitude and stiffness of the nonlinear foundation, on the nonlinear resonance of FG nanobeams is clarified. At last, the conclusion is given.
2. Mathematical formulation
5
2.1. Functionally graded materials
An FG nanobeam with length L, width b, and thickness h resting on the nonlinear elastic foundation is shown in Fig. 1. Following Şimşek [49], the nanobeam has immovable simple supports at both ends. The FG nanobeam is assumed to be made of ceramic and metals, which are temperature-dependent [61]. Then, the material parameter P, such as Young’s modulus E, thermal expansion α, Poisson's ratio υ and mass density ρ, are functions of temperature and with the form [61]
(1)
P = P0 ( P− 1T − 1 + 1 + P1T + P2 T 2 + P3 T 3 )
Here P0, P-1, P1, P2, P3 are temperature-dependent coefficients.
Fig. 1. Geometry of a temperature-dependent functionally graded nanobeam resting on the nonlinear elastic foundation subjected to harmonic forces.
Because the material parameters are gradient distributed, the physical neutral surface no longer coincides with the geometrical one. Two different coordinates, x0zm and x’0’zp, which are defined at the geometrical and physical neutral surfaces, respectively, are introduced and presented in Fig. 1. Then points in the two coordinates have the relation
zm = z p + hp ,
where h p = ∫
h/2
− h/ 2
(2)
x = x'
E ( zm , T ) zm dz m
∫
h/2
− h/ 2
E ( zm , T ) dz m is the derivation between the physical neutral surface
6
and the geometrical one.
Assume that the effective material properties vary continuously along the thickness direction according to power-law (PFGM) or sigmoid (SFGM) functions [62]. Referring to Fig. 1, the volume fraction of PFGM is defined as
V1 = (
z p + hp h
1 + )k 2
h h z p ∈ [ − − hp , − h p ] 2 2
(3)
where k is the non-homogeneity index. Then, the material parameters for PFGM along the thickness direction are expressed as Peff ( z p , T ) = Pc (T )V1 + Pm (T )(1 − V1 ) = Pc (T )(
z p + hp h
(4)
z p + hp 1 k 1 + )k + Pm (T )[1 − ( + ) ] 2 2 h
where Pc and Pm denote the temperature-dependent material parameters of ceramic and metal, respectively.
In the same way, the material parameters for SFGM are expressed as
Peff ( z p , T ) = Pc (T )V2 + Pm (T )(1 − V2 ) =
Pc (T ) 2( z p + h p ) 1 2( z p + hp ) h [ + 1]k + Pm (T ){1 − [ + 1]k } z p ∈ [− − hp , −h p ] 2 h 2 h 2
(5)
Peff ( z p , T ) = Pc (T )V3 + Pm (T )(1 − V3 ) 2( z p + h p ) k 2( z p + h p ) k P (T ) 1 ] }+ m [1 − ] = Pc (T ){1 − [1 − 2 h 2 h
h z p ∈ [ − hp , − h p ] 2
(6)
where the volume fraction is defined as 1 2( z p + hp ) h V2 = [ z p ∈ [− − hp , − hp ]; + 1]k , 2 h 2 2( z p + h p ) k 1 h ] , z p ∈ [ −h p , − hp ] V3 = 1 − [1 − 2 h 2
7
(7)
Eqs. (4) to (6) indicate that the non-homogeneity index and temperature variation have determinant effects on the derivation between the physical and geometrical neutral surfaces.
Assume that the temperature T varies linearly through the thickness direction as
1 z p + hp ) T = Tm + ∆T ( + 2 h
(8)
where ∆T=Tc-Tm, with Tc and Tm the temperatures of the top (full-ceramic) and the bottom surfaces (full-metal), respectively. Fig. 2 gives the variation of the neutral surface derivation with respect to the non-homogeneity index and temperature variation. The bottom surface temperature Tm is set as 300K and temperature-dependent coefficients P0, P-1, P1, P2, P3 are given in Table 1. Table 1 Temperature-dependent coefficients for ceramic and metals [51, 61].
Material
Properties
P-1
P0
P1
P2
P3
SUS304
E(Pa)
0
201.04e+9
3.079e-4
-6.534e-7
0
ρ(kg/m 3)
0
8166
0
0
0
α(K-1)
0
12.33e-6
8.086e-4
0
0
υ
0
0.3262
-2.002e-4
3.797e-7
0
E(Pa)
0
348.43e+9
-3.07e-4
2.160e-7
-8.946e-11
ρ(kg/m 3)
0
2370
0
0
0
α(K-1)
0
5.8723e-6
9.095e-4
0
0
υ
0
0.24
0
0
0
Si3N4
For PFGM nanobeams, along with the increase of the non-homogeneity index k, the physical neutral 8
surface firstly moves away from the geometrical one, and then moves back again when k is greater than a certain value (see, k=2 in the present discussion). For SFGM nanobeams, the physical neutral surface always moves away from the geometrical one when the non-homogeneity index is smaller than a certain value (see, k=4 in the present discussion), after which the neutral surface derivation is insensitive to the non-homogeneity index anymore. Along with the increase of the temperature variation, the deviation is reduced.
Fig. 2. The effects of non-homogeneity index k and temperature variation ∆T on the neutral surface derivation.
The variation of Young’s modulus along thickness direction for PFGM and SFGM nanobeams with respect to temperature variation is plotted in Fig. 3, in which black and red lines are corresponding to
∆T=0K and ∆T=300K, respectively. It is seen that the Young’s modulus of PFGM and SFGM nanobeams are reduced along with the increase of the temperature variation. Because the physical neutral surface no longer coincides with the geometrical one, and the deviation is temperature dependent, the Young’s moduli of SFGM nanobeams with different k are no longer equal to each other at zp=0 and display nonsymmetrical variation along the beam thickness direction.
9
Fig. 3. The variation of Young’s modulus through the thickness direction for PFGM and SFGM nanobeams with different temperature variation ∆T.
2.2. Nonlocal strain gradient theory
According to the nonlocal strain gradient theory developed by Lim et al. [32], the total stress ςxx , the classical nonlocal stress σxx(0) and the higher-order stress σxx(1) are defined as
(12)
ς xx = σ xx( 0 ) − ∇ σ xx(1)
L
σ xx(0) = E ( z p ) ∫ β 0 ( x, x ′, e0 a ) ε xx′ ( x ′)dx′
(13)
0
L
σ xx(1) = l 2 E ( z p ) ∫ β1 ( x, x ′, e1a ) ε xx′ , x ( x ′)dx ′
(14)
0
where ∇ = ∂ / ∂ x . e0a and e1a, which are named as nonlocal parameters, are introduced to determine the significance of lower and higher order nonlocal elastic stress fields, respectively. l, which is named as material length scale parameter, is introduced to determine the significance of higher-order strain gradient field. εxx and ε xx′ are the strains at point x and neighboring point x′ , respectively. Correspondingly, ′ , x = ∂ε xx ′ / ∂x are the strain gradient at point x and neighboring point x′ , respectively. ε xx , x = ∂ε xx / ∂x and ε xx
Following the assumptions given by Lim et al. [32] that if the nonlocal attenuation functions β0(x, x ′ , e0a) and β1(x, x′ , e1a) meet the conditions given by Eringen [22] and e=e0=e1, the nonlocal strain gradient 10
constitutive relation is with the form
[1 − (ea) 2 ∇ 2 ]ζ xx = E ( z p )(1 − l 2 ∇2 )ε xx
(15a)
where the Laplacian operator ∇ 2 is equal to ∂ 2 / ∂ x 2 . Then, Eq. (15a) can be rewritten as
[1 − (ea ) 2
2 ∂2 2 ∂ ] ζ ( )(1 )ε xx = E z − l xx p ∂x 2 ∂x 2
(15b)
Eq. (15b) can be reduced to the nonlocal constitutive equation without taking into consideration of the material length scale parameter (l=0), that is
[1 − (ea ) 2
∂2 ]ζ xx = E ( z p )ε xx ∂x 2
(16)
When the nonlocal parameter ea=0, Eq. (15b) is reduced to the constitutive equation based on the strain gradient theory, that is
ζ xx = E ( z p )(1 − l 2
∂2 )ε xx ∂x 2
(17)
2.3. The governing equation of FG nanobeams
For an Euler–Bernoulli beam, the von Kármán type nonlinear strain-displacement relation is expressed as
ε xx =
∂u 1 ∂w 2 ∂2 w + ( ) − zp 2 ∂x ∂x 2 ∂x
(18)
where u is the axial displacement and w the transverse displacement.
