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Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory
Composite Structures 94 (2012) 2038–2047
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Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory Liao-Liang Ke ⇑, Yue-Sheng Wang, Zheng-Dao Wang Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, PR China
a r t i c l e
i n f o
Article history: Available online 2 February 2012 Keywords: Piezoelectric materials Nonlinear vibration Nonlocal theory Nanobeams Size effect
a b s t r a c t This paper investigates the nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory and Timoshenko beam theory. The piezoelectric nanobeam is subjected to an applied voltage and a uniform temperature change. The nonlinear governing equations and boundary conditions are derived by using the Hamilton principle and discretized by using the differential quadrature (DQ) method. A direct iterative method is employed to determine the nonlinear frequencies and mode shapes of the piezoelectric nanobeams. A detailed parametric study is conducted to study the influences of the nonlocal parameter, temperature change and external electric voltage on the size-dependent nonlinear vibration characteristics of the piezoelectric nanobeams. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Since Wang and his co-authors [1] reported the ZnO piezoelectric nanostructures in Science magazine, various piezoelectric nanomaterials (e.g. ZnO, ZnS, PZT, GaN, BaTiO3, etc.) and their nanostructures (e.g. nanowires, nanobelts, nanorings, nanohelices, etc.) have received considerable attention from research communities [2–4]. It is found that the piezoelectric nanostructures have the novel electrical, mechanical and other physical and chemical properties than their bulk counterparts [2,4], and have the potential applications in many nanodevices, such as nanoresonators [5], field effect transistors [6], light emitting diodes [7], chemical sensors [8] and nanogenerators [9]. In these nanodevices, the dimension may vary from several hundred nanometers to just a few nanometers. On this scale, the size effects become very obvious. Many experiments and atomistic simulations [10–13] have demonstrated the size-dependent deformation behavior of the piezoelectric nanostructures. Using the piezoresponse force microscopy, Zhao et al. [10] observed that the effective piezoelectric coefficient of ZnO nanowires was much larger than that of the bulk ZnO. Chen et al. [11] reported a size dependence of the Young’s modulus in ZnO nanowires. They found that the Young’s modulus for the nanowires with diameters smaller than about 120 nm was increasing dramatically with the decreasing diameters. These studies gave important evidences of the sizedependent material properties of the piezoelectric nanostructures, and hence the size effect must be taken into account in the related theoretical and experimental studies. Because the classical continuum theory is a scale independent theory, many high-order theories, such as the strain gradient
⇑ Corresponding author. Tel.: +86 10 51684070; fax: +86 10 51682094. E-mail address: [email protected] (L.-L. Ke). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2012.01.023
theory, couple stress theory, micropolar theory, nonlocal elasticity theory, etc., are developed to characterize the size effect in the nanostructures by introducing an intrinsic length scale. Among them, the nonlocal elasticity theory developed by Eringen [14– 16] has been widely accepted and applied to analyze the size effect of the nanostructures. The nonlocal elasticity theory contains the information about the forces between atoms, and introduces the internal length scale into the constitutive equations as a material parameter. In recent years, the studies of nanostructures by using the nonlocal elasticity theory have been an area of active research. Based on this theory, the nonlocal nanobeam model, nonlocal nanoplate model and nonlocal nanoshell model were developed and employed to discuss the bending [17,18], buckling [19–21], linear vibration [22–25], nonlinear vibration [26–28], postbuckling [29,30] and wave propagation characteristics [31–33] of the carbon nanotubes, graphene sheets, mass sensors, nanowires, etc. For the use of the nonlocal theory, the identification of the nonlocal parameter is significant to properly understand the size effect of the nanostructures. Many investigators proposed that the nonlocal parameter should be less than 2 nm based on the molecular mechanics and molecular dynamic simulations [34–36]. More in detail, Narendar and Gopalakrishnan [37] tabulated the values of the nonlocal parameter obtained by various investigators using different methods. It is worth to mention that Narendar et al. [36] predicted the nonlocal scaling parameter for armchair and zigzag single-walled carbon nanotubes based on the molecular structural mechanics, nonlocal elasticity and wave propagation. They found that the nonlocal parameter is a function of diameter of carbon nanotube. Their work is very important for understanding the dependence of the nonlocal parameter on the size of the nanostructures. The above investigations [17–37] on the nanostructures are based on the nonlocal elasticity theory, which is unsuitable for a
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direct use in the piezoelectric materials. To the best of authors’ knowledge, no literature is available for the piezoelectric nanostructures based on the nonlocal theory. So far, only a few works were reported for the piezoelectric nanostructures based on the surface elasticity theory. The effect of the surface piezoelectricity on the electromechanical behavior of a piezoelectric ring was discussed by Huang and Yu [38]. Yan and Jiang [39] studied the vibration and buckling behaviors of piezoelectric nanobeams with the surface effect. They [40] also analyzed the surface effect on the electromechanical coupling and bending behaviors of piezoelectric nanowires. Shen and Hu [41] established a variational principle for nanosized dielectrics that includes the flexoelectric and surface effects. It should be pointed out that most nanodevices with piezoelectric nanowires or nanobelts as fundamental elements are beambased [39] structures. Therefore, investigating the mechanical behaviors of the piezoelectric nanobeams is of primary importance in the design of the nanodevices. The simplest beam model is the Euler beam model. However, the Euler beam model is inaccurate to predict the mechanical behaviors of the short beam because the effect of the transverse shear deformation becomes significant and cannot be neglected for the beam with smaller length-to-thickness ratios. In practical applications, the short nanobeams have been used as nanoweezers [42] and atomic force microscope tips [43]. In order to incorporate the effect of the transverse shear deformation, the Timoshenko beam model should be used for a better estimation of the mechanical behaviors of the short nanobeams. In this paper, the nonlinear vibration of the piezoelectric nanobeams is first investigated based on the nonlocal elasticity theory and Timoshenko beam theory. The piezoelectric nanobeam is subjected to an applied voltage and a uniform temperature change. The governing equations and boundary conditions are derived by using the Hamilton’s principle. The differential quadrature (DQ) method is employed to discretize the governing equations which are then solved by a direct iterative method to obtain the nonlinear frequencies and mode shapes of the piezoelectric nanobeams with different boundary conditions. The influences of the nonlocal parameter, temperature change and external electric voltage on the nonlinear vibration characteristics of the piezoelectric nanobeams are discussed in details. 2. Nonlocal theory for the piezoelectric materials In Eringen’s nonlocal elasticity theory [14–16], the stress at a point x in a body depends not only on the strain at that point but also on those at all points of the body. The nonlocal elasticity theory can satisfactorily explain some phenomena related to atomic and molecular scales such as high frequency vibration and wave dispersion. Recently, Zhou and his co-authors [44–46] extended the Eringen’s nonlocal elasticity theory to the piezoelectric materials. Mathematically, the basic equations for a homogeneous and nonlocal piezoelectric solid without body force can be written as
rij ¼ Di ¼
Z ZV
0
0
0
0
aðjx xj; sÞ½C ijkl ekl ðx Þ ekij Ek ðx Þ kij DTdx ;
ð1Þ
aðjx0 xj; sÞ½eikl ekl ðx0 Þ 2ik Ek ðx0 Þ þ pi DTdx0 ;
ð2Þ
electric potential. a(jx0 xj, s) in Eqs. (1) and (2) is the nonlocal attenuation function which incorporates into the constitutive equations at the reference point x produced by the local strain at the source x0 ; jx0 xj is the Euclidean distance. s = e0a/l is defined as the scale coefficient that incorporates the small scale factor, where e0 is a material constant determined experimentally or approximated by matching the dispersion curves of the plane waves with those of the atomic lattice dynamics; and a and l are the internal (e.g. lattice parameter, granular size) and external characteristic lengths (e.g. crack length, wavelength) of the nanostructures, respectively. Referring to Eringen [15], it is possible to represent the integral constitutive relations in an equivalent differential form as
rij ðe0 aÞ2 r2 rij ¼ C ijkl ekl ekij Ek kij DT;
ð5Þ
Di ðe0 aÞ2 r2 Di ¼ eikl ekl þ 2ik Ek þ pi DT;
ð6Þ
2
where r is the Laplace operator; e0a is the scale coefficient revealing the size effect on the response of structures in nanosize. 3. Nonlinear vibration analysis of the piezoelectric nanobeam By using the above nonlocal theory for the piezoelectric materials, this section will analyze the nonlinear vibration of the piezoelectric nanobeam under the thermo-electrical loading. Fig. 1 shows a piezoelectric nanobeam with length L and thickness h subjected to an applied voltage U(x, z, t) and a uniform temperature change DT. The poling direction of the piezoelectric medium is parallel to the positive z-axis, where (x, z) is the coordinate system shown in the figure. Based on the Timoshenko beam theory, the displacements of an arbitrary point in the beam along the x- and z-axes, denoted by ux(x, z, t) and uz (x, z, t) respectively, are
ux ðx; z; tÞ ¼ Uðx; tÞ þ zWðx; tÞ;
uz ðx; z; tÞ ¼ Wðx; tÞ;
where U(x, t) and W(x, t) are displacement components in the midplane; W(x, t) is the rotation of beam cross-section; and t is the time. In addition to the displacement field, we need to assume the distribution of the electric potential for the present piezoelectric nanobeam model. Recently, Jiang and Yan [47] studied the electromechanical response of a curved piezoelectric nanobeam with the consideration of the surface effects. In their work, they assumed a linear distribution of the electric potential in the thickness direction of the piezoelectric nanobeams. Though this assumption violates the Maxwell equation, their results are reasonable for the electromechanical response of the piezoelectric nanobeams. In the present analysis, we follow Wang [48] and assume the electric potential as a combination of a cosine and linear variation, which satisfies the Maxwell equation. It can be written as
Uðx; z; tÞ ¼ cosðbzÞ/ðx; tÞ þ
2zV 0 iXt e ; h
1 2
eij ¼ ðui;j þ uj;i Þ; Ei ¼ U;i
ð3Þ ð4Þ
where rij, eij, Di, Ei and ui are the components of the stress, strain, electric field, electric displacement and displacement, respectively; Cijkl, ekij, 2ik, kij, pi and q are the elastic constants, piezoelectric constants, dielectric constants, thermal moduli, pyroelectric constants and mass density, respectively; DT is the temperature change; U is the
ð8Þ
where b = p/h; /(x, t) is the spatial and time variation of the electric potential in the x-direction; V0 is the external electric voltage; X is the natural frequency of the piezoelectric nanobeam. Note that / (x, t) must satisfy the electric boundary conditions.