The first variation of the strain energy is expressed as
11
L
L
A
0
δUm = ∫ (σ xx(0)δε xx + σ xx(1) ∇δε xx )dV = ∫ (σ xx(0) −∇σ xx(1) ) δε xx dV + ∫σ xx(1)δε xx dA = ∫ ζ xxδε xx dV + ∫ σ xx(1)δε xx dA V
A
V
0
V L
∂δ u ∂w ∂δ w ∂ δw ∂u ∂w ∂δ w ∂ δ w = ∫ [N( + ) − M 2 ]dx + N (1) (δ + ) − M (1) 0 ∂x ∂x ∂x ∂x ∂ x ∂ x ∂ x ∂x2 0 2
L
2
(19)
where
N = ∫ ζ xx dA, A
M = ∫ z pζ xx dA A
N (1) = ∫ σ xx(1) dA A
M (1) = ∫ z pσ xx(1) dA A
(20)
Substituting Eqs. (15) and (18) into Eq. (20), we have
N − ( ea )
2
M − ( ea )
∂2 N ∂ 2 ∂u 1 ∂w ∂2 ∂ 2w = A11 (1 − l 2 2 )[ + ( ) 2 ] − B11 (1 − l 2 2 ) 2 2 ∂x ∂x ∂x 2 ∂x ∂x ∂x 2
(21)
∂2 M ∂ 2 ∂u 1 ∂w ∂2 ∂ 2 w = B11 (1 − l 2 2 )[ + ( ) 2 ] − D11 (1 − l 2 2 ) 2 2 ∂x ∂x ∂x 2 ∂x ∂x ∂x
(22)
where
A11 = ∫ E ( z p )dA = b ∫
h / 2 − hp
− h / 2 − hp
A
D11 = ∫ z 2p E ( z p )dA = b ∫
E ( z p )dz p
h / 2 − hp
− h / 2 − hp
A
B11 = ∫ z p E ( z p )dA = b ∫
h / 2− hp
− h / 2 − hp
A
z p E ( z p )dz p
(23)
z 2p E ( z p )dz p
The first variation of the additional strain energy induced by thermal loading can be expressed as
L
δ U ad = ∫ NT 0
∂w ∂δ w dx ∂x ∂x
(24)
where
NT = − ∫ E ( z p )α ( z p )[T ( z p ) - T0 ]dA = −b ∫ A
h / 2 − hp
− h / 2− hp
E ( z p )α ( z p )[T ( z p ) - T0 ]dz p
with T0 the reference temperature.
The first variation of the kinetic energy is expressed as [48]
12
(25)
L
δ S = ∫ Im 0
L ∂u ∂δ u ∂w ∂δ w dx + ∫ I m dx 0 ∂t ∂t ∂t ∂t
(26)
where the mass moment of inertia Im is given as
I m = ∫ ρ ( z p )dA = b ∫
h / 2 − hp
− h / 2 − hp
A
(27)
ρ ( z p )dz p
The first variation of the work done by nonlinear elastic foundation and harmonic force is expressed as
L
δ W = ∫ −kL wδ w − ks 0
∂w ∂δ w − k NL w3δ w + F cos(Ω t)δ wdx ∂x ∂x
(28)
Here kL, ks and kNL are, respectively, linear, shear, and nonlinear coefficients of the nonlinear elastic foundation. F denotes the excitation amplitude.
Using Hamilton’s principle, we have
t
∫ δ (U 0
m
(29)
+ U ad − S − W )dt = 0
Then, the equations of motion are derived as
δu :
δw:
∂N ∂ 2u = Im 2 ∂t ∂x
(30)
∂2M ∂ ∂w ∂2 w ∂2w ∂2 w 3 + ( N ) − k w + k − k w + N + F cos( Ω t) = I L s NL T m ∂x 2 ∂x ∂x 2 ∂x 2 ∂t 2 ∂x
(31)
Moreover, the corresponding boundary conditions (x=0 or x=L) are expressed as
δ u : N = 0 or u = 0
(32)
∂u : N (1) = 0 or ∂x
(33)
δ
δw:
∂u =0 ∂x
∂M ∂w ∂w ∂w +N + ks + NT = 0 or ∂x ∂x ∂x ∂x
w=0
13
(34)
δ
∂w ∂w : M = N (1) ∂x ∂x
δ
∂2 w : M (1) = 0 or ∂x 2
∂w =0 ∂x
or
(35)
∂2 w =0 ∂x 2
(36)
Ignoring the axial inertial [49], Eq. (30) gives
∂N =0 ∂x
(37)
By combining Eqs. (21) and (37), we can obtain the axial force N as
N = N c = A11 (1 − l 2
∂ 2 ∂u 1 ∂w 2 ∂2 ∂2 w )[ + ( ) ] − B11 (1 − l 2 2 ) 2 2 ∂x ∂x 2 ∂x ∂x ∂x
(38)
where Nc denotes a constant value. For a nanobeam with immovable ends, the boundary conditions can be obtained as [49]
u (0, t ) = u ( L , t ) =
∂ 2 u (0, t ) ∂ 2 u ( L , t ) = =0 ∂ x2 ∂ x2
(39)
It is worth to note that the boundary conditions related to the second derivative of the axial displacement are due to the strain gradient theory [23]. By integrating each side of Eq. (38) over the domain [0, L] and using the above boundary conditions, we have
N=
A11 2L
∫
L
0
(
A l2 ∂w 2 ) dx − 11 ∂x L
∫
L
0
[
2 B11 L ∂w ∂ 3 w ∂ 2 w 2 ∂2w 2 ∂ dx l dx + ( ) ] − (1 − ) ∂x ∂x3 ∂x 2 L ∫0 ∂x 2 ∂x 2
(40)
By combining Eqs. (22), (31), and (40), we can obtain the moment M as
M =(
B112 ∂ 2 w ∂ 2 w NB − D11 )(1 − l 2 2 ) 2 + 11 A11 ∂x ∂x A11
∂2 w ∂2 w + (ea ) [ I m 2 + k L w − (ks + NT + N ) 2 + k NL w3 − F cos(Ω t)] ∂t ∂x
(41)
2
Inserting Eqs. (40) and (41) into Eq. (31), we obtain the governing equation of motion for the nonlocal strain gradient Euler–Bernoulli beam, that is, 14
( D11 −
2 B112 ∂2 ∂4w ∂2w ∂2 w ∂2 w 2 ∂ )(1 − l 2 2 ) 4 + [1 − ( ea ) ][ I m 2 + k L w − k s 2 + k NL w3 − N T ] 2 A11 ∂x ∂x ∂x ∂t ∂x ∂x 2
4 ∂2w A L ∂w A l2 2 ∂ w ]{ 11 ∫ ( ) 2 dx − 11 − ( ea ) 2 4 ∂x ∂x 2 L 0 ∂x L = F cos(Ω t)
−[
∫
L
0
[
∂w ∂ 3 w ∂ 2 w 2 B + ( 2 ) ]dx − 11 ∂x ∂x 3 ∂x L
∫
L
0
(1 − l 2
∂2 ∂2w ) dx} ∂x 2 ∂x 2
(42)
For the sake of simplification, the following dimensionless quantities are defined, that is,
xˆ =
x L
KL =
wˆ = kL L4 Em I
w r= r Ks =
ks L2 Em I
I A
zˆ =
zp h
tˆ = t
N L2 Nˆ T = T Em I
Em 0 I
µ=
ρm AL4
K NL =
kNL r 2 L4 Em 0 I
A B r D11 FL4 ( Aˆ11 , Bˆ11 , Dˆ 11 ) = ( 11 , 11 ) Fˆ = Em 0 A Em 0 I Em 0 I rEm 0 I
ea l η= L L ˆ =Ω/ Ω
Em 0 I
ρm0 AL4
(43)
I Iˆ0 = m ρm A
where A=bh and I=bh3/12 are the area and moment of inertia of the cross section of the beam, respectively. Em0 is equal to Young's modulus of the metal at T=0K.
Then, the governing equation is rewritten in the following dimensionless form as
( Dˆ11 −
ˆ Bˆ112 ∂ 2 ∂ 4 wˆ ∂wˆ ∂2 w )(1 −η 2 2 ) 4 + K NL wˆ 3 − K NL µ 2 [6wˆ ( )2 + 3wˆ 2 2 ] ∂xˆ ∂xˆ ∂xˆ ∂xˆ Aˆ11
3 2 4 ˆ ˆ 2 ˆ ˆ ∂ 2 ∂ 2 wˆ Aˆ11 1 ∂wˆ 2 ∂2w 2 ∂ w ˆ η 2 1[ ∂wˆ ∂ w ˆ 1( ∂ w ˆ ˆ ) { ( ) dx A ( ) ] dx B )dxˆ} − + − − η 11 11 ∫0 2 2 3 2 2 ∫ ∫ 0 0 ∂xˆ ∂xˆ 2 ∂xˆ ∂xˆ ∂xˆ ∂xˆ ∂xˆ ∂xˆ 4 ∂2 ∂ 2 wˆ ∂2 wˆ ∂2 wˆ ˆ tˆ) +(1 − µ 2 2 )( Iˆ0 2 + K L wˆ − K s 2 − Nˆ T 2 ) = Fˆ cos(Ω ∂xˆ ∂tˆ ∂xˆ ∂xˆ
−(1 − µ 2
(44)
The Galerkin method is adopted to degenerate the above partial differential equation into a time-dependent ordinary differential equation. The transverse displacement can be assumed as the following form
wˆ ( xˆ , tˆ) = p (tˆ)φ ( xˆ ), φ ( xˆ ) = sin(π xˆ )
(45)
in which φ ( xˆ ) denotes a trial function which meets the boundary conditions and p (tˆ ) stands for an undetermined time-dependent function. Inserting Eq. (45) into Eq. (44), multiplying each side of Eq. (44) 15
with the trial function φ ( xˆ ) and finally integrating the resulting equation from 0 to 1, we can obtain a second-order ordinary differential equation as
ˆ tˆ ) p (tˆ) + γ 1 p (tˆ ) + γ 2 p 2 (tˆ ) + γ 3 p 3 (tˆ ) = fˆ cos(Ω
(46)
where
fˆ =
1 Fˆ ∫ φ d xˆ 0
1 Iˆ0 ∫ φ (φ − µ 2φ ′′) d xˆ
,
0
1 1 Bˆ 2 1 Bˆ 2 1 ( Dˆ11 − 11 )∫ φ (4)φ d xˆ − η 2 ( Dˆ11 − 11 ) ∫ φ (6)φ d xˆ − ( K s + Nˆ T ) ∫ φ ′′φ d xˆ + µ 2 ( K s + Nˆ T ) ∫ φ (4)φ d xˆ 0 0 0 0 ˆ ˆ A11 A11 K + L, γ1 = ˆI 1φ (φ − µ 2φ ′′) d xˆ Iˆ0
∫
0 0
γ2 =
1 1 1 1 Bˆ11 ( ∫ φ ′′ d xˆ − η 2 ∫ φ (4) d xˆ )(∫ φ ′′φ d xˆ − µ 2 ∫ φ (4)φ d xˆ ) 0
0
0
0
1
Iˆ0 ∫ φ (φ − µ 2φ ′′) d xˆ
,
0
1 1 1 1 1 Aˆ11 2 1 (4) ( µ ∫ φ φ d xˆ − ∫ φ ′′φ d xˆ ) ∫ (φ ′) 2 d xˆ + K NL [ ∫ (φ )4 d xˆ − 6µ 2 ∫ φ 2 (φ ′) 2 d xˆ − 3µ 2 ∫ φ 3φ ′′ d ˆx] 0 0 0 0 0 0 γ3 = 2 1 Iˆ0 ∫ φ (φ − µ 2φ ′′) d xˆ 0
1 1 1 1 1 1 η Aˆ11[ ∫ φ ′′′φ ′ d xˆ + ∫ (φ ′′)2 d xˆ ]∫ φ ′′φ d xˆ − µ 2η 2 Aˆ11[ ∫ φ ′′′φ ′ d xˆ + ∫ (φ ′′)2 d xˆ ]∫ φ (4)φ d xˆ 2
+
0
0
0
0
0
1 Iˆ0 ∫ φ (φ − µ 2φ ′′) d xˆ
0
.
0
Eq. (46) is the governing equation referring to the physical neutral surface. If hp=0, Eq. (46) is reduced to the governing equation referring to the geometrical neutral surface.
2.4. Solution procedure
ˆ is supposed to close to the linear Considering the primary resonance, the excitation frequency Ω natural frequency ω0. Then, the relation between the excitation frequency and the linear natural frequency is defined as [63]
16
ˆ = ω + ε 2λ Ω 0
(47)
where λ denotes the detuning parameter and ε represents a small parameter.
Assuming that p(t)=εg(t) and fˆ =ε3f, the governing equation, Eq. (46), can be rewritten as
ˆ tˆ ) g(tˆ ) + γ 1 g (tˆ ) + εγ 2 g 2 (tˆ ) + ε 2 γ 3 g 3 (tˆ ) = ε 2 f cos(Ω
(48)
Using the multiple scale method, g can be expanded as
g ( H 0 , H 1 , H 2 ) = g 0 ( H 0 , H 1 , H 2 ) + ε g1 ( H 0 , H1 , H 2 ) + ε 2 g 2 ( H 0 , H1 , H 2 ) + O (ε 3 )
(49)
where H 0 = tˆ , H1 = ε tˆ , H 2 = ε 2 tˆ .
The operation of time differentiation is given as
d / dt = ∂ / ∂H 0 + ε ∂ / ∂H1 + ε 2 ∂ / ∂H 2 + O (ε 3 ) = D0 + ε D1 + ε 2 D2 + O(ε 3 ) d 2 / dt 2 = d (∂ / ∂H 0 + ε ∂ / ∂H1 + ε 2 ∂ / ∂H 2 + O(ε 3 )) / dt 2 0
2
2 1
(50)
3
= D + 2ε D0 D1 + ε ( D + 2 D0 D2 ) + O(ε )
where Di = ∂ / ∂H i .
Substituting Eqs. (49) and (50) into Eq. (48), we have
ε 0 : D02 g 0 + ω02 g 0 = 0
(51)
ε 1 : D02 g1 + ω02 g1 = −2 D0 D1 g 0 − γ 2 g 02
(52)
ε 2 : D02 g 2 + ω02 g 2 = f cos(ω0 H 0 + λ H 2 ) − 2 D0 D1 g1 − 2 D0 D2 g 0 − D12 g 0 − 2γ 2 g 0 g1 − γ 3 g 03
(53)
The solution of Eq. (51) has the form
(54)
g 0 = R ( H 1 , H 2 )ei ω0 H 0 + R ( H 1 , H 2 )e − i ω0 H0 17
where R denotes an undetermined complex function with respect to H1 and H2. By inserting Eq. (54) into Eq. (52) and eliminating secular terms, we have
(55)
D1 R = 0
Hence, R is independent of H1. The solution of Eq. (52) is given as
g1 =
γ 2 R 2 2 iω H ( e − RR ) + cc ω02 3 0
(56)
0
where cc stands for the complex conjugate of the preceding terms.
Inserting Eq. (54) and Eq. (56) into Eq. (53) and eliminating secular terms, we have
2iω0 R ′ + (3γ 3 −
10γ 22 2 1 ) R R − feiλ H 2 = 0 3ω02 2
(57)
To solve Eq. (57), let R have the following polar form
R=
ae i θ 2
(58)
where θ and a are both real functions with respect to H2. Introducing Eq. (58) into (57) and decomposing into real and imaginary parts, we obtain
a′ =
aθ ′ =
f sin(ψ ) 2ω0
(59)
1 f cos(ψ ) (9γ 3ω02 − 10γ 22 )a 3 − 3 24ω0 2ω0
(60)
where
(61)
ψ = λH2 −θ
Inserting Eq. (61) into Eq. (60) yields 18
a′ =
f sin(ψ ) 2ω0
aψ ′ = λ a −
(62)
1 f cos(ψ ) (9γ 3ω02 − 10γ 22 )a 3 + 24ω03 2ω0
(63)
Considering steady-state conditions a ′ = 0 and ψ ′ = 0 , we have
[λ a −
f 2 1 (9γ 3ω02 − 10γ 22 ) a 3 ]2 = ( ) 24ω03 2ω0
(64)
By now, we obtain the frequency response equation for the response amplitude a as the function of the excitation amplitude f and detuning parameter λ.
3. Numerical Results In this section, the influence of the deviation between the physical and geometrical neutral surfaces on the forced vibration of embedded temperature-dependent PFGM and SFGM nanobeams are studied based on NSGT. The geometric parameters are taken as L=10nm, h=0.5nm, b=2h. The temperature rises gradually from the bottom surface as Tm-T0=5K, where the reference temperature T0 is assumed to be 300K. In order to evaluate accuracy of our model, comparing calculation is made for free vibration [64] and nonlinear free vibration [49] cases. The results given in Fig.4 show that the results predicted by our model have an excellent coincidence with those of Şimşek [49] and Ebrahimi et al. [64], which verifies the accuracy of the present model.
19
Fig. 4. Comparison of (a) the nondimensional natural frequency with different temperature variations; and (b) the nonlinear frequency ratio with different non-homogeneity indexes.
3.1 Effect of the neutral surface deviation due to non-homogeneity index
The effect of the neutral surface deviation due to the non-homogeneity index on frequency response curves of PFGM and SFGM nanobeams are shown in Fig. 5. In the calculation, we have µ=0.25, η=0.25,
∆T=150K, KL=10, Ks=10, KNL=5 and f=0.5. It is seen that the system always presents a hardening effect (bend to the right) when the coefficient of the cubic term (the nonlinear term) in Eq. (48) is positive. For PFGM nanobeams, the hardening-type behavior of the system increases with the decrease of non-homogeneity index. When k is small, see, k=0.2, the effect of the neutral surface derivation could be ignored. But along with increase of non-homogeneity index k, the neutral surface derivation should be considered, which leads to the underestimation of the hardening behavior of the system, that is, the resonance frequency shifts to higher value when physical neutral surface is referred. The underestimation is firstly increased and then decreased with an increase of non-homogeneity index k. For SFGM nanobeams, the hardening behavior of the system is increased with decrease of non-homogeneity index, which is weakened when the physical neutral axis is referred.
20
Fig. 5. Effects of the non-homogeneity index and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. 3.2 Effect of neutral surface derivation due to temperature variation
The effect of the neutral surface deviation due to temperature variation on the frequency response curves of PFGM and SFGM nanobeams is plotted in Fig. 6. In the calculation we have µ=0.25, η=0.25, k=5, KL=10, Ks=10, KNL=5 and f=0.5. It is observed that the frequency responses of PFGM and SFGM nanobeams are temperature-dependent. Along with the increase of temperature variation, the hardening behavior is intensified. The underestimation due to the neutral surface shifting becomes larger with increasing of temperature variation. It is noticed that PFGM nanobeams are more sensitive to the variation of the temperature, whilst SFGM nanobeams are more affected by the neutral surface deviation.
Fig. 6. Effects of temperature variation and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. 21
3.3. Effects of the nonlocal and material length scale parameters
To present the size effect on the forced vibration of PFGM and SFGM nanobeams, the variation of the nonlocal and material length scale parameters, and the neutral surface deviation on the frequency response curves are shown in Figs. 7 and 8, respectively. In the calculation, we have k=5, ∆T=150K, KL=10, Ks=10, KNL=5 and f=0.5. It is observed that increasing the nonlocal parameter strengthens the hardening behavior whilst the inverse trend is observed for material length scale parameter. Larger material length scale parameter leads the resonances of the nanobeams at lower frequencies. It is noticed that when the size effect is ignored, the influence of the neutral surface derivation on the vibration of nanobeams can be ignored for PFGM nanobeams. However, larger nonlocal and material length scale parameters will lead to increased underestimation of resonance frequencies due to the neutral surface shifting, especially for SFGM nanobeams which display significant sensitivity to the variation of the material length scale parameter.
Fig. 7. Effects of the material length scale parameter and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams.