V
rij;j ¼ qu€ i ; Di;i ¼ 0;
ð7Þ
z
z
Fig. 1. Schematic configuration of a piezoelectric nanobeam.
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The von Kármán type nonlinear strain–displacement relations are given by
exx ¼
2 @U 1 @W @W @W þ þ W: þz ;c ¼ @x 2 @x @x @x xz
ð9Þ
@U @/ @U 2V 0 iXt ; Ez ¼ ¼ cosðbzÞ ¼ b sinðbzÞ/ e ; @x @x @z h
ð10Þ
The strain energy PS of the piezoelectric nanobeams is given by
PS ¼
1 2
Z
Z
L
0
1 2
Z
h=2
¼
1 2
ðrxx exx þ rxz cxz Dx Ex Dz Ez Þdz dx;
ð11Þ
0
Z
h=2
1 2
Z
L
0
" 2 # @U 1 @W @W @W þ rxx z rxx þ þ W dz dx þ rxz @x 2 @x @x @x Z h=2 2V 0 iXt Dx cosðbzÞ Dz b sinðbzÞ/ þ e h h=2
(
@/ dz dx @x
Z
L
"
"
@/ dz dx; @x
where the normal resultant force Nx, the bending moment Mx, and the transverse shear force Qx are calculated from
Z
@M x @2W Q x ¼ I3 2 ; @x @t Z h=2 @Dx cosðbzÞ þ b sinðbzÞDz dz ¼ 0: @x h=2
h=2
rxx dz; Mx ¼
h=2
Z
h=2
rxx zdz; Q x ¼
h=2
Z
h=2
rxz dz:
ð13Þ
h=2
2 2 # Z " 2 1 L @U @W @W dx; PK ¼ I1 þ I1 þI3 2 0 @t @t @t
ð24Þ
W ¼ 0 or Q x ¼ 0; x ¼ 0; L;
ð25Þ
W ¼ 0 or M x ¼ 0; x ¼ 0; L;
ð26Þ
/ ¼ 0 or
Z
h=2
Dx cosðbzÞdz ¼ 0;
x ¼ 0; L:
@2r rxx ðe0 aÞ2 2xx ¼ c11 exx e31 Ez k1 DT; @x 2 2 @ rxz rxz ðe0 aÞ ¼ c44 cxz e15 Ex ; @x2 @ 2 Dx Dx ðe0 aÞ2 ¼ e15 cxz þ 211 Ex ; @x2 2 @ Dz Dz ðe0 aÞ2 ¼ e31 exx þ 233 Ez þ p1 DT: @x2
" 2 # @ 2 Nx @U 1 @W þ 2e31 V 0 k1 hDT; þ Nx ðe0 aÞ ¼ A11 @x 2 @x @x2 @ 2 Mx @W ¼ D11 þ F 31 /; @x2 @x 2 @ Qx @W @/ ks E15 ¼ k A W Q x ðe0 aÞ2 þ ; s 44 @x @x @x2 " # Z h=2 @ 2 Dx @W @/ cosðbzÞdz ¼ E þ X 11 þ ; Dx ðe0 aÞ2 W 15 2 @x @x @x h=2 " # Z h=2 @ 2 Dz @W b sinðbzÞdz ¼ F 31 Dz ðe0 aÞ2 X 33 /; @x2 @x h=2 Mx ðe0 aÞ2
ð28Þ ð29Þ ð30Þ ð31Þ ð32Þ
where ks is the shear correction factor depending on the shape of the cross-section of the beam; and
Z
h=2
c11 dz;
D11 ¼
h=2
ð14Þ A44 ¼
Z
h=2
c44 dz;
F 31 ¼
h=2
E15 ¼
Z
Z Z
h=2
c11 z2 dz;
h=2 h=2
e31 b sinðbzÞzdz;
e15 cosðbzÞdz;
X 11 ¼
h=2
X 33 ¼
Z
h=2
ð33Þ
h=2
h=2
Z
h=2
211 cos2 ðbzÞdz;
h=2
233 ½b sinðbzÞ2 dz:
ð34Þ
h=2
ð15Þ ð16Þ
By substituting Eqs. (20)–(22) into Eqs. (28)–(30), the explicit expressions of the nonlocal normal resultant force Nx, bending moment Mx and shear force Qx can be obtained as
ð17Þ ð18Þ
" Nx ¼ A11
Consider the Hamilton’s principle
Mx ¼ D11
t
ðdPS dPK Þdt ¼ 0:
ð27Þ
h=2
A11 ¼
where {I1, I3} = {qh, qh3/12}. For a beam type structure, both thickness and width are much smaller than its length. Therefore, for beams with the transverse motion in the x–z plane, the nonlocal constitutive relations (5) and (6) can be approximated to one-dimensional form as
0
ð23Þ
U ¼ 0 or Nx ¼ 0; x ¼ 0; L;
The kinetic energy PK can be calculated from
Z
ð22Þ
The corresponding boundary conditions at the beam ends (x = 0, L) require
2
ð12Þ
Nx ¼
d/ :
ð21Þ
From Eqs. (15)–(18), we have
2 # # @U 1 @W @W @W þ þ W dz dx Nx þ Qx þ Mx @x 2 @x @x @x 0 Z Z 1 L h=2 2V 0 iXt Dx cosðbzÞ Dz b sinðbzÞ/ þ e þ 2 0 h=2 h
ð20Þ
L
h=2
þ
dW :
h=2
Which, by submitting Eqs. (9) and (10) into Eq. (11), can be rewritten as
PS ¼
@Nx @2U ¼ I1 2 @x @t @Q x @ @W @2W Nx ¼ I1 ; dW : þ @x @x @x @t2
dU :
According to Eq. (8), the electric fields can be expressed as
Ex ¼
Then substituting Eqs. (12) and (14) into Eq. (19), integrating it by parts and setting the coefficients of dU, dW, d W and d/ to zero lead to the equations of motion as
ð19Þ
2 # @U 1 @W @3U þ NE þ NT þ ðe0 aÞ2 I1 þ ; @x 2 @x @x@t2
ð35Þ
" # @W @3W @2W @ @W N ; þ I þ F 31 / þ ðe0 aÞ2 I3 1 x @x @x @x @t 2 @x@t 2 ð36Þ
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Q x ¼ ks A44
"
@W @/ @3W @2 @W þ W ks E15 þ ðe0 aÞ2 I1 Nx @x @x @x @x@t2 @x2
!
# ;
ð37Þ where NT = k1hDT and NE = 2e31V0 are the normal force induced by the temperature change DT and external electric voltage V0, respectively. Note that the shear correction factor, which is quite complex involving integral expressions, not only depends on the material and geometric parameters but also on the loading and boundary conditions. For a macroscale structure with a rectangular cross-section, it is suitable to take the shear correction factor as 5/6. However, the shear correction factor for a macroscale structures may not be applicable to a nanoscale structure duo to the size-dependent material properties. So far, no experimental or theoretical results of the shear correction factor are available for nanostructures. Therefore in the present paper, following Reddy [49], we take the shear correction factor for the nanobeams approximately as the same value (i.e. ks = 5/6) as that for the macroscale beams. Unlike Nx, Mx and Qx in Eqs. (35)–(37), the electric displacements Dx and Dz cannot be obtained explicitly by substituting Eq. (23) into Eqs. (31) and (32) because there are two unknowns but only one equilibrium Eq. (23). However, we can re-express Eq. (23) in the form of U, W, W and / using Eqs. (31) and (32). Then, with the help of Eqs. (35)–(37) and Eqs. (31) and (32), the nonlinear governing Eqs. (20)–(23) for the piezoelectric nanobeams can be written as
@ 2 U @W @ 2 W þ @x2 @x @x2
A11
!
" # 2 @2 2@ U ¼ I1 2 U ðe0 aÞ ; @x2 @t
ð38Þ
ðX 11 ; X 33 Þ ¼
i¼
t L
sffiffiffiffiffiffiffi I1 : A11
! X 11 /20 X 33 /20 ; ; 2 A11 A11 h
ðI1 ; I3 Þ ¼
NT ¼
I 1 I3 ; ; I 1 I 1 h2 ð44bÞ
khDT ; A11
NE ¼
2e31 V 0 ; A11 ð44cÞ
Eqs. (38)–(41) can be expressed in the dimensionless form as
! ! 4 1 @w @ 2 w @2 2@ u þ ¼ I1 2 u l ; @i @f2 g @f @f2 @f2
@2u
A11
! @w @2u @2w ks E15 2 þ ðNT þ NE Þ 2 2 @f @f @f @f " # @4w @2 @2w l2 ðNT þ NE Þ 4 þ Z 1 l2 Z 2 ¼ I1 2 w l2 2 ; @s @f @f @2w
ks A44
@u @w 2 k A g g w þ F 31 g þ ks E15 g þ s 44 @f @f @f2 " # 2 2 @ @ w ¼ I 3 2 w l2 2 ; @s @f
D11
@2W @W @2/ @2W ks E15 2 þ ðNT þ NE Þ 2 þ @x2 @x @x @x
ð45Þ
þg
ð46Þ
@2w
! @w @2w @w @2u þ X 11 2 X 33 g2 / ¼ 0; þ E15 F 31 g þg 2 @f @f @f @f
!
ks A44
A11 A44 D11 ; ; ; A11 A11 A11 h2 E15 /0 F 31 /0 E15 ; F 31 ¼ ; ; A11 h A11 h
ðA11 ; A44 ; D11 Þ ¼
!