22
Fig. 8. Effects of the nonlocal parameter and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. 3.4. Effect of nonlinear elastic foundation
The effect of nonlinear elastic foundation on the frequency responses of PFGM and SFGM nanobeams are plotted in Fig. 9. Here µ=0.25, η=0.25, k=5, ∆T=150K and f=0.5 are used. As shown in Fig. 9, with the increase of the shear and linear stiffness coefficients, the hardening response of the system is reduced. However, larger nonlinear stiffness coefficient leads the shifting of the resonance frequencies to the higher ones. This indicates that the nonlinearity term is dependent on the nonlinear stiffness and the linear one is related to the linear and shear coefficients. The underestimation due to the neutral surface shifting becomes smaller when stiffness coefficients of the nonlinear elastic foundation are increased, especially for PFGM nanobeams, the effect of the neutral surface deviation could be ignored when the stiffness coefficients of the nonlinear elastic foundation are larger enough.
23
Fig. 9. Effects of nonlinear elastic foundation and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams.
4. Conclusions
In this paper, the forced vibrations of temperature-dependent PFGM and SFGM nanobeams resting on a nonlinear elastic foundation are investigated by using the nonlocal strain gradient theory. Referring to the physical neutral surface, the governing equation of motion is derived based on Euler-Bernoulli beam 24
theory together with the von-Kármán’s geometric nonlinearity. Closed-form solution is obtained for the forced vibration by using multiple time scale method. The influences of the neutral surface deviation due to the material gradient and temperature sensitivity, the size effects, as well as the nonlinear elastic foundation, on the nonlinear resonances of nanobeams are discussed in detail. Summarizing results above, we conclude that
(1) For a PFGM or SFGM embedded nanobeam, along with the increase of the non-homogeneity index, material length scale parameter, shear and linear foundation stiffness coefficients, the resonance frequency of the nanobeam will shift to lower ones. However, the increase of the temperature variation, nonlocal parameter and nonlinear foundation stiffness coefficient will strengthen the hardening responses of the nanobeam. The larger the excitation amplitude is, the wider the resonant region is.
(2) If the neutral surface shifting due to the material gradient and temperature variation is ignored, the hardening behavior of the beam will be underestimated. The increase of the non-homogeneity index and temperature variation strengthens this underestimation. It is noticed that stronger size effects and weaker nonlinear foundation will amplify the effects of the neutral surface derivation, especially when the effect of the higher order strain field is dominated (larger material length scale parameter). It should be pointed out that the resonance of a SFGM nanobeam is more affected by the position of the neutral surface. When the size effect and temperature variation are not considered, the influence of the neutral surface shifting could be ignored for a PFGM nanobeam, which is insensitive to the stiffness of nonlinear foundation.
These findings are meaning for the mechanical analysis and design of temperature-dependent FG nanobeams.
25
Acknowledgements
The second author thanks the support from the Fundamental Research Funds for the Central Universities of China (2014JBZ014). Supports from the National Natural Science Foundation of China (11772044), and National Basic Research Program of China (973Program) (2015CB057800), are acknowledged.
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33
LIST OF FIGURES AND TABLE CAPTIONS
Fig. 1. Geometry of a temperature-dependent functionally graded nanobeam resting on the nonlinear elastic foundation subjected to harmonic forces. Fig. 2. The effects of non-homogeneity index k and temperature variation ∆T on the neutral surface derivation. Fig. 3. The variation of Young’s modulus through the thickness direction for PFGM and SFGM nanobeams with different temperature variation ∆T.
Fig. 4. Comparison of (a) the nondimensional natural frequency with different temperature variations; and (b) the nonlinear frequency ratio with different non-homogeneity indexes. Fig. 5. Effects of the non-homogeneity index and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. Fig. 6. Effects of temperature variation and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. Fig. 7. Effects of the material length scale parameter and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. Fig. 8. Effects of the nonlocal parameter and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. Fig. 9. Effects of nonlinear elastic foundation and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. Table 1: Temperature-dependent coefficients for ceramic and metals
34
Fig. 1. Geometry of a temperature-dependent functionally graded nanobeam resting on the nonlinear elastic foundation subjected to harmonic forces.
35
Fig. 2. The effects of non-homogeneity index k and temperature variation ∆T on the derivation between the physical neutral surfaces.
36
Fig. 3. The variation of Young’s modulus through the thickness direction of PFGM and SFGM nanobeams with different temperature variation ∆T .
37
Fig. 4. Comparison of (a) the nondimensional natural frequency with different temperature variations; and (b) the nonlinear frequency ratio with different non-homogeneity indexes.
38
Fig. 5. Effects of the non-homogeneity index and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams.
39
Fig. 6. Effects of temperature variation and neutral surface deviation on the frequency response curves of 40
PFGM and SFGM nanobeams.
41
Fig. 7. Effects of the material length scale parameter and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams.
42
Fig. 8. Effects of the nonlocal parameter and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams.
43
Fig. 9. Effects of nonlinear elastic foundation and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams.
44
Table 1 Temperature-dependent coefficients for ceramic and metals
Material
Properties
P-1
P0
P1
P2
P3
SUS304
E(Pa)
0
201.04e+9
3.079e-4
-6.534e-7
0
ρ(kg/m3)
0
8166
0
0
0
α(K-1)
0
12.33e-6
8.086e-4
0
0
υ
0
0.3262
-2.002e-4
3.797e-7
0
E(Pa)
0
348.43e+9
-3.07e-4
2.160e-7
-8.946e-11
ρ(kg/m3)
0
2370
0
0
0
α(K-1)
0
5.8723e-6
9.095e-4
0
0
υ
0
0.24
0
0
0
Si3N4
45
S0263-8223(17)30095-8 https://doi.org/10.1016/j.compstruct.2017.10.058 COST 9033
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
11 January 2017 13 October 2017 18 October 2017
Please cite this article as: Tang, Y-g., Liu, Y., Zhao, D., Effects of neutral surface deviation on nonlinear resonance of embedded temperature-dependent functionally graded nanobeams, Composite Structures (2017), doi: https:// doi.org/10.1016/j.compstruct.2017.10.058
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Effects of neutral surface deviation on nonlinear resonance of embedded temperature-dependent functionally graded nanobeams
Yu-gang Tang, Ying Liu*, Dong Zhao
Department of Mechanics, School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, PR China *Email: [email protected]; Tel: 86-10-51688763; Fax: 86-10-51682094
Abstract
In this paper, forced vibration of functionally graded (FG) nanobeams resting on the nonlinear elastic foundations are investigated based on the nonlocal strain gradient theory. The material parameters of FG nanobeams are assumed to be temperature-dependent and change continuously along the thickness direction according to the power-law function (PFGM) or sigmoid function (SFGM). Based on the Euler–Bernoulli beam theory and von-Kármán geometric nonlinearity, the governing equations of motion are derived by considering the deviation between the geometrical and physical neutral surfaces. Closed-form approximate solution for nonlinear forced vibration of a FG nanobeam is derived by using multiple time scale method. The results show that decrease of non-homogeneity index and material length scale parameter, or increase of temperature variation and nonlocal parameter will increase the resonance frequencies of FG nanobeams. The effect of the in-coincidence of physical and geometrical neutral surfaces on the nonlinear resonance of the nanobeams could not be ignored, especially for SFGM nanobeams with larger non-homogeneity index and stronger size effects, embedded in a softer medium
1
with obvious temperature variation.
Keywords: Functionally graded nanobeam; Nonlocal strain gradient theory; Physical neutral surface; Nonlinear forced vibration; Temperature dependent
1. Introduction
Composites have been widely used in different fields and their mechanical properties are always the attention of scientists [1-10]. As one kind of new improved composite materials that has smooth and continuous material parameter variation throughout certain dimension(s), functionally graded material (FGM) has attracted great attention due to their exceptional mechanical and thermal properties such as higher fracture toughness, enhanced thermal resistance and improved stress spreading, which promise prospective applications in micro/nanoelectromechanical systems (MEMS/NEMS) [11-16].
Both experiments and simulations have showed that size effects play a significant role on the performance of small-size structures [17-21]. Due to the limitation of classical elasticity theory (CET) in description size effects, several non-classical continuum theories such as nonlocal elasticity theory (NET) [22] and strain gradient theory (SGT) [23, 24] have been proposed to determine the size-dependent deformation of small-size structures. In contrast to the classical continuum theory, the nonlocal elasticity theory [22] considers the size effect by assuming that the stress at a reference point is a function of strains at all points in the body. Numerous works has been done based on NET to consider size effects of nanostructures [25-27]. Results show that although NET is suitable to capture softening stiffness, the capability in prediction the stiffness enhancement observed in the experimental or theoretical studies is weak. Strain gradient theory (SGT) considers the effects of stiffness enhancement. Based on SGT, Akgöz 2
and Civalek [28] investigated the static bending and free vibration behavior of simply supported microbeams. Ghorbanpour Arani et al. [29] analyzed the nonlinear vibration of a piezoelectric nanobeam. Ghayesh et al. [30] investigated the nonlinear forced vibration of a microbeam. More recently, Thai et al. [31] presented a review of continuum mechanics models for size effects on the mechanical response of beams and plates and also pointed out that NET could only predicted the softening stiffness, while SGT was good at prediction the stiffness enhancement. To address this issue, by combining NET and SGT together, Lim et al. [32] developed a nonlocal strain gradient theory (NSGT) by considering both of the nonlocal parameter and the material gradient length scale parameter, which can well predict the stiffness softening as well as stiffness enhancement simultaneously. By adopting NSGT, Tang et al. [33, 34], and Li et al. [35] investigated the size effects on the wave propagation in viscoelastic SWCNTs. Li et al. [36] developed a size-dependent model to study the buckling analysis of nonlinear Euler-Bernoulli beam. It is seen that more and more works have been done based on NSGT to reflect size effects of micro/nano-structures.
As basic elements in engineering application, mechanical performances of FG structures (beam, plate, etc.) are always the frontier [37-40]. For a FG beam, based on the classical continuum theory, Bourada et al. [41] performed an investigation on the bending and vibration of FG trigonometric higher-order beams including the shear deformation effect. Calim [42] discussed the free and forced vibrations of axially FG Timoshenko beams embedded in the viscoelastic foundation. Considering temperature variation, Ansari et al. [43] studied nonlinear forced vibration of FG nanobeams based on surface elasticity theory. Uymaz [44] investigated the forced vibration of FG beams based on NET. Niknam and Aghdam [45] investigated the large amplitude free vibration and buckling of a nonlocal FG beam resting on nonlinear elastic 3
foundation by adopting a semi analytical approach. El-Borgi et al. [46] explored the nonlocal effect on the free and forced vibrations of FG Euler–Bernoulli nanobeams resting on a nonlinear elastic foundation. Bouafia et al. [47] analyzed the bending and free flexural vibration behaviors of FG nanobeams based on a nonlocal quasi-3D theory. Thai et al. [48] investigated size effects on the static bending, buckling and free vibration behaviors of FG sandwich microbeams based on the modified couple stress theory. Based on NSGT, Şimşek [49] developed a novel size-dependent beam model to analyze the nonlinear free vibration of FG nanobeams. Trinh et al. [50] discussed the vibration and buckling behaviors of PFGM beams under mechanical and thermal loads based on a high-order beam theory. Ebrahimi and Salari [51] studied the thermal effects on the buckling and free vibration of PFGM nanobeams based on the nonlocal Timoshenko beam model. Ghadiri and Shafiei [52] explored thermal vibration of rotating PFGM microbeams based on modified couple stress theory. Referring to the modified SGT, Shenas et al. [53] investigated the vibration characteristics of rotating pre-twisted PFGM microbeams in the thermal environment. Ebrahimi and Barati [54] developed a quasi-3D beam model to study wave propagation in PFGM nanobeams in thermal environments based on NSGT.
It should be pointed out that in the above discussion, the geometrical symmetrical axis of the beams were taken as the neutral surface in the equation establishment. In fact, for a FG structure, the gradient distribution of materials in space definitely causes the deviation of the geometrical neutral axis from the physical one, which has attracted more and more attention in recent years. Bellifa et al. [55] proposed a simple shear deformation theory to study bending and dynamic behaviors of PFGM plates with the neutral surface position concept. Eltaher et al. [56] investigated the free vibration of FG macro/nanobeams based on the nonlocal continuum model considering the physical neutral axis. Barretta 4
et al. [57] developed a new Eringen gradient elasticity model to study the bending behaviors of FG Timoshenko nanobeams based on the physical neutral surface. Ahouel et al. [58] proposed a nonlocal trigonometric shear deformation beam theory to examine the bending, buckling, and vibration behaviors of PFGM nanobeams including neutral surface position concept. On the basis of the modified couple stress theory and the neutral surface, Al-Basyouni et al. [59] presented the size-dependent bending and vibration of PFGM microbeams. Ebrahimi and Barati [60] investigated the vibration characteristics of hydro-thermally affected PFGM viscoelastic nanobeams embedded in viscoelastic foundation based on NSGT. However, to per author’s best knowledge, the influence of the physical neutral surface derivation from geometrical one due to the temperature-dependent material gradient on the nonlinear resonance of FG nanobeams has not yet been fully addressed, especially when the nonlocal stress and higher order strain field are both considered.
The objective of the present manuscript is to study the forced vibration of temperature-dependent PFGM and SFGM nanobeams resting on the nonlinear elastic foundation by adopting NSGT. Firstly, referring to physical neutral surface, governing equations of motion are derived based on Euler–Bernoulli beam theory together with von-Kármán’s geometric nonlinearity. Then the closed-form approximate solution for the nonlinear forced vibration is presented based on the multiple time scale method. The influence of the neutral surface deviation caused by the temperature-dependent gradient, small scale parameters, as well as force amplitude and stiffness of the nonlinear foundation, on the nonlinear resonance of FG nanobeams is clarified. At last, the conclusion is given.
2. Mathematical formulation
5
2.1. Functionally graded materials
An FG nanobeam with length L, width b, and thickness h resting on the nonlinear elastic foundation is shown in Fig. 1. Following Şimşek [49], the nanobeam has immovable simple supports at both ends. The FG nanobeam is assumed to be made of ceramic and metals, which are temperature-dependent [61]. Then, the material parameter P, such as Young’s modulus E, thermal expansion α, Poisson's ratio υ and mass density ρ, are functions of temperature and with the form [61]
(1)
P = P0 ( P− 1T − 1 + 1 + P1T + P2 T 2 + P3 T 3 )
Here P0, P-1, P1, P2, P3 are temperature-dependent coefficients.
Fig. 1. Geometry of a temperature-dependent functionally graded nanobeam resting on the nonlinear elastic foundation subjected to harmonic forces.
Because the material parameters are gradient distributed, the physical neutral surface no longer coincides with the geometrical one. Two different coordinates, x0zm and x’0’zp, which are defined at the geometrical and physical neutral surfaces, respectively, are introduced and presented in Fig. 1. Then points in the two coordinates have the relation
zm = z p + hp ,
where h p = ∫
h/2
− h/ 2
(2)
x = x'
E ( zm , T ) zm dz m
∫
h/2
− h/ 2
E ( zm , T ) dz m is the derivation between the physical neutral surface
6
and the geometrical one.
Assume that the effective material properties vary continuously along the thickness direction according to power-law (PFGM) or sigmoid (SFGM) functions [62]. Referring to Fig. 1, the volume fraction of PFGM is defined as
V1 = (
z p + hp h
1 + )k 2
h h z p ∈ [ − − hp , − h p ] 2 2
(3)
where k is the non-homogeneity index. Then, the material parameters for PFGM along the thickness direction are expressed as Peff ( z p , T ) = Pc (T )V1 + Pm (T )(1 − V1 ) = Pc (T )(
z p + hp h
(4)
z p + hp 1 k 1 + )k + Pm (T )[1 − ( + ) ] 2 2 h
where Pc and Pm denote the temperature-dependent material parameters of ceramic and metal, respectively.
In the same way, the material parameters for SFGM are expressed as
Peff ( z p , T ) = Pc (T )V2 + Pm (T )(1 − V2 ) =
Pc (T ) 2( z p + h p ) 1 2( z p + hp ) h [ + 1]k + Pm (T ){1 − [ + 1]k } z p ∈ [− − hp , −h p ] 2 h 2 h 2
(5)
Peff ( z p , T ) = Pc (T )V3 + Pm (T )(1 − V3 ) 2( z p + h p ) k 2( z p + h p ) k P (T ) 1 ] }+ m [1 − ] = Pc (T ){1 − [1 − 2 h 2 h
h z p ∈ [ − hp , − h p ] 2
(6)
where the volume fraction is defined as 1 2( z p + hp ) h V2 = [ z p ∈ [− − hp , − hp ]; + 1]k , 2 h 2 2( z p + h p ) k 1 h ] , z p ∈ [ −h p , − hp ] V3 = 1 − [1 − 2 h 2
7
(7)
Eqs. (4) to (6) indicate that the non-homogeneity index and temperature variation have determinant effects on the derivation between the physical and geometrical neutral surfaces.
Assume that the temperature T varies linearly through the thickness direction as
1 z p + hp ) T = Tm + ∆T ( + 2 h
(8)
where ∆T=Tc-Tm, with Tc and Tm the temperatures of the top (full-ceramic) and the bottom surfaces (full-metal), respectively. Fig. 2 gives the variation of the neutral surface derivation with respect to the non-homogeneity index and temperature variation. The bottom surface temperature Tm is set as 300K and temperature-dependent coefficients P0, P-1, P1, P2, P3 are given in Table 1. Table 1 Temperature-dependent coefficients for ceramic and metals [51, 61].
Material
Properties
P-1
P0
P1
P2
P3
SUS304
E(Pa)
0
201.04e+9
3.079e-4
-6.534e-7
0
ρ(kg/m 3)
0
8166
0
0
0
α(K-1)
0
12.33e-6
8.086e-4
0
0
υ
0
0.3262
-2.002e-4
3.797e-7
0
E(Pa)
0
348.43e+9
-3.07e-4
2.160e-7
-8.946e-11
ρ(kg/m 3)
0
2370
0
0
0
α(K-1)
0
5.8723e-6
9.095e-4
0
0
υ
0
0.24
0
0
0
Si3N4
For PFGM nanobeams, along with the increase of the non-homogeneity index k, the physical neutral 8
surface firstly moves away from the geometrical one, and then moves back again when k is greater than a certain value (see, k=2 in the present discussion). For SFGM nanobeams, the physical neutral surface always moves away from the geometrical one when the non-homogeneity index is smaller than a certain value (see, k=4 in the present discussion), after which the neutral surface derivation is insensitive to the non-homogeneity index anymore. Along with the increase of the temperature variation, the deviation is reduced.
Fig. 2. The effects of non-homogeneity index k and temperature variation ∆T on the neutral surface derivation.
The variation of Young’s modulus along thickness direction for PFGM and SFGM nanobeams with respect to temperature variation is plotted in Fig. 3, in which black and red lines are corresponding to
∆T=0K and ∆T=300K, respectively. It is seen that the Young’s modulus of PFGM and SFGM nanobeams are reduced along with the increase of the temperature variation. Because the physical neutral surface no longer coincides with the geometrical one, and the deviation is temperature dependent, the Young’s moduli of SFGM nanobeams with different k are no longer equal to each other at zp=0 and display nonsymmetrical variation along the beam thickness direction.