ð47Þ
ð48Þ
where
@4W ðe0 aÞ ðNT þ NE Þ þ Z 1 ðe0 aÞ2 Z 2 @x4 " # 2 @2 2@ W ¼ I1 2 W ðe0 aÞ ; @x2 @t 2
Z1 ¼ ð39Þ
@2W @W @/ @/ þ F 31 þ þ ks E15 k A W s 44 @x @x @x @x2 " # @2 @2W ¼ I3 2 W ðe0 aÞ2 2 ; @x @t
D11
F 31
ð40Þ
! @W @2W @W @2/ þ X 11 2 X 33 / ¼ 0; þ þ E15 2 @x @x @x @x
ð41Þ
" # 2 A11 @ 2 u @w 3 @w @ 2 w @u @ 2 w ; þ þ g @f2 @f 2g @f @f2 @f @f2
! A11 @ 4 u @w @3u @2w @ 2 u @ 3 w @u @ 4 w þ3 3 Z2 ¼ þ3 2 þ @f @f4 g @f4 @f @f @f2 @f @f3 2 3 !3 2 A11 4 @ 2 w @w @ 2 w @ 3 w 3 @w @ 4 w5 : þ 2 3 þ9 þ @f @f2 @f3 2 @f g @f2 @f4
ð49Þ
ð50Þ
Assume that the electric potential is zero at the ends of the nanobeam. Then, the associated boundary conditions can be written in the dimensionless form as
u ¼ w ¼ w ¼ u ¼ 0;
ð51Þ
for a clamped end, and
where
" # 2 @ 2 U @W 3 @W @ 2 W @U @ 2 W Z 1 ¼ A11 þ þ ; @x2 @x 2 @x @x2 @x @x2
ð42Þ
! @ 4 U @W @3U @2W @ 2 U @ 3 W @U @ 4 W þ 3 þ 3 þ @x4 @x @x3 @x2 @x2 @x3 @x @x4 2 3 !3 2 4 2 2 3 @ W @W @ W @ W 3 @W @ W 5: þ A11 43 þ9 þ @x2 @x @x2 @x3 2 @x @x4
Z 2 ¼ A11
! 2 @w @w @2u 2 @ 2 þ F 31 g/ þ l þ I1 gw l I1 g 2 D11 I3 @f @f @ i2 @f (" ) 2 # 2 2 @ w l A11 @ @u 1 @w @w 2 ¼ 0; þ l ðNT þ NE Þg 2 @f g @f @f 2g @f @f u ¼ w ¼ u ¼ 0;
ð43Þ
ð52Þ
for a hinged end. 4. Solution method
Introducing the following dimensionless quantities
x W f¼ ; w¼ ; L h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /0 ¼ e33 =A11 ;
w ¼ W;
L g¼ ; h
e a l¼ 0 ; L
/ u¼ ; /0 ð44aÞ
The DQ method [50] is used to solve the nonlinear governing equations (45)–(48) with the associated boundary conditions of the piezoelectric nanobeams. According to the DQ method, we discretize the beam domain by N nodes along the x-axis, and
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designate the values of u, w, w and u and their kth partial derivatives with respect to x can be approximated by N X
fu; w; w; ug ¼
lm ðfÞfum ðfm ; iÞ; wm ðfm ; iÞ; wm ðfm ; iÞ; um ðfm ; iÞg;
m¼1
ð53Þ @k k
@f
fu; w;w; ugjx¼xi ¼
N X
ðkÞ
C im fum ðfm ; iÞ;wm ðfm ; iÞ; wm ðfm ; iÞ; um ðfm ; iÞg;
m¼1
with the over dot denoting the partial derivative with respect to the dimensionless time i. The associated boundary conditions can be handled in the same way. For a hinged–hinged (H–H) piezoelectric nanobeam, we have
!2 N N X 3 X ð1Þ ð2Þ C 1m wm C 1m wm 2g m¼1 g m¼1 m¼1 m¼1 # " N N N N X X X X ð1Þ ð2Þ ð1Þ ð2Þ € 2 €1 C 1m um C 1m wm D11 C 1m wm þ l I3 C 1m w þ m þ I1 gw
F 31 gu1
m¼1
ð54Þ ðkÞ
where lm(x) is the Lagrange interpolation polynomials; and C im is the weighting coefficients whose recursive formula can be found in Shu [50]. The cosine pattern is used to generate the DQ point system
1 pði 1Þ 1 cos ; fi ¼ 2 N1
i ¼ 1; 2; . . . N:
ð55Þ
Applying the relationships (53) and (54) to Eqs. (45)–(48), one obtains a set of nonlinear ordinary differential equations ! N X
A11
N 1X
ð2Þ
C im um þ
g m¼1
m¼1 2
€i l ¼ I1 u
N X
N X
ð1Þ
C im wm
N X
ks A44
ð2Þ
C im wm þ g
m¼1
N X
ð56Þ
; ! ð1Þ
C im wm
ks E15
m¼1
þ ðNT þ NE Þ
N X
N X
ð2Þ
C im wm l2 ðNT þ NE Þ
m¼1
D11
ð2Þ
C im wm ks A44 g
m¼1
N X
þ ðF 31 g þ ks E15 gÞ
N X
ð1Þ C im
N X
ð1Þ
C im wm þ g2 wi
um ¼ I 3
F 31 g
ð1Þ C im wm
þ E15
m¼1
þ X 11
ð2Þ C im wm
þg
m¼1 N X
N X
N X
# ð2Þ € C im w m
ð58Þ
;
ð59Þ
where
Z 1i ¼
g
þ
m¼1 N X
m¼1
ð1Þ
C im um
m¼1
Z 2i ¼
2 A11 4
g2
N X
3
N X
N X
N X m¼1
!2
N X m¼1
ð2Þ
!3 ð2Þ C im wm
þ9
ð60Þ
ð3Þ
m¼1
C im um
ð1Þ C im um
N X
!2
N X
ð2Þ
C im wm þ 3
m¼1
m¼1
3
ð4Þ C im wm 5
ð4Þ C im wm
# ;
N X
ð2Þ
C im wm
m¼1
þ
m¼1
N X
N X
ð1Þ
C im wm
N X m¼1
" N A11 X
g ð2Þ
N X
ð3Þ
C im wm
m¼1
ð4Þ
C im um
m¼1
C im um
ð2Þ
C Nm um
N X
m¼1 N X
ð1Þ
C Nm wm þ
m¼1
ð2Þ
€ m ðNT þ NE Þg C Nm u
m¼1
N X
#
m¼1
ð2Þ
C Nm wm ¼ 0;
m¼1
uN ¼ wN ¼ uN ¼ 0;
at f ¼ 1:
ð63Þ
2 N A11 4X
at f ¼ 0;
ð64Þ
!2 N N X 3 X ð1Þ ð2Þ F 31 guN þ C Nm wm C Nm wm 2 g m¼1 g m¼1 m¼1 m¼1 # " N N N N X X X X ð1Þ ð2Þ ð1Þ ð2Þ € 2 €N C Nm um C Nm wm D11 C Nm wm þ l I3 C Nm w þ m þ I 1 gw ð2Þ C Nm um
N X
m¼1 N X
ð1Þ C Nm wm
m¼1
ð2Þ
€ m ðNT þ NE Þg C Nm u
N X
#
m¼1
ð2Þ
C Nm wm ¼ 0;
m¼1
uN ¼ wN ¼ uN ¼ 0;
at f ¼ 1:
ð65Þ
fui gT ; fwi gT ; fwi gT ; fui gT
T
;
i ¼ 1; 2; . . . N;
ð66Þ
N X
N X
€ ¼ 0; ðKL þ KNL Þd þ Md
ð67Þ
where M is the mass matrix; KL is the stiffness matrix; and KNL is nonlinear stiffness matrix that is the function of d. KL, KNL and M are 4N 4N matrices. Expand the dynamic displacement vector d in the form of
ð2Þ C im wm
C im wm ;
m¼1
m¼1
þ
#
N 3 X ð1Þ þ C wm 2g m¼1 im
m¼1
N 3 X ð1Þ C wm þ 2 m¼1 im
þ3
ð1Þ C im wm
ð62Þ
Eqs. (56)–(59), together with the boundary conditions, can be expressed in the matrix form as
m¼1
ð2Þ
N X
ð2Þ
C 1m wm ¼ 0;
!2 N N X 3 X ð1Þ ð2Þ C Nm wm C Nm wm 2g m¼1 g m¼1 m¼1 m¼1 # " N N N N X X X X ð1Þ ð2Þ ð1Þ ð2Þ € 2 €N C Nm um C Nm wm D11 C Nm wm þ l I3 C Nm w þ m þ I 1 gw
F 31 guN
d¼
ð1Þ C im wm
C im um X 33 g2 u ¼ 0;
ð2Þ C im um
m¼1
By denoting the unknown dynamic displacement vector as
!
m¼1
2 N A11 4X
#
at f ¼ 0;
m¼1
m¼1
N X
2 N A11 4X
N X m¼1
u1 ¼ w1 ¼ u1 ¼ 0;
l2 I 1 g
€ i l2 w
ð2Þ
€ m ðNT þ NE Þg C 1m u
m¼1
m¼1
!
m¼1 N X
ð4Þ
C im wm þ Z 1i l2 Z 2i ð57Þ
"
ð1Þ
C 1m wm þ
m¼1
u1 ¼ w1 ¼ w1 ¼ u1 ¼ 0;
m¼1
m¼1
N X
For a clamped–hinged (C–H) piezoelectric nanobeam, we have ð2Þ
C im um
m¼1
" # N X @2 ð2Þ € i l2 €m ; ¼ I1 2 w C im w @s m¼1 N X
l2 I 1 g
l2 I 1 g
m¼1
ð2Þ
C 1m um
m¼1 N X
m¼1
m¼1
!
ð2Þ €m C im u
ð2Þ
C im wm
2 N A11 4X
ð1Þ
C im wm
m¼1
d ¼ d eixi ;
ð68Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi where x ¼ XL I1 =A11 represents the dimensionless natural fren T T quency of the piezoelectric nanobeams; and d ¼ ui ; wi ; T T T wi ; ui g is the vibration mode shape vector. Then substitution of Eq. (68) into Eq. (67) yields the nonlinear eigenvalue equation
ðKL þ KNL Þd x2 Md ¼ 0:
ð69Þ
The above nonlinear equations can be solved through a direct iterative process below:
ð3Þ
C im wm
m¼1
ð61Þ
Step 1: By neglecting the nonlinear matrix KNL, a linear eigenvalue (xl) and the associated eigenvector are obtained
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L.-L. Ke et al. / Composite Structures 94 (2012) 2038–2047 Table 1 Material properties of PZT-4 [51,52]. c11 (GPa)
e31 (C/m2)
c44 (GPa)
132
26
e15 (C/m2) 14.1
4.1
211 (C/Vm) 5.841 10
from Eq. (69). The eigenvector is then appropriately scaled up such that the maximum transverse displacement is equal to a given vibration amplitude wmax . Note that wmax ¼ wð0:5Þ for the H–H nanobeam and wmax ¼ wð0:57Þ for the C–H piezoelectric nanobeam. Step 2: Calculate KNL by using the above eigenvector. Then a new eigenvalue and the associated eigenvector are obtained from the updated eigensystem (69). Step 3: The eigenvector is scaled up again and step 2 is repeated until the relative error between the given vibration amplitude and the maximum transverse displacement wmax is within 0.1%.