9
Fig. 3. The variation of Young’s modulus through the thickness direction for PFGM and SFGM nanobeams with different temperature variation ∆T.
2.2. Nonlocal strain gradient theory
According to the nonlocal strain gradient theory developed by Lim et al. [32], the total stress ςxx , the classical nonlocal stress σxx(0) and the higher-order stress σxx(1) are defined as
(12)
ς xx = σ xx( 0 ) − ∇ σ xx(1)
L
σ xx(0) = E ( z p ) ∫ β 0 ( x, x ′, e0 a ) ε xx′ ( x ′)dx′
(13)
0
L
σ xx(1) = l 2 E ( z p ) ∫ β1 ( x, x ′, e1a ) ε xx′ , x ( x ′)dx ′
(14)
0
where ∇ = ∂ / ∂ x . e0a and e1a, which are named as nonlocal parameters, are introduced to determine the significance of lower and higher order nonlocal elastic stress fields, respectively. l, which is named as material length scale parameter, is introduced to determine the significance of higher-order strain gradient field. εxx and ε xx′ are the strains at point x and neighboring point x′ , respectively. Correspondingly, ′ , x = ∂ε xx ′ / ∂x are the strain gradient at point x and neighboring point x′ , respectively. ε xx , x = ∂ε xx / ∂x and ε xx
Following the assumptions given by Lim et al. [32] that if the nonlocal attenuation functions β0(x, x ′ , e0a) and β1(x, x′ , e1a) meet the conditions given by Eringen [22] and e=e0=e1, the nonlocal strain gradient 10
constitutive relation is with the form
[1 − (ea) 2 ∇ 2 ]ζ xx = E ( z p )(1 − l 2 ∇2 )ε xx
(15a)
where the Laplacian operator ∇ 2 is equal to ∂ 2 / ∂ x 2 . Then, Eq. (15a) can be rewritten as
[1 − (ea ) 2
2 ∂2 2 ∂ ] ζ ( )(1 )ε xx = E z − l xx p ∂x 2 ∂x 2
(15b)
Eq. (15b) can be reduced to the nonlocal constitutive equation without taking into consideration of the material length scale parameter (l=0), that is
[1 − (ea ) 2
∂2 ]ζ xx = E ( z p )ε xx ∂x 2
(16)
When the nonlocal parameter ea=0, Eq. (15b) is reduced to the constitutive equation based on the strain gradient theory, that is
ζ xx = E ( z p )(1 − l 2
∂2 )ε xx ∂x 2
(17)
2.3. The governing equation of FG nanobeams
For an Euler–Bernoulli beam, the von Kármán type nonlinear strain-displacement relation is expressed as
ε xx =
∂u 1 ∂w 2 ∂2 w + ( ) − zp 2 ∂x ∂x 2 ∂x
(18)
where u is the axial displacement and w the transverse displacement.
The first variation of the strain energy is expressed as
11
L
L
A
0
δUm = ∫ (σ xx(0)δε xx + σ xx(1) ∇δε xx )dV = ∫ (σ xx(0) −∇σ xx(1) ) δε xx dV + ∫σ xx(1)δε xx dA = ∫ ζ xxδε xx dV + ∫ σ xx(1)δε xx dA V
A
V
0
V L
∂δ u ∂w ∂δ w ∂ δw ∂u ∂w ∂δ w ∂ δ w = ∫ [N( + ) − M 2 ]dx + N (1) (δ + ) − M (1) 0 ∂x ∂x ∂x ∂x ∂ x ∂ x ∂ x ∂x2 0 2
L
2
(19)
where
N = ∫ ζ xx dA, A
M = ∫ z pζ xx dA A
N (1) = ∫ σ xx(1) dA A
M (1) = ∫ z pσ xx(1) dA A
(20)
Substituting Eqs. (15) and (18) into Eq. (20), we have
N − ( ea )
2
M − ( ea )
∂2 N ∂ 2 ∂u 1 ∂w ∂2 ∂ 2w = A11 (1 − l 2 2 )[ + ( ) 2 ] − B11 (1 − l 2 2 ) 2 2 ∂x ∂x ∂x 2 ∂x ∂x ∂x 2
(21)
∂2 M ∂ 2 ∂u 1 ∂w ∂2 ∂ 2 w = B11 (1 − l 2 2 )[ + ( ) 2 ] − D11 (1 − l 2 2 ) 2 2 ∂x ∂x ∂x 2 ∂x ∂x ∂x
(22)
where
A11 = ∫ E ( z p )dA = b ∫
h / 2 − hp
− h / 2 − hp
A
D11 = ∫ z 2p E ( z p )dA = b ∫
E ( z p )dz p
h / 2 − hp
− h / 2 − hp
A
B11 = ∫ z p E ( z p )dA = b ∫
h / 2− hp
− h / 2 − hp
A
z p E ( z p )dz p
(23)
z 2p E ( z p )dz p
The first variation of the additional strain energy induced by thermal loading can be expressed as
L
δ U ad = ∫ NT 0
∂w ∂δ w dx ∂x ∂x
(24)
where
NT = − ∫ E ( z p )α ( z p )[T ( z p ) - T0 ]dA = −b ∫ A
h / 2 − hp
− h / 2− hp
E ( z p )α ( z p )[T ( z p ) - T0 ]dz p
with T0 the reference temperature.
The first variation of the kinetic energy is expressed as [48]
12
(25)
L
δ S = ∫ Im 0
L ∂u ∂δ u ∂w ∂δ w dx + ∫ I m dx 0 ∂t ∂t ∂t ∂t
(26)
where the mass moment of inertia Im is given as
I m = ∫ ρ ( z p )dA = b ∫
h / 2 − hp
− h / 2 − hp
A
(27)
ρ ( z p )dz p
The first variation of the work done by nonlinear elastic foundation and harmonic force is expressed as
L
δ W = ∫ −kL wδ w − ks 0
∂w ∂δ w − k NL w3δ w + F cos(Ω t)δ wdx ∂x ∂x
(28)
Here kL, ks and kNL are, respectively, linear, shear, and nonlinear coefficients of the nonlinear elastic foundation. F denotes the excitation amplitude.
Using Hamilton’s principle, we have
t
∫ δ (U 0
m
(29)
+ U ad − S − W )dt = 0
Then, the equations of motion are derived as
δu :
δw:
∂N ∂ 2u = Im 2 ∂t ∂x
(30)
∂2M ∂ ∂w ∂2 w ∂2w ∂2 w 3 + ( N ) − k w + k − k w + N + F cos( Ω t) = I L s NL T m ∂x 2 ∂x ∂x 2 ∂x 2 ∂t 2 ∂x
(31)
Moreover, the corresponding boundary conditions (x=0 or x=L) are expressed as
δ u : N = 0 or u = 0
(32)
∂u : N (1) = 0 or ∂x
(33)
δ
δw:
∂u =0 ∂x
∂M ∂w ∂w ∂w +N + ks + NT = 0 or ∂x ∂x ∂x ∂x
w=0
13
(34)
δ
∂w ∂w : M = N (1) ∂x ∂x
δ
∂2 w : M (1) = 0 or ∂x 2
∂w =0 ∂x
or
(35)
∂2 w =0 ∂x 2
(36)
Ignoring the axial inertial [49], Eq. (30) gives
∂N =0 ∂x
(37)
By combining Eqs. (21) and (37), we can obtain the axial force N as
N = N c = A11 (1 − l 2
∂ 2 ∂u 1 ∂w 2 ∂2 ∂2 w )[ + ( ) ] − B11 (1 − l 2 2 ) 2 2 ∂x ∂x 2 ∂x ∂x ∂x
(38)
where Nc denotes a constant value. For a nanobeam with immovable ends, the boundary conditions can be obtained as [49]
u (0, t ) = u ( L , t ) =
∂ 2 u (0, t ) ∂ 2 u ( L , t ) = =0 ∂ x2 ∂ x2
(39)
It is worth to note that the boundary conditions related to the second derivative of the axial displacement are due to the strain gradient theory [23]. By integrating each side of Eq. (38) over the domain [0, L] and using the above boundary conditions, we have
N=
A11 2L
∫
L
0
(
A l2 ∂w 2 ) dx − 11 ∂x L
∫
L
0
[
2 B11 L ∂w ∂ 3 w ∂ 2 w 2 ∂2w 2 ∂ dx l dx + ( ) ] − (1 − ) ∂x ∂x3 ∂x 2 L ∫0 ∂x 2 ∂x 2
(40)
By combining Eqs. (22), (31), and (40), we can obtain the moment M as
M =(
B112 ∂ 2 w ∂ 2 w NB − D11 )(1 − l 2 2 ) 2 + 11 A11 ∂x ∂x A11
∂2 w ∂2 w + (ea ) [ I m 2 + k L w − (ks + NT + N ) 2 + k NL w3 − F cos(Ω t)] ∂t ∂x
(41)
2
Inserting Eqs. (40) and (41) into Eq. (31), we obtain the governing equation of motion for the nonlocal strain gradient Euler–Bernoulli beam, that is, 14
( D11 −
2 B112 ∂2 ∂4w ∂2w ∂2 w ∂2 w 2 ∂ )(1 − l 2 2 ) 4 + [1 − ( ea ) ][ I m 2 + k L w − k s 2 + k NL w3 − N T ] 2 A11 ∂x ∂x ∂x ∂t ∂x ∂x 2
4 ∂2w A L ∂w A l2 2 ∂ w ]{ 11 ∫ ( ) 2 dx − 11 − ( ea ) 2 4 ∂x ∂x 2 L 0 ∂x L = F cos(Ω t)
−[
∫
L
0
[
∂w ∂ 3 w ∂ 2 w 2 B + ( 2 ) ]dx − 11 ∂x ∂x 3 ∂x L
∫
L
0
(1 − l 2
∂2 ∂2w ) dx} ∂x 2 ∂x 2
(42)
For the sake of simplification, the following dimensionless quantities are defined, that is,
xˆ =
x L
KL =
wˆ = kL L4 Em I
w r= r Ks =
ks L2 Em I
I A
zˆ =
zp h
tˆ = t
N L2 Nˆ T = T Em I
Em 0 I
µ=
ρm AL4
K NL =
kNL r 2 L4 Em 0 I
A B r D11 FL4 ( Aˆ11 , Bˆ11 , Dˆ 11 ) = ( 11 , 11 ) Fˆ = Em 0 A Em 0 I Em 0 I rEm 0 I
ea l η= L L ˆ =Ω/ Ω
Em 0 I
ρm0 AL4
(43)
I Iˆ0 = m ρm A
where A=bh and I=bh3/12 are the area and moment of inertia of the cross section of the beam, respectively. Em0 is equal to Young's modulus of the metal at T=0K.