9
233 (N/m2 K)
k1 (N/m2 K)
9
5
7.124 10
4.738 10
p1 (C/m2 K) 2.5 10
5
q (kg/m3) 7500
experiments or molecular dynamic (MD) simulation. However, it is lack of such works on the piezoelectric nanomaterials at present. Therefore, in order to quantitatively understand the thermoelectric-mechanical responses of the piezoelectric nanobeams, we choose the material properties of the macroscopic piezoelectric materials for the case study. The effects of the dimensionless nonlcoal parameter l, temperature change DT and external electric voltage V0 on the nonlinear vibration frequencies and mode shapes of the piezoelectric nanobeams are discussed in detail. In Figs. 2–4, the linear fundamental frequencies xl are also given. Unless otherwise stated, it is assumed that the length of the nanobeam L = 12 nm, thickness h = 2 nm and shear correction factor ks = 5/6.
5. Numerical results
5.1. Comparison and convergence studies
In this section, we present the numerical results for the nonlinear vibration of the hinged–hinged (H–H) and clamped–hinged (C–H) piezoelectric nanobeams. The nanobeam is made of PZT-4 with the thermo-electric-mechanical material properties listed in Table 1 [51,52], which are taken from the macroscopic piezoelectric materials. Actually, the materials properties must be sizedependent in the piezoelectric nanomaterials and their thermoelectric-mechanical material properties should be determined by
Table 2 lists the dimensionless linear and nonlinear frequencies of the piezoelectric nanobeams with varying total number of nodes (N) in the DQ method (here we take wmax ¼ 0:5; DT ¼ 0 C; V 0 ¼ 0V and l = 0.1). It is seen that the results become closer to each other as N increases and those with N = 14 and 16 are identical. Hence, N = 14 is used in all of the following calculations. Table 3 compares the dimensionless linear and nonlinear frequencies of the piezoelectric nanobeams obtained from both
(a)
2.0
(a)
H-H:
1.6
o
ΔT = 0 C
1.8
(ωl = 0.4263) o
ΔT = 100 C (ωl = 0.4222) o
ωnl / ω l
ωnl / ω l
H-H:
μ = 0.00 (ωl = 0.4468) μ = 0.05 (ωl = 0.4413) μ = 0.10 (ωl = 0.4263) μ = 0.15 (ωl = 0.4042)
1.8
2.0
1.4
1.2
ΔT = 200 C (ωl = 0.4180)
1.6
o
ΔT = 300 C (ωl = 0.4138)
1.4
1.2
1.0 0.0
0.2
0.4
0.6
1.0 0.0
0.8
0.2
0.4
*
w max
1.5 1.4
ωnl / ωl
(b)
1.6
C-H: μ = 0.00 (ωl = 0.6323) μ = 0.05 (ωl = 0.6239) μ = 0.10 (ωl = 0.6006) μ = 0.15 (ωl = 0.5668)
1.4
0.8
(ωl = 0.6006)
1.1
1.1
0.6
0.8
w*max Fig. 2. The effect of the nonlocal parameter l on nonlinear frequency ratio versus amplitude curves of the piezoelectric nanobeams with DT = 0 °C and V0 = 0: (a) H– H; and (b) C–H.
o
ΔT = 200 C (ωl = 0.5938) o
1.3 1.2
0.4
o
ΔT = 0 C o
1.2
0.2
0.6
C-H: ΔT = 100 C (ωl = 0.5972)
1.3
1.0 0.0
0.8
1.6 1.5
ωnl / ωl
(b)
0.6
w*max
1.0 0.0
ΔT = 300 C (ωl = 0.5904)
0.2
0.4
w*max Fig. 3. The effect of the temperature change DT (°C) on nonlinear frequency ratio versus amplitude curves of the piezoelectric nanobeams with l = 0.1 and V0 = 0: (a) H–H; and (b) C–H.
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L.-L. Ke et al. / Composite Structures 94 (2012) 2038–2047
(a)
(a)
2.0
2.0
H-H:
1.6
L / h = 6 (ωl = 0.4263) L / h = 8 (ωl = 0.3290) L / h = 12 (ωl = 0.2243) L / h = 20 (ωl = 0.1362)
1.8
ωnl/ ωl
ωnl / ω l
H-H: V 0 = 0.1 V ( ωl = 0.3893) V 0 = 0.05 V ( ωl = 0.4082) V0 = 0 V ( ωl = 0.4263) V 0 = -0.05 V ( ωl = 0.4436) V 0 = -0.1 V ( ωl = 0.4602)
1.8
1.4
1.6
1.4
1.2
1.2
1.0 0.0
0.2
0.4
0.6
1.0 0.0
0.8
0.2
w max
(b) 1.6
(b)
0.8
0.6
0.8
C-H: 1.5
V 0 = 0.1 V ( ωl = 0.5707) V 0 = 0.05 V ( ωl = 0.5858) V0 = 0 V ( ωl = 0.6006) V 0 = -0.05 V ( ωl = 0.6149) V 0 = -0.1 V ( ωl = 0.6289)
1.4
ωnl/ ωl
ωnl / ω l
0.6
1.6
C-H:
1.4
0.4
w*max
*
1.2
L / h = 6 (ωl = 0.6006) L / h = 8 (ωl = 0.4806) L / h = 12 (ωl = 0.3378) L / h = 20 (ωl = 0.2088)
1.3 1.2 1.1
1.0 0.0
0.2
0.4
0.6
1.0 0.0
0.8
0.2
*
w max Fig. 4. The effect of the external electric voltage V0 on nonlinear frequency ratio versus amplitude curves of the piezoelectric nanobeams with l = 0.1 and DT = 0 °C: (a) H–H; and (b) C–H.
Table 2 Dimensionless linear and nonlinear frequencies of the H–H and C–H piezoelectric nanobeams with varying total numbers of the nodes N (wmax ¼ 0:5; DT ¼ 0 C; V 0 ¼ 0 and l = 0.1). N
5 6 8 10 14 16
H–H
0.4
w*max
C–H
xl
xnl
xl
xnl
0.41974 0.42447 0.42629 0.42626 0.42626 0.42626
0.47030 0.58486 0.58903 0.58909 0.58905 0.58905
0.59911 0.59891 0.60060 0.60056 0.60057 0.60057
– 0.65786 0.73393 0.73865 0.73852 0.73855
nonlocal Euler beam model and Timoshenko beam model with wmax ¼ 0:5; h ¼ 2 nm; DT ¼ 0 C; V 0 ¼ 0V and l = 0.1. The governing equations and boundary conditions of the piezoelectric
Fig. 5. The effect of the length-to-thickness ratioL/h on nonlinear frequency ratio versus amplitude curves of the piezoelectric nanobeams with l = 0.1, V0 = 0, DT = 0 and h = 2 nm: (a) H–H; and (b) C–H.
nanobeams based on the Euler beam theory and nonlocal theory are given in Appendix A. The difference between the results obtained from the Euler beam model and Timoshenko beam model is observed to be quite small when L/h P 16 but relatively large when L/h < 16. This is because the transverse shear deformation which tends to be more significant for the beams with smaller slenderness ratio is included in the Timoshenko beam theory but neglected in the Euler beam theory. 5.2. Nonlinear vibration analysis Fig. 2 plots the effect of the nonlocal parameter l on the nonlinear frequency ratio versus the amplitude curves of the piezoelectric nanobeams with DT = 0 °C and V0 = 0. Note that the nonlocal parameter l = 0 corresponds to the classical naobeams without the nonlocal effect. Obviously, the nonlinear frequency ratio increases as the vibration amplitude increases for both H–H and
Table 3 Comparison of the dimensionless linear and nonlinear frequencies obtained from Euler beam model and Timoshenko beam model with wmax ¼ 0:5; h ¼ 2 nm; DT ¼ 0 C; V 0 ¼ 0V and l = 0.1. L/h
Euler beam (H–H)
xl 6 8 10 16 20 30
0.4570 0.3428 0.2742 0.1714 0.1371 0.0914
Timoshenko beam (H–H)
Euler beam (C–H)
Timoshenko beam (C–H)
xnl
xl
xnl
xl
xnl
xl
xnl
0.6140 0.4608 0.3687 0.2305 0.1844 0.1229
0.4263 0.3290 0.2670 0.1696 0.1362 0.0911
0.5891 0.4495 0.3628 0.2290 0.1836 0.1227
0.7077 0.5310 0.4250 0.2658 0.2127 0.1420
0.8456 0.6354 0.5088 0.3184 0.2548 0.1700
0.6006 0.4806 0.3976 0.2585 0.2088 0.1406
0.7385 0.5833 0.4800 0.3105 0.2505 0.1685
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L.-L. Ke et al. / Composite Structures 94 (2012) 2038–2047
(a)
(a)
0.5
0.00006
0.4 0.00004
ϕ*
w*
0.3
H-H: μ = 0.00 μ = 0.05 μ = 0.10 μ = 0.15
0.2
0.1
0.0 0.0
0.2
0.4
0.6
H-H: μ = 0.00 μ = 0.05 μ = 0.10 μ = 0.15
0.00002
0.8
0.00000 0.0
1.0
0.2
0.4
x/L
(b)
(b)
0.5
0.6
0.8
1.0
x/L
0.4
0.00009
0.00006
ϕ*
w*
0.3
C-H: μ = 0.00 μ = 0.05 μ = 0.10 μ = 0.15
0.2
0.1
0.0 0.0
0.2
0.4
0.6
C-H:
0.00003
μ = 0.00 μ = 0.05 μ = 0.10 μ = 0.15
0.00000
0.8
1.0
-0.00003 0.0
C–H piezoelectric nanobeams. The nonlocal parameter has a significant effect on the nonlinear vibration behavior. At a given vibration amplitude, an increase in the nonlocal parameter leads to both smaller linear and nonlinear frequencies but a higher nonlinear frequency ratio. The reason is that the presence of the nonlocal effect tends to decrease the stiffness of the nanobeams and hence decreases the values of the linear and nonlinear frequencies. The C–H piezoelectric nanobeam has the higher linear and nonlinear frequencies than the H–H one since the end support is the stronger for the C–H piezoelectric nanobeam than for the H–H piezoelectric nanobeam. Fig. 3 shows the effect of the temperature change DT(°C) on the curves of the nonlinear frequency ratio versus the amplitude for the piezoelectric nanobeams with l = 0.1 and V0 = 0. With the increase of the temperature change, the linear and nonlinear frequencies decrease for both H–H and C–H piezoelectric nanobeams. The reason is that a larger temperature change results in more reduction in the nanobeam stiffness, and hence leads to lower linear and nonlinear frequencies of the nanobeams. At a given vibration amplitude, the nonlinear frequency ratio increases as the temperature change increases. The effect of the external electric voltage V0 on the curves of the nonlinear frequency ratio versus the amplitude for the piezoelectric nanobeams is depicted in Fig. 4 with l = 0.1 and DT = 0°C. It is observed that the positive/negative voltage decreases/increases the linear and nonlinear frequencies of the nanobeams. This is due to the fact that the axial compressive and tensile forces are generated in the nanobeams by the applied positive and negative voltages, respectively. At a given vibration amplitude, it is found that a change in the external electric voltage from 0.1V to 0.1V leads to the decrease of the nonlinear frequency ratio. Comparing
0.4
0.6
0.8
1.0
x/L
x/L Fig. 6. The effect of nonlocal parameter on the nonlinear mode shapes (deflection w⁄) of the piezoelectric nanobeams with DT = 0 °C and V0 = 0: (a) H–H; and (b) C–H.
0.2
Fig. 7. The effect of nonlocal parameter on the nonlinear mode shapes (electrical potential /⁄) of the piezoelectric nanobeams with DT = 0 °C and V0 = 0: (a) H–H; and (b) C–H.
the effects of the temperature change and external electric voltage on the nonlinear vibration behaviors of the piezoelectric nanobeams in Figs. 3 and 4, we find that the external electric voltage has the greater effect than the temperature change. This phenomenon can be explained from the expressions of NT and NE in which the coefficients of NT and NE have the relation of j k1hj < j2e31j. Fig. 5 plots the effect of the length-to-thickness ratio L/h on the curves of the nonlinear frequency ratio versus the amplitude for the piezoelectric nanobeams with l = 0.1, V0 = 0, DT = 0 and h = 2 nm. Both linear and nonlinear frequency ratios decrease as the length-to-thickness ratio increases. The effect of slenderness ratio on the nonlinear frequency ratio tends to be very weak for the long beams (L/h P 12). The effect of the nonlocal parameter on the nonlinear mode shapes (the dimensionless displacement w⁄ and electric potential u⁄) of the piezoelectric nanobeams are plotted in Figs. 6 and 7 with DT = 0 °C and V0 = 0. The maximum displacement and electric potential occur at the midpoint of the H–H piezoelectric nanobeams, buts slightly deviate from the center of the C–H piezoelectric nanobeams. For the H–H piezoelectric nanobeam, the nonlocal parameter nearly has no effect on the mode shape w⁄, while has a slight effect on the mode shape u⁄. However, the nonlocal parameter has a relatively large effect on the mode shapes (w⁄ and u⁄) for the C–H piezoelectric nanobeam, especially for the mode shape of the electric potential. 6. Conclusions This paper investigates the nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory and Timoshenko beam theory. The nonlinear governing equations and boundary
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L.-L. Ke et al. / Composite Structures 94 (2012) 2038–2047
conditions are discretized by using the differential quadrature (DQ) method. A direct iterative method is employed to determine the nonlinear frequencies and mode shapes of the piezoelectric nanobeams. The influences of the nonlocal parameter, temperature change and external electric voltage on the nonlinear vibration characteristics of the piezoelectric nanobeams are discussed in detail. Numerical results show that: (1) An increase in the nonlocal parameter leads to both smaller linear and nonlinear frequencies but a higher nonlinear frequency ratio; (2) The nonlinear frequency ratio increases with the increase of the temperature change; (3) A change in the external electric voltage from a positive value to a negative value leads to the decrease of the nonlinear frequency ratio; (4) The nonlocal parameter has a strong effect on the mode shapes for the C–H piezoelectric nanobeam than those for the H–H piezoelectric nanobeam.
nonlinear governing equations (A3)–(A5) for the piezoelectric nanobeams based on the nonlocal Euler beam model can be written as
A11
F 31
@ 2 U @W @ 2 W þ @x2 @x @x2
Appendix A Based on the Euler beam theory, the displacements of an arbitrary point in the beam along the x- and z-axes, denoted by ux (x, z, t) and uz(x, z, t) respectively, are
uz ðx; z; tÞ ¼ Wðx; tÞ:
ðA1Þ
The distribution of the electric potential for the piezoelectric nanobeam is given in Eq. (8). The von Kármán type nonlinear strain–displacement relationship is written as
ex ¼
" # 2 @2 2@ U ; U ðe aÞ 0 @x2 @t 2
ðA10Þ
@4W þ Z 1 ðe0 aÞ2 Z 2 @x4 " # 2 @2 2@ W ¼ I1 2 W ðe0 aÞ ; @x2 @t ðe0 aÞ2 ðNT þ N E Þ
F 31
@2W @2/ þ X 33 / X 11 2 ¼ 0: @x2 @x
ðA11Þ
ðA12Þ
The associated boundary conditions can be written as
@W ¼ / ¼ 0; @x
ðA13Þ
for a clamped end and
The work described in this paper was supported by National Natural Science Foundation of China (No. 11002019), Ph.D. Programs Foundation of Ministry of Education of China (No. 20100009120018) and Fundamental Research Funds for the Central Universities (No. 2009JBM073).
@W ; ux ðx; z; tÞ ¼ Uðx; tÞ z @x
¼ I1
@2/ @2W @2W D11 þ ðNT þ N E Þ 2 2 2 @x @x @x
U¼W ¼ Acknowledgements
!
2 @U @ 2 W 1 @W z 2 þ : @x @x 2 @x
ðA2Þ
By using the Hamilton’s principle, the equations of motion of the Euler nanobeam are derived as
@Nx @2U ¼ I1 2 @x @t @ 2 Mx @ @W @2W Nx ¼ I1 þ ; dW : 2 @x @x @x @t 2 Z h=2 @Dx d/ : cosðbzÞ þ b sinðbzÞDz dz ¼ 0: @x h=2
dU :
ðA3Þ ðA4Þ ðA5Þ
The corresponding boundary conditions at beam ends (x = 0, L) require
U ¼ 0 or Nx ¼ 0; x ¼ 0; L; @M x @W W ¼ 0 or ¼ 0; x ¼ 0; L; þ Nx @x @x @W ¼ 0 or M x ¼ 0; x ¼ 0; L; @x Z h=2 Dx cosðbzÞdz ¼ 0; x ¼ 0; L: / ¼ 0 or
ðA6Þ ðA7Þ ðA8Þ ðA9Þ
h=2
Similar with the derivation in the nonlocal Timoshenko beam model, we can re-express Eqs. (A3)–(A5) in the form of U, W and /. Then, the
" # @2W @2W @ @W 2 Nx ¼ 0; Mx ¼ F 31 / D11 þ ðe0 aÞ I1 @x2 @x @x @t 2 U ¼ W ¼ / ¼ 0;
ðA14Þ
for a hinged end. Then, The DQ method is used to solve the nonlinear equations (A10)–(A12) and the associated boundary of the piezoelectric nanobeams.
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Contents lists available at SciVerse ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory Liao-Liang Ke ⇑, Yue-Sheng Wang, Zheng-Dao Wang Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, PR China
a r t i c l e
i n f o
Article history: Available online 2 February 2012 Keywords: Piezoelectric materials Nonlinear vibration Nonlocal theory Nanobeams Size effect
a b s t r a c t This paper investigates the nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory and Timoshenko beam theory. The piezoelectric nanobeam is subjected to an applied voltage and a uniform temperature change. The nonlinear governing equations and boundary conditions are derived by using the Hamilton principle and discretized by using the differential quadrature (DQ) method. A direct iterative method is employed to determine the nonlinear frequencies and mode shapes of the piezoelectric nanobeams. A detailed parametric study is conducted to study the influences of the nonlocal parameter, temperature change and external electric voltage on the size-dependent nonlinear vibration characteristics of the piezoelectric nanobeams. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Since Wang and his co-authors [1] reported the ZnO piezoelectric nanostructures in Science magazine, various piezoelectric nanomaterials (e.g. ZnO, ZnS, PZT, GaN, BaTiO3, etc.) and their nanostructures (e.g. nanowires, nanobelts, nanorings, nanohelices, etc.) have received considerable attention from research communities [2–4]. It is found that the piezoelectric nanostructures have the novel electrical, mechanical and other physical and chemical properties than their bulk counterparts [2,4], and have the potential applications in many nanodevices, such as nanoresonators [5], field effect transistors [6], light emitting diodes [7], chemical sensors [8] and nanogenerators [9]. In these nanodevices, the dimension may vary from several hundred nanometers to just a few nanometers. On this scale, the size effects become very obvious. Many experiments and atomistic simulations [10–13] have demonstrated the size-dependent deformation behavior of the piezoelectric nanostructures. Using the piezoresponse force microscopy, Zhao et al. [10] observed that the effective piezoelectric coefficient of ZnO nanowires was much larger than that of the bulk ZnO. Chen et al. [11] reported a size dependence of the Young’s modulus in ZnO nanowires. They found that the Young’s modulus for the nanowires with diameters smaller than about 120 nm was increasing dramatically with the decreasing diameters. These studies gave important evidences of the sizedependent material properties of the piezoelectric nanostructures, and hence the size effect must be taken into account in the related theoretical and experimental studies. Because the classical continuum theory is a scale independent theory, many high-order theories, such as the strain gradient
⇑ Corresponding author. Tel.: +86 10 51684070; fax: +86 10 51682094. E-mail address: [email protected] (L.-L. Ke). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2012.01.023
theory, couple stress theory, micropolar theory, nonlocal elasticity theory, etc., are developed to characterize the size effect in the nanostructures by introducing an intrinsic length scale. Among them, the nonlocal elasticity theory developed by Eringen [14– 16] has been widely accepted and applied to analyze the size effect of the nanostructures. The nonlocal elasticity theory contains the information about the forces between atoms, and introduces the internal length scale into the constitutive equations as a material parameter. In recent years, the studies of nanostructures by using the nonlocal elasticity theory have been an area of active research. Based on this theory, the nonlocal nanobeam model, nonlocal nanoplate model and nonlocal nanoshell model were developed and employed to discuss the bending [17,18], buckling [19–21], linear vibration [22–25], nonlinear vibration [26–28], postbuckling [29,30] and wave propagation characteristics [31–33] of the carbon nanotubes, graphene sheets, mass sensors, nanowires, etc. For the use of the nonlocal theory, the identification of the nonlocal parameter is significant to properly understand the size effect of the nanostructures. Many investigators proposed that the nonlocal parameter should be less than 2 nm based on the molecular mechanics and molecular dynamic simulations [34–36]. More in detail, Narendar and Gopalakrishnan [37] tabulated the values of the nonlocal parameter obtained by various investigators using different methods. It is worth to mention that Narendar et al. [36] predicted the nonlocal scaling parameter for armchair and zigzag single-walled carbon nanotubes based on the molecular structural mechanics, nonlocal elasticity and wave propagation. They found that the nonlocal parameter is a function of diameter of carbon nanotube. Their work is very important for understanding the dependence of the nonlocal parameter on the size of the nanostructures. The above investigations [17–37] on the nanostructures are based on the nonlocal elasticity theory, which is unsuitable for a