Then, the governing equation is rewritten in the following dimensionless form as
( Dˆ11 −
ˆ Bˆ112 ∂ 2 ∂ 4 wˆ ∂wˆ ∂2 w )(1 −η 2 2 ) 4 + K NL wˆ 3 − K NL µ 2 [6wˆ ( )2 + 3wˆ 2 2 ] ∂xˆ ∂xˆ ∂xˆ ∂xˆ Aˆ11
3 2 4 ˆ ˆ 2 ˆ ˆ ∂ 2 ∂ 2 wˆ Aˆ11 1 ∂wˆ 2 ∂2w 2 ∂ w ˆ η 2 1[ ∂wˆ ∂ w ˆ 1( ∂ w ˆ ˆ ) { ( ) dx A ( ) ] dx B )dxˆ} − + − − η 11 11 ∫0 2 2 3 2 2 ∫ ∫ 0 0 ∂xˆ ∂xˆ 2 ∂xˆ ∂xˆ ∂xˆ ∂xˆ ∂xˆ ∂xˆ 4 ∂2 ∂ 2 wˆ ∂2 wˆ ∂2 wˆ ˆ tˆ) +(1 − µ 2 2 )( Iˆ0 2 + K L wˆ − K s 2 − Nˆ T 2 ) = Fˆ cos(Ω ∂xˆ ∂tˆ ∂xˆ ∂xˆ
−(1 − µ 2
(44)
The Galerkin method is adopted to degenerate the above partial differential equation into a time-dependent ordinary differential equation. The transverse displacement can be assumed as the following form
wˆ ( xˆ , tˆ) = p (tˆ)φ ( xˆ ), φ ( xˆ ) = sin(π xˆ )
(45)
in which φ ( xˆ ) denotes a trial function which meets the boundary conditions and p (tˆ ) stands for an undetermined time-dependent function. Inserting Eq. (45) into Eq. (44), multiplying each side of Eq. (44) 15
with the trial function φ ( xˆ ) and finally integrating the resulting equation from 0 to 1, we can obtain a second-order ordinary differential equation as
ˆ tˆ ) p (tˆ) + γ 1 p (tˆ ) + γ 2 p 2 (tˆ ) + γ 3 p 3 (tˆ ) = fˆ cos(Ω
(46)
where
fˆ =
1 Fˆ ∫ φ d xˆ 0
1 Iˆ0 ∫ φ (φ − µ 2φ ′′) d xˆ
,
0
1 1 Bˆ 2 1 Bˆ 2 1 ( Dˆ11 − 11 )∫ φ (4)φ d xˆ − η 2 ( Dˆ11 − 11 ) ∫ φ (6)φ d xˆ − ( K s + Nˆ T ) ∫ φ ′′φ d xˆ + µ 2 ( K s + Nˆ T ) ∫ φ (4)φ d xˆ 0 0 0 0 ˆ ˆ A11 A11 K + L, γ1 = ˆI 1φ (φ − µ 2φ ′′) d xˆ Iˆ0
∫
0 0
γ2 =
1 1 1 1 Bˆ11 ( ∫ φ ′′ d xˆ − η 2 ∫ φ (4) d xˆ )(∫ φ ′′φ d xˆ − µ 2 ∫ φ (4)φ d xˆ ) 0
0
0
0
1
Iˆ0 ∫ φ (φ − µ 2φ ′′) d xˆ
,
0
1 1 1 1 1 Aˆ11 2 1 (4) ( µ ∫ φ φ d xˆ − ∫ φ ′′φ d xˆ ) ∫ (φ ′) 2 d xˆ + K NL [ ∫ (φ )4 d xˆ − 6µ 2 ∫ φ 2 (φ ′) 2 d xˆ − 3µ 2 ∫ φ 3φ ′′ d ˆx] 0 0 0 0 0 0 γ3 = 2 1 Iˆ0 ∫ φ (φ − µ 2φ ′′) d xˆ 0
1 1 1 1 1 1 η Aˆ11[ ∫ φ ′′′φ ′ d xˆ + ∫ (φ ′′)2 d xˆ ]∫ φ ′′φ d xˆ − µ 2η 2 Aˆ11[ ∫ φ ′′′φ ′ d xˆ + ∫ (φ ′′)2 d xˆ ]∫ φ (4)φ d xˆ 2
+
0
0
0
0
0
1 Iˆ0 ∫ φ (φ − µ 2φ ′′) d xˆ
0
.
0
Eq. (46) is the governing equation referring to the physical neutral surface. If hp=0, Eq. (46) is reduced to the governing equation referring to the geometrical neutral surface.
2.4. Solution procedure
ˆ is supposed to close to the linear Considering the primary resonance, the excitation frequency Ω natural frequency ω0. Then, the relation between the excitation frequency and the linear natural frequency is defined as [63]
16
ˆ = ω + ε 2λ Ω 0
(47)
where λ denotes the detuning parameter and ε represents a small parameter.
Assuming that p(t)=εg(t) and fˆ =ε3f, the governing equation, Eq. (46), can be rewritten as
ˆ tˆ ) g(tˆ ) + γ 1 g (tˆ ) + εγ 2 g 2 (tˆ ) + ε 2 γ 3 g 3 (tˆ ) = ε 2 f cos(Ω
(48)
Using the multiple scale method, g can be expanded as
g ( H 0 , H 1 , H 2 ) = g 0 ( H 0 , H 1 , H 2 ) + ε g1 ( H 0 , H1 , H 2 ) + ε 2 g 2 ( H 0 , H1 , H 2 ) + O (ε 3 )
(49)
where H 0 = tˆ , H1 = ε tˆ , H 2 = ε 2 tˆ .
The operation of time differentiation is given as
d / dt = ∂ / ∂H 0 + ε ∂ / ∂H1 + ε 2 ∂ / ∂H 2 + O (ε 3 ) = D0 + ε D1 + ε 2 D2 + O(ε 3 ) d 2 / dt 2 = d (∂ / ∂H 0 + ε ∂ / ∂H1 + ε 2 ∂ / ∂H 2 + O(ε 3 )) / dt 2 0
2
2 1
(50)
3
= D + 2ε D0 D1 + ε ( D + 2 D0 D2 ) + O(ε )
where Di = ∂ / ∂H i .
Substituting Eqs. (49) and (50) into Eq. (48), we have
ε 0 : D02 g 0 + ω02 g 0 = 0
(51)
ε 1 : D02 g1 + ω02 g1 = −2 D0 D1 g 0 − γ 2 g 02
(52)
ε 2 : D02 g 2 + ω02 g 2 = f cos(ω0 H 0 + λ H 2 ) − 2 D0 D1 g1 − 2 D0 D2 g 0 − D12 g 0 − 2γ 2 g 0 g1 − γ 3 g 03
(53)
The solution of Eq. (51) has the form
(54)
g 0 = R ( H 1 , H 2 )ei ω0 H 0 + R ( H 1 , H 2 )e − i ω0 H0 17
where R denotes an undetermined complex function with respect to H1 and H2. By inserting Eq. (54) into Eq. (52) and eliminating secular terms, we have
(55)
D1 R = 0
Hence, R is independent of H1. The solution of Eq. (52) is given as
g1 =
γ 2 R 2 2 iω H ( e − RR ) + cc ω02 3 0
(56)
0
where cc stands for the complex conjugate of the preceding terms.
Inserting Eq. (54) and Eq. (56) into Eq. (53) and eliminating secular terms, we have
2iω0 R ′ + (3γ 3 −
10γ 22 2 1 ) R R − feiλ H 2 = 0 3ω02 2
(57)
To solve Eq. (57), let R have the following polar form
R=
ae i θ 2
(58)
where θ and a are both real functions with respect to H2. Introducing Eq. (58) into (57) and decomposing into real and imaginary parts, we obtain
a′ =
aθ ′ =
f sin(ψ ) 2ω0
(59)
1 f cos(ψ ) (9γ 3ω02 − 10γ 22 )a 3 − 3 24ω0 2ω0
(60)
where
(61)
ψ = λH2 −θ
Inserting Eq. (61) into Eq. (60) yields 18
a′ =
f sin(ψ ) 2ω0
aψ ′ = λ a −
(62)
1 f cos(ψ ) (9γ 3ω02 − 10γ 22 )a 3 + 24ω03 2ω0
(63)
Considering steady-state conditions a ′ = 0 and ψ ′ = 0 , we have
[λ a −
f 2 1 (9γ 3ω02 − 10γ 22 ) a 3 ]2 = ( ) 24ω03 2ω0
(64)
By now, we obtain the frequency response equation for the response amplitude a as the function of the excitation amplitude f and detuning parameter λ.
3. Numerical Results In this section, the influence of the deviation between the physical and geometrical neutral surfaces on the forced vibration of embedded temperature-dependent PFGM and SFGM nanobeams are studied based on NSGT. The geometric parameters are taken as L=10nm, h=0.5nm, b=2h. The temperature rises gradually from the bottom surface as Tm-T0=5K, where the reference temperature T0 is assumed to be 300K. In order to evaluate accuracy of our model, comparing calculation is made for free vibration [64] and nonlinear free vibration [49] cases. The results given in Fig.4 show that the results predicted by our model have an excellent coincidence with those of Şimşek [49] and Ebrahimi et al. [64], which verifies the accuracy of the present model.
19
Fig. 4. Comparison of (a) the nondimensional natural frequency with different temperature variations; and (b) the nonlinear frequency ratio with different non-homogeneity indexes.
3.1 Effect of the neutral surface deviation due to non-homogeneity index
The effect of the neutral surface deviation due to the non-homogeneity index on frequency response curves of PFGM and SFGM nanobeams are shown in Fig. 5. In the calculation, we have µ=0.25, η=0.25,
∆T=150K, KL=10, Ks=10, KNL=5 and f=0.5. It is seen that the system always presents a hardening effect (bend to the right) when the coefficient of the cubic term (the nonlinear term) in Eq. (48) is positive. For PFGM nanobeams, the hardening-type behavior of the system increases with the decrease of non-homogeneity index. When k is small, see, k=0.2, the effect of the neutral surface derivation could be ignored. But along with increase of non-homogeneity index k, the neutral surface derivation should be considered, which leads to the underestimation of the hardening behavior of the system, that is, the resonance frequency shifts to higher value when physical neutral surface is referred. The underestimation is firstly increased and then decreased with an increase of non-homogeneity index k. For SFGM nanobeams, the hardening behavior of the system is increased with decrease of non-homogeneity index, which is weakened when the physical neutral axis is referred.
20
Fig. 5. Effects of the non-homogeneity index and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. 3.2 Effect of neutral surface derivation due to temperature variation
The effect of the neutral surface deviation due to temperature variation on the frequency response curves of PFGM and SFGM nanobeams is plotted in Fig. 6. In the calculation we have µ=0.25, η=0.25, k=5, KL=10, Ks=10, KNL=5 and f=0.5. It is observed that the frequency responses of PFGM and SFGM nanobeams are temperature-dependent. Along with the increase of temperature variation, the hardening behavior is intensified. The underestimation due to the neutral surface shifting becomes larger with increasing of temperature variation. It is noticed that PFGM nanobeams are more sensitive to the variation of the temperature, whilst SFGM nanobeams are more affected by the neutral surface deviation.
Fig. 6. Effects of temperature variation and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. 21
3.3. Effects of the nonlocal and material length scale parameters
To present the size effect on the forced vibration of PFGM and SFGM nanobeams, the variation of the nonlocal and material length scale parameters, and the neutral surface deviation on the frequency response curves are shown in Figs. 7 and 8, respectively. In the calculation, we have k=5, ∆T=150K, KL=10, Ks=10, KNL=5 and f=0.5. It is observed that increasing the nonlocal parameter strengthens the hardening behavior whilst the inverse trend is observed for material length scale parameter. Larger material length scale parameter leads the resonances of the nanobeams at lower frequencies. It is noticed that when the size effect is ignored, the influence of the neutral surface derivation on the vibration of nanobeams can be ignored for PFGM nanobeams. However, larger nonlocal and material length scale parameters will lead to increased underestimation of resonance frequencies due to the neutral surface shifting, especially for SFGM nanobeams which display significant sensitivity to the variation of the material length scale parameter.
Fig. 7. Effects of the material length scale parameter and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams.
22
Fig. 8. Effects of the nonlocal parameter and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. 3.4. Effect of nonlinear elastic foundation
The effect of nonlinear elastic foundation on the frequency responses of PFGM and SFGM nanobeams are plotted in Fig. 9. Here µ=0.25, η=0.25, k=5, ∆T=150K and f=0.5 are used. As shown in Fig. 9, with the increase of the shear and linear stiffness coefficients, the hardening response of the system is reduced. However, larger nonlinear stiffness coefficient leads the shifting of the resonance frequencies to the higher ones. This indicates that the nonlinearity term is dependent on the nonlinear stiffness and the linear one is related to the linear and shear coefficients. The underestimation due to the neutral surface shifting becomes smaller when stiffness coefficients of the nonlinear elastic foundation are increased, especially for PFGM nanobeams, the effect of the neutral surface deviation could be ignored when the stiffness coefficients of the nonlinear elastic foundation are larger enough.
23
Fig. 9. Effects of nonlinear elastic foundation and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams.
4. Conclusions
In this paper, the forced vibrations of temperature-dependent PFGM and SFGM nanobeams resting on a nonlinear elastic foundation are investigated by using the nonlocal strain gradient theory. Referring to the physical neutral surface, the governing equation of motion is derived based on Euler-Bernoulli beam 24
theory together with the von-Kármán’s geometric nonlinearity. Closed-form solution is obtained for the forced vibration by using multiple time scale method. The influences of the neutral surface deviation due to the material gradient and temperature sensitivity, the size effects, as well as the nonlinear elastic foundation, on the nonlinear resonances of nanobeams are discussed in detail. Summarizing results above, we conclude that
(1) For a PFGM or SFGM embedded nanobeam, along with the increase of the non-homogeneity index, material length scale parameter, shear and linear foundation stiffness coefficients, the resonance frequency of the nanobeam will shift to lower ones. However, the increase of the temperature variation, nonlocal parameter and nonlinear foundation stiffness coefficient will strengthen the hardening responses of the nanobeam. The larger the excitation amplitude is, the wider the resonant region is.
(2) If the neutral surface shifting due to the material gradient and temperature variation is ignored, the hardening behavior of the beam will be underestimated. The increase of the non-homogeneity index and temperature variation strengthens this underestimation. It is noticed that stronger size effects and weaker nonlinear foundation will amplify the effects of the neutral surface derivation, especially when the effect of the higher order strain field is dominated (larger material length scale parameter). It should be pointed out that the resonance of a SFGM nanobeam is more affected by the position of the neutral surface. When the size effect and temperature variation are not considered, the influence of the neutral surface shifting could be ignored for a PFGM nanobeam, which is insensitive to the stiffness of nonlinear foundation.
These findings are meaning for the mechanical analysis and design of temperature-dependent FG nanobeams.
25
Acknowledgements
The second author thanks the support from the Fundamental Research Funds for the Central Universities of China (2014JBZ014). Supports from the National Natural Science Foundation of China (11772044), and National Basic Research Program of China (973Program) (2015CB057800), are acknowledged.
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33
LIST OF FIGURES AND TABLE CAPTIONS
Fig. 1. Geometry of a temperature-dependent functionally graded nanobeam resting on the nonlinear elastic foundation subjected to harmonic forces. Fig. 2. The effects of non-homogeneity index k and temperature variation ∆T on the neutral surface derivation. Fig. 3. The variation of Young’s modulus through the thickness direction for PFGM and SFGM nanobeams with different temperature variation ∆T.
Fig. 4. Comparison of (a) the nondimensional natural frequency with different temperature variations; and (b) the nonlinear frequency ratio with different non-homogeneity indexes. Fig. 5. Effects of the non-homogeneity index and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. Fig. 6. Effects of temperature variation and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. Fig. 7. Effects of the material length scale parameter and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. Fig. 8. Effects of the nonlocal parameter and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. Fig. 9. Effects of nonlinear elastic foundation and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams. Table 1: Temperature-dependent coefficients for ceramic and metals
34
Fig. 1. Geometry of a temperature-dependent functionally graded nanobeam resting on the nonlinear elastic foundation subjected to harmonic forces.
35
Fig. 2. The effects of non-homogeneity index k and temperature variation ∆T on the derivation between the physical neutral surfaces.
36
Fig. 3. The variation of Young’s modulus through the thickness direction of PFGM and SFGM nanobeams with different temperature variation ∆T .
37
Fig. 4. Comparison of (a) the nondimensional natural frequency with different temperature variations; and (b) the nonlinear frequency ratio with different non-homogeneity indexes.
38
Fig. 5. Effects of the non-homogeneity index and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams.
39
Fig. 6. Effects of temperature variation and neutral surface deviation on the frequency response curves of 40
PFGM and SFGM nanobeams.
41
Fig. 7. Effects of the material length scale parameter and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams.
42
Fig. 8. Effects of the nonlocal parameter and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams.
43
Fig. 9. Effects of nonlinear elastic foundation and neutral surface deviation on the frequency response curves of PFGM and SFGM nanobeams.
44
Table 1 Temperature-dependent coefficients for ceramic and metals
Material
Properties
P-1
P0
P1
P2
P3
SUS304
E(Pa)
0
201.04e+9
3.079e-4
-6.534e-7
0
ρ(kg/m3)
0
8166
0
0
0
α(K-1)
0
12.33e-6
8.086e-4
0
0
υ
0
0.3262
-2.002e-4
3.797e-7
0
E(Pa)
0
348.43e+9
-3.07e-4
2.160e-7
-8.946e-11
ρ(kg/m3)
0
2370
0
0
0
α(K-1)
0
5.8723e-6
9.095e-4
0
0
υ
0
0.24
0
0
0
Si3N4
45