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direct use in the piezoelectric materials. To the best of authors’ knowledge, no literature is available for the piezoelectric nanostructures based on the nonlocal theory. So far, only a few works were reported for the piezoelectric nanostructures based on the surface elasticity theory. The effect of the surface piezoelectricity on the electromechanical behavior of a piezoelectric ring was discussed by Huang and Yu [38]. Yan and Jiang [39] studied the vibration and buckling behaviors of piezoelectric nanobeams with the surface effect. They [40] also analyzed the surface effect on the electromechanical coupling and bending behaviors of piezoelectric nanowires. Shen and Hu [41] established a variational principle for nanosized dielectrics that includes the flexoelectric and surface effects. It should be pointed out that most nanodevices with piezoelectric nanowires or nanobelts as fundamental elements are beambased [39] structures. Therefore, investigating the mechanical behaviors of the piezoelectric nanobeams is of primary importance in the design of the nanodevices. The simplest beam model is the Euler beam model. However, the Euler beam model is inaccurate to predict the mechanical behaviors of the short beam because the effect of the transverse shear deformation becomes significant and cannot be neglected for the beam with smaller length-to-thickness ratios. In practical applications, the short nanobeams have been used as nanoweezers [42] and atomic force microscope tips [43]. In order to incorporate the effect of the transverse shear deformation, the Timoshenko beam model should be used for a better estimation of the mechanical behaviors of the short nanobeams. In this paper, the nonlinear vibration of the piezoelectric nanobeams is first investigated based on the nonlocal elasticity theory and Timoshenko beam theory. The piezoelectric nanobeam is subjected to an applied voltage and a uniform temperature change. The governing equations and boundary conditions are derived by using the Hamilton’s principle. The differential quadrature (DQ) method is employed to discretize the governing equations which are then solved by a direct iterative method to obtain the nonlinear frequencies and mode shapes of the piezoelectric nanobeams with different boundary conditions. The influences of the nonlocal parameter, temperature change and external electric voltage on the nonlinear vibration characteristics of the piezoelectric nanobeams are discussed in details. 2. Nonlocal theory for the piezoelectric materials In Eringen’s nonlocal elasticity theory [14–16], the stress at a point x in a body depends not only on the strain at that point but also on those at all points of the body. The nonlocal elasticity theory can satisfactorily explain some phenomena related to atomic and molecular scales such as high frequency vibration and wave dispersion. Recently, Zhou and his co-authors [44–46] extended the Eringen’s nonlocal elasticity theory to the piezoelectric materials. Mathematically, the basic equations for a homogeneous and nonlocal piezoelectric solid without body force can be written as
rij ¼ Di ¼
Z ZV
0
0
0
0
aðjx xj; sÞ½C ijkl ekl ðx Þ ekij Ek ðx Þ kij DTdx ;
ð1Þ
aðjx0 xj; sÞ½eikl ekl ðx0 Þ 2ik Ek ðx0 Þ þ pi DTdx0 ;
ð2Þ
electric potential. a(jx0 xj, s) in Eqs. (1) and (2) is the nonlocal attenuation function which incorporates into the constitutive equations at the reference point x produced by the local strain at the source x0 ; jx0 xj is the Euclidean distance. s = e0a/l is defined as the scale coefficient that incorporates the small scale factor, where e0 is a material constant determined experimentally or approximated by matching the dispersion curves of the plane waves with those of the atomic lattice dynamics; and a and l are the internal (e.g. lattice parameter, granular size) and external characteristic lengths (e.g. crack length, wavelength) of the nanostructures, respectively. Referring to Eringen [15], it is possible to represent the integral constitutive relations in an equivalent differential form as
rij ðe0 aÞ2 r2 rij ¼ C ijkl ekl ekij Ek kij DT;
ð5Þ
Di ðe0 aÞ2 r2 Di ¼ eikl ekl þ 2ik Ek þ pi DT;
ð6Þ
2
where r is the Laplace operator; e0a is the scale coefficient revealing the size effect on the response of structures in nanosize. 3. Nonlinear vibration analysis of the piezoelectric nanobeam By using the above nonlocal theory for the piezoelectric materials, this section will analyze the nonlinear vibration of the piezoelectric nanobeam under the thermo-electrical loading. Fig. 1 shows a piezoelectric nanobeam with length L and thickness h subjected to an applied voltage U(x, z, t) and a uniform temperature change DT. The poling direction of the piezoelectric medium is parallel to the positive z-axis, where (x, z) is the coordinate system shown in the figure. Based on the Timoshenko beam theory, the displacements of an arbitrary point in the beam along the x- and z-axes, denoted by ux(x, z, t) and uz (x, z, t) respectively, are
ux ðx; z; tÞ ¼ Uðx; tÞ þ zWðx; tÞ;
uz ðx; z; tÞ ¼ Wðx; tÞ;
where U(x, t) and W(x, t) are displacement components in the midplane; W(x, t) is the rotation of beam cross-section; and t is the time. In addition to the displacement field, we need to assume the distribution of the electric potential for the present piezoelectric nanobeam model. Recently, Jiang and Yan [47] studied the electromechanical response of a curved piezoelectric nanobeam with the consideration of the surface effects. In their work, they assumed a linear distribution of the electric potential in the thickness direction of the piezoelectric nanobeams. Though this assumption violates the Maxwell equation, their results are reasonable for the electromechanical response of the piezoelectric nanobeams. In the present analysis, we follow Wang [48] and assume the electric potential as a combination of a cosine and linear variation, which satisfies the Maxwell equation. It can be written as
Uðx; z; tÞ ¼ cosðbzÞ/ðx; tÞ þ
2zV 0 iXt e ; h
1 2
eij ¼ ðui;j þ uj;i Þ; Ei ¼ U;i
ð3Þ ð4Þ
where rij, eij, Di, Ei and ui are the components of the stress, strain, electric field, electric displacement and displacement, respectively; Cijkl, ekij, 2ik, kij, pi and q are the elastic constants, piezoelectric constants, dielectric constants, thermal moduli, pyroelectric constants and mass density, respectively; DT is the temperature change; U is the
ð8Þ
where b = p/h; /(x, t) is the spatial and time variation of the electric potential in the x-direction; V0 is the external electric voltage; X is the natural frequency of the piezoelectric nanobeam. Note that / (x, t) must satisfy the electric boundary conditions.
V
rij;j ¼ qu€ i ; Di;i ¼ 0;
ð7Þ
z
z
Fig. 1. Schematic configuration of a piezoelectric nanobeam.
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The von Kármán type nonlinear strain–displacement relations are given by
exx ¼
2 @U 1 @W @W @W þ þ W: þz ;c ¼ @x 2 @x @x @x xz
ð9Þ
@U @/ @U 2V 0 iXt ; Ez ¼ ¼ cosðbzÞ ¼ b sinðbzÞ/ e ; @x @x @z h
ð10Þ
The strain energy PS of the piezoelectric nanobeams is given by
PS ¼
1 2
Z
Z
L
0
1 2
Z
h=2
¼
1 2
ðrxx exx þ rxz cxz Dx Ex Dz Ez Þdz dx;
ð11Þ
0
Z
h=2
1 2
Z
L
0
" 2 # @U 1 @W @W @W þ rxx z rxx þ þ W dz dx þ rxz @x 2 @x @x @x Z h=2 2V 0 iXt Dx cosðbzÞ Dz b sinðbzÞ/ þ e h h=2
(
@/ dz dx @x
Z
L
"
"
@/ dz dx; @x
where the normal resultant force Nx, the bending moment Mx, and the transverse shear force Qx are calculated from
Z
@M x @2W Q x ¼ I3 2 ; @x @t Z h=2 @Dx cosðbzÞ þ b sinðbzÞDz dz ¼ 0: @x h=2
h=2
rxx dz; Mx ¼
h=2
Z
h=2
rxx zdz; Q x ¼
h=2
Z
h=2
rxz dz:
ð13Þ
h=2
2 2 # Z " 2 1 L @U @W @W dx; PK ¼ I1 þ I1 þI3 2 0 @t @t @t
ð24Þ
W ¼ 0 or Q x ¼ 0; x ¼ 0; L;
ð25Þ
W ¼ 0 or M x ¼ 0; x ¼ 0; L;
ð26Þ
/ ¼ 0 or
Z
h=2
Dx cosðbzÞdz ¼ 0;
x ¼ 0; L:
@2r rxx ðe0 aÞ2 2xx ¼ c11 exx e31 Ez k1 DT; @x 2 2 @ rxz rxz ðe0 aÞ ¼ c44 cxz e15 Ex ; @x2 @ 2 Dx Dx ðe0 aÞ2 ¼ e15 cxz þ 211 Ex ; @x2 2 @ Dz Dz ðe0 aÞ2 ¼ e31 exx þ 233 Ez þ p1 DT: @x2
" 2 # @ 2 Nx @U 1 @W þ 2e31 V 0 k1 hDT; þ Nx ðe0 aÞ ¼ A11 @x 2 @x @x2 @ 2 Mx @W ¼ D11 þ F 31 /; @x2 @x 2 @ Qx @W @/ ks E15 ¼ k A W Q x ðe0 aÞ2 þ ; s 44 @x @x @x2 " # Z h=2 @ 2 Dx @W @/ cosðbzÞdz ¼ E þ X 11 þ ; Dx ðe0 aÞ2 W 15 2 @x @x @x h=2 " # Z h=2 @ 2 Dz @W b sinðbzÞdz ¼ F 31 Dz ðe0 aÞ2 X 33 /; @x2 @x h=2 Mx ðe0 aÞ2
ð28Þ ð29Þ ð30Þ ð31Þ ð32Þ
where ks is the shear correction factor depending on the shape of the cross-section of the beam; and
Z
h=2
c11 dz;
D11 ¼
h=2
ð14Þ A44 ¼
Z
h=2
c44 dz;
F 31 ¼
h=2
E15 ¼
Z
Z Z
h=2
c11 z2 dz;
h=2 h=2
e31 b sinðbzÞzdz;
e15 cosðbzÞdz;
X 11 ¼
h=2
X 33 ¼
Z
h=2
ð33Þ
h=2
h=2
Z
h=2
211 cos2 ðbzÞdz;
h=2
233 ½b sinðbzÞ2 dz:
ð34Þ
h=2
ð15Þ ð16Þ
By substituting Eqs. (20)–(22) into Eqs. (28)–(30), the explicit expressions of the nonlocal normal resultant force Nx, bending moment Mx and shear force Qx can be obtained as
ð17Þ ð18Þ
" Nx ¼ A11
Consider the Hamilton’s principle
Mx ¼ D11
t
ðdPS dPK Þdt ¼ 0:
ð27Þ
h=2
A11 ¼
where {I1, I3} = {qh, qh3/12}. For a beam type structure, both thickness and width are much smaller than its length. Therefore, for beams with the transverse motion in the x–z plane, the nonlocal constitutive relations (5) and (6) can be approximated to one-dimensional form as
0
ð23Þ
U ¼ 0 or Nx ¼ 0; x ¼ 0; L;
The kinetic energy PK can be calculated from
Z
ð22Þ
The corresponding boundary conditions at the beam ends (x = 0, L) require
2
ð12Þ
Nx ¼
d/ :
ð21Þ
From Eqs. (15)–(18), we have
2 # # @U 1 @W @W @W þ þ W dz dx Nx þ Qx þ Mx @x 2 @x @x @x 0 Z Z 1 L h=2 2V 0 iXt Dx cosðbzÞ Dz b sinðbzÞ/ þ e þ 2 0 h=2 h
ð20Þ
L
h=2
þ
dW :
h=2
Which, by submitting Eqs. (9) and (10) into Eq. (11), can be rewritten as
PS ¼
@Nx @2U ¼ I1 2 @x @t @Q x @ @W @2W Nx ¼ I1 ; dW : þ @x @x @x @t2
dU :
According to Eq. (8), the electric fields can be expressed as
Ex ¼
Then substituting Eqs. (12) and (14) into Eq. (19), integrating it by parts and setting the coefficients of dU, dW, d W and d/ to zero lead to the equations of motion as
ð19Þ
2 # @U 1 @W @3U þ NE þ NT þ ðe0 aÞ2 I1 þ ; @x 2 @x @x@t2
ð35Þ
" # @W @3W @2W @ @W N ; þ I þ F 31 / þ ðe0 aÞ2 I3 1 x @x @x @x @t 2 @x@t 2 ð36Þ
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Q x ¼ ks A44
"
@W @/ @3W @2 @W þ W ks E15 þ ðe0 aÞ2 I1 Nx @x @x @x @x@t2 @x2
!
# ;
ð37Þ where NT = k1hDT and NE = 2e31V0 are the normal force induced by the temperature change DT and external electric voltage V0, respectively. Note that the shear correction factor, which is quite complex involving integral expressions, not only depends on the material and geometric parameters but also on the loading and boundary conditions. For a macroscale structure with a rectangular cross-section, it is suitable to take the shear correction factor as 5/6. However, the shear correction factor for a macroscale structures may not be applicable to a nanoscale structure duo to the size-dependent material properties. So far, no experimental or theoretical results of the shear correction factor are available for nanostructures. Therefore in the present paper, following Reddy [49], we take the shear correction factor for the nanobeams approximately as the same value (i.e. ks = 5/6) as that for the macroscale beams. Unlike Nx, Mx and Qx in Eqs. (35)–(37), the electric displacements Dx and Dz cannot be obtained explicitly by substituting Eq. (23) into Eqs. (31) and (32) because there are two unknowns but only one equilibrium Eq. (23). However, we can re-express Eq. (23) in the form of U, W, W and / using Eqs. (31) and (32). Then, with the help of Eqs. (35)–(37) and Eqs. (31) and (32), the nonlinear governing Eqs. (20)–(23) for the piezoelectric nanobeams can be written as
@ 2 U @W @ 2 W þ @x2 @x @x2
A11
!
" # 2 @2 2@ U ¼ I1 2 U ðe0 aÞ ; @x2 @t
ð38Þ
ðX 11 ; X 33 Þ ¼
i¼
t L
sffiffiffiffiffiffiffi I1 : A11
! X 11 /20 X 33 /20 ; ; 2 A11 A11 h
ðI1 ; I3 Þ ¼
NT ¼
I 1 I3 ; ; I 1 I 1 h2 ð44bÞ
khDT ; A11
NE ¼
2e31 V 0 ; A11 ð44cÞ
Eqs. (38)–(41) can be expressed in the dimensionless form as
! ! 4 1 @w @ 2 w @2 2@ u þ ¼ I1 2 u l ; @i @f2 g @f @f2 @f2
@2u
A11
! @w @2u @2w ks E15 2 þ ðNT þ NE Þ 2 2 @f @f @f @f " # @4w @2 @2w l2 ðNT þ NE Þ 4 þ Z 1 l2 Z 2 ¼ I1 2 w l2 2 ; @s @f @f @2w
ks A44
@u @w 2 k A g g w þ F 31 g þ ks E15 g þ s 44 @f @f @f2 " # 2 2 @ @ w ¼ I 3 2 w l2 2 ; @s @f
D11
@2W @W @2/ @2W ks E15 2 þ ðNT þ NE Þ 2 þ @x2 @x @x @x
ð45Þ
þg
ð46Þ
@2w
! @w @2w @w @2u þ X 11 2 X 33 g2 / ¼ 0; þ E15 F 31 g þg 2 @f @f @f @f
!
ks A44
A11 A44 D11 ; ; ; A11 A11 A11 h2 E15 /0 F 31 /0 E15 ; F 31 ¼ ; ; A11 h A11 h
ðA11 ; A44 ; D11 Þ ¼
!
ð47Þ
ð48Þ
where
@4W ðe0 aÞ ðNT þ NE Þ þ Z 1 ðe0 aÞ2 Z 2 @x4 " # 2 @2 2@ W ¼ I1 2 W ðe0 aÞ ; @x2 @t 2
Z1 ¼ ð39Þ
@2W @W @/ @/ þ F 31 þ þ ks E15 k A W s 44 @x @x @x @x2 " # @2 @2W ¼ I3 2 W ðe0 aÞ2 2 ; @x @t
D11
F 31
ð40Þ
! @W @2W @W @2/ þ X 11 2 X 33 / ¼ 0; þ þ E15 2 @x @x @x @x
ð41Þ
" # 2 A11 @ 2 u @w 3 @w @ 2 w @u @ 2 w ; þ þ g @f2 @f 2g @f @f2 @f @f2
! A11 @ 4 u @w @3u @2w @ 2 u @ 3 w @u @ 4 w þ3 3 Z2 ¼ þ3 2 þ @f @f4 g @f4 @f @f @f2 @f @f3 2 3 !3 2 A11 4 @ 2 w @w @ 2 w @ 3 w 3 @w @ 4 w5 : þ 2 3 þ9 þ @f @f2 @f3 2 @f g @f2 @f4
ð49Þ
ð50Þ
Assume that the electric potential is zero at the ends of the nanobeam. Then, the associated boundary conditions can be written in the dimensionless form as
u ¼ w ¼ w ¼ u ¼ 0;
ð51Þ
for a clamped end, and
where
" # 2 @ 2 U @W 3 @W @ 2 W @U @ 2 W Z 1 ¼ A11 þ þ ; @x2 @x 2 @x @x2 @x @x2
ð42Þ
! @ 4 U @W @3U @2W @ 2 U @ 3 W @U @ 4 W þ 3 þ 3 þ @x4 @x @x3 @x2 @x2 @x3 @x @x4 2 3 !3 2 4 2 2 3 @ W @W @ W @ W 3 @W @ W 5: þ A11 43 þ9 þ @x2 @x @x2 @x3 2 @x @x4
Z 2 ¼ A11
! 2 @w @w @2u 2 @ 2 þ F 31 g/ þ l þ I1 gw l I1 g 2 D11 I3 @f @f @ i2 @f (" ) 2 # 2 2 @ w l A11 @ @u 1 @w @w 2 ¼ 0; þ l ðNT þ NE Þg 2 @f g @f @f 2g @f @f u ¼ w ¼ u ¼ 0;
ð43Þ
ð52Þ
for a hinged end. 4. Solution method
Introducing the following dimensionless quantities
x W f¼ ; w¼ ; L h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /0 ¼ e33 =A11 ;
w ¼ W;
L g¼ ; h
e a l¼ 0 ; L
/ u¼ ; /0 ð44aÞ
The DQ method [50] is used to solve the nonlinear governing equations (45)–(48) with the associated boundary conditions of the piezoelectric nanobeams. According to the DQ method, we discretize the beam domain by N nodes along the x-axis, and
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L.-L. Ke et al. / Composite Structures 94 (2012) 2038–2047
designate the values of u, w, w and u and their kth partial derivatives with respect to x can be approximated by N X
fu; w; w; ug ¼
lm ðfÞfum ðfm ; iÞ; wm ðfm ; iÞ; wm ðfm ; iÞ; um ðfm ; iÞg;
m¼1
ð53Þ @k k
@f
fu; w;w; ugjx¼xi ¼
N X
ðkÞ
C im fum ðfm ; iÞ;wm ðfm ; iÞ; wm ðfm ; iÞ; um ðfm ; iÞg;
m¼1
with the over dot denoting the partial derivative with respect to the dimensionless time i. The associated boundary conditions can be handled in the same way. For a hinged–hinged (H–H) piezoelectric nanobeam, we have
!2 N N X 3 X ð1Þ ð2Þ C 1m wm C 1m wm 2g m¼1 g m¼1 m¼1 m¼1 # " N N N N X X X X ð1Þ ð2Þ ð1Þ ð2Þ € 2 €1 C 1m um C 1m wm D11 C 1m wm þ l I3 C 1m w þ m þ I1 gw
F 31 gu1
m¼1
ð54Þ ðkÞ
where lm(x) is the Lagrange interpolation polynomials; and C im is the weighting coefficients whose recursive formula can be found in Shu [50]. The cosine pattern is used to generate the DQ point system
1 pði 1Þ 1 cos ; fi ¼ 2 N1
i ¼ 1; 2; . . . N:
ð55Þ
Applying the relationships (53) and (54) to Eqs. (45)–(48), one obtains a set of nonlinear ordinary differential equations ! N X
A11
N 1X
ð2Þ
C im um þ
g m¼1
m¼1 2
€i l ¼ I1 u
N X
N X
ð1Þ
C im wm
N X
ks A44
ð2Þ
C im wm þ g
m¼1
N X
ð56Þ
; ! ð1Þ
C im wm
ks E15
m¼1
þ ðNT þ NE Þ
N X
N X
ð2Þ
C im wm l2 ðNT þ NE Þ
m¼1
D11
ð2Þ
C im wm ks A44 g
m¼1
N X
þ ðF 31 g þ ks E15 gÞ
N X
ð1Þ C im
N X
ð1Þ
C im wm þ g2 wi
um ¼ I 3
F 31 g
ð1Þ C im wm
þ E15
m¼1
þ X 11
ð2Þ C im wm
þg
m¼1 N X
N X
N X
# ð2Þ € C im w m
ð58Þ
;
ð59Þ
where
Z 1i ¼
g
þ
m¼1 N X
m¼1
ð1Þ
C im um
m¼1
Z 2i ¼
2 A11 4
g2
N X
3
N X
N X
N X m¼1
!2
N X m¼1
ð2Þ
!3 ð2Þ C im wm
þ9
ð60Þ
ð3Þ
m¼1
C im um
ð1Þ C im um
N X
!2
N X
ð2Þ
C im wm þ 3
m¼1
m¼1
3
ð4Þ C im wm 5
ð4Þ C im wm
# ;
N X
ð2Þ
C im wm
m¼1
þ
m¼1
N X
N X
ð1Þ
C im wm
N X m¼1
" N A11 X
g ð2Þ
N X
ð3Þ
C im wm
m¼1
ð4Þ
C im um
m¼1
C im um
ð2Þ
C Nm um
N X
m¼1 N X
ð1Þ
C Nm wm þ
m¼1
ð2Þ
€ m ðNT þ NE Þg C Nm u
m¼1
N X
#
m¼1
ð2Þ
C Nm wm ¼ 0;
m¼1
uN ¼ wN ¼ uN ¼ 0;
at f ¼ 1:
ð63Þ
2 N A11 4X
at f ¼ 0;
ð64Þ
!2 N N X 3 X ð1Þ ð2Þ F 31 guN þ C Nm wm C Nm wm 2 g m¼1 g m¼1 m¼1 m¼1 # " N N N N X X X X ð1Þ ð2Þ ð1Þ ð2Þ € 2 €N C Nm um C Nm wm D11 C Nm wm þ l I3 C Nm w þ m þ I 1 gw ð2Þ C Nm um
N X
m¼1 N X
ð1Þ C Nm wm
m¼1
ð2Þ
€ m ðNT þ NE Þg C Nm u
N X
#
m¼1
ð2Þ
C Nm wm ¼ 0;
m¼1
uN ¼ wN ¼ uN ¼ 0;
at f ¼ 1:
ð65Þ
fui gT ; fwi gT ; fwi gT ; fui gT
T
;
i ¼ 1; 2; . . . N;
ð66Þ
N X
N X
€ ¼ 0; ðKL þ KNL Þd þ Md
ð67Þ
where M is the mass matrix; KL is the stiffness matrix; and KNL is nonlinear stiffness matrix that is the function of d. KL, KNL and M are 4N 4N matrices. Expand the dynamic displacement vector d in the form of
ð2Þ C im wm
C im wm ;
m¼1
m¼1
þ
#
N 3 X ð1Þ þ C wm 2g m¼1 im
m¼1
N 3 X ð1Þ C wm þ 2 m¼1 im
þ3
ð1Þ C im wm
ð62Þ
Eqs. (56)–(59), together with the boundary conditions, can be expressed in the matrix form as
m¼1
ð2Þ
N X
ð2Þ
C 1m wm ¼ 0;
!2 N N X 3 X ð1Þ ð2Þ C Nm wm C Nm wm 2g m¼1 g m¼1 m¼1 m¼1 # " N N N N X X X X ð1Þ ð2Þ ð1Þ ð2Þ € 2 €N C Nm um C Nm wm D11 C Nm wm þ l I3 C Nm w þ m þ I 1 gw
F 31 guN
d¼
ð1Þ C im wm
C im um X 33 g2 u ¼ 0;
ð2Þ C im um
m¼1
By denoting the unknown dynamic displacement vector as
!
m¼1
2 N A11 4X
#
at f ¼ 0;
m¼1
m¼1
N X
2 N A11 4X
N X m¼1
u1 ¼ w1 ¼ u1 ¼ 0;
l2 I 1 g
€ i l2 w
ð2Þ
€ m ðNT þ NE Þg C 1m u
m¼1
m¼1
!
m¼1 N X
ð4Þ
C im wm þ Z 1i l2 Z 2i ð57Þ
"
ð1Þ
C 1m wm þ
m¼1
u1 ¼ w1 ¼ w1 ¼ u1 ¼ 0;
m¼1
m¼1
N X
For a clamped–hinged (C–H) piezoelectric nanobeam, we have ð2Þ
C im um
m¼1
" # N X @2 ð2Þ € i l2 €m ; ¼ I1 2 w C im w @s m¼1 N X
l2 I 1 g
l2 I 1 g
m¼1
ð2Þ
C 1m um
m¼1 N X
m¼1
m¼1
!
ð2Þ €m C im u
ð2Þ
C im wm
2 N A11 4X
ð1Þ
C im wm
m¼1
d ¼ d eixi ;
ð68Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi where x ¼ XL I1 =A11 represents the dimensionless natural fren T T quency of the piezoelectric nanobeams; and d ¼ ui ; wi ; T T T wi ; ui g is the vibration mode shape vector. Then substitution of Eq. (68) into Eq. (67) yields the nonlinear eigenvalue equation
ðKL þ KNL Þd x2 Md ¼ 0:
ð69Þ
The above nonlinear equations can be solved through a direct iterative process below:
ð3Þ
C im wm
m¼1
ð61Þ
Step 1: By neglecting the nonlinear matrix KNL, a linear eigenvalue (xl) and the associated eigenvector are obtained
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L.-L. Ke et al. / Composite Structures 94 (2012) 2038–2047 Table 1 Material properties of PZT-4 [51,52]. c11 (GPa)
e31 (C/m2)
c44 (GPa)
132
26
e15 (C/m2) 14.1
4.1
211 (C/Vm) 5.841 10
from Eq. (69). The eigenvector is then appropriately scaled up such that the maximum transverse displacement is equal to a given vibration amplitude wmax . Note that wmax ¼ wð0:5Þ for the H–H nanobeam and wmax ¼ wð0:57Þ for the C–H piezoelectric nanobeam. Step 2: Calculate KNL by using the above eigenvector. Then a new eigenvalue and the associated eigenvector are obtained from the updated eigensystem (69). Step 3: The eigenvector is scaled up again and step 2 is repeated until the relative error between the given vibration amplitude and the maximum transverse displacement wmax is within 0.1%.
9
233 (N/m2 K)
k1 (N/m2 K)
9
5
7.124 10
4.738 10
p1 (C/m2 K) 2.5 10
5
q (kg/m3) 7500
experiments or molecular dynamic (MD) simulation. However, it is lack of such works on the piezoelectric nanomaterials at present. Therefore, in order to quantitatively understand the thermoelectric-mechanical responses of the piezoelectric nanobeams, we choose the material properties of the macroscopic piezoelectric materials for the case study. The effects of the dimensionless nonlcoal parameter l, temperature change DT and external electric voltage V0 on the nonlinear vibration frequencies and mode shapes of the piezoelectric nanobeams are discussed in detail. In Figs. 2–4, the linear fundamental frequencies xl are also given. Unless otherwise stated, it is assumed that the length of the nanobeam L = 12 nm, thickness h = 2 nm and shear correction factor ks = 5/6.
5. Numerical results
5.1. Comparison and convergence studies
In this section, we present the numerical results for the nonlinear vibration of the hinged–hinged (H–H) and clamped–hinged (C–H) piezoelectric nanobeams. The nanobeam is made of PZT-4 with the thermo-electric-mechanical material properties listed in Table 1 [51,52], which are taken from the macroscopic piezoelectric materials. Actually, the materials properties must be sizedependent in the piezoelectric nanomaterials and their thermoelectric-mechanical material properties should be determined by
Table 2 lists the dimensionless linear and nonlinear frequencies of the piezoelectric nanobeams with varying total number of nodes (N) in the DQ method (here we take wmax ¼ 0:5; DT ¼ 0 C; V 0 ¼ 0V and l = 0.1). It is seen that the results become closer to each other as N increases and those with N = 14 and 16 are identical. Hence, N = 14 is used in all of the following calculations. Table 3 compares the dimensionless linear and nonlinear frequencies of the piezoelectric nanobeams obtained from both
(a)
2.0
(a)
H-H:
1.6
o
ΔT = 0 C
1.8
(ωl = 0.4263) o
ΔT = 100 C (ωl = 0.4222) o
ωnl / ω l
ωnl / ω l
H-H:
μ = 0.00 (ωl = 0.4468) μ = 0.05 (ωl = 0.4413) μ = 0.10 (ωl = 0.4263) μ = 0.15 (ωl = 0.4042)
1.8
2.0
1.4
1.2
ΔT = 200 C (ωl = 0.4180)
1.6
o
ΔT = 300 C (ωl = 0.4138)
1.4
1.2
1.0 0.0
0.2
0.4
0.6
1.0 0.0
0.8
0.2
0.4
*
w max
1.5 1.4
ωnl / ωl
(b)
1.6
C-H: μ = 0.00 (ωl = 0.6323) μ = 0.05 (ωl = 0.6239) μ = 0.10 (ωl = 0.6006) μ = 0.15 (ωl = 0.5668)
1.4
0.8
(ωl = 0.6006)
1.1
1.1
0.6
0.8
w*max Fig. 2. The effect of the nonlocal parameter l on nonlinear frequency ratio versus amplitude curves of the piezoelectric nanobeams with DT = 0 °C and V0 = 0: (a) H– H; and (b) C–H.
o
ΔT = 200 C (ωl = 0.5938) o
1.3 1.2
0.4
o
ΔT = 0 C o
1.2
0.2
0.6
C-H: ΔT = 100 C (ωl = 0.5972)
1.3
1.0 0.0
0.8
1.6 1.5
ωnl / ωl
(b)
0.6
w*max
1.0 0.0
ΔT = 300 C (ωl = 0.5904)
0.2
0.4
w*max Fig. 3. The effect of the temperature change DT (°C) on nonlinear frequency ratio versus amplitude curves of the piezoelectric nanobeams with l = 0.1 and V0 = 0: (a) H–H; and (b) C–H.
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L.-L. Ke et al. / Composite Structures 94 (2012) 2038–2047
(a)
(a)
2.0
2.0
H-H:
1.6
L / h = 6 (ωl = 0.4263) L / h = 8 (ωl = 0.3290) L / h = 12 (ωl = 0.2243) L / h = 20 (ωl = 0.1362)
1.8
ωnl/ ωl
ωnl / ω l
H-H: V 0 = 0.1 V ( ωl = 0.3893) V 0 = 0.05 V ( ωl = 0.4082) V0 = 0 V ( ωl = 0.4263) V 0 = -0.05 V ( ωl = 0.4436) V 0 = -0.1 V ( ωl = 0.4602)
1.8
1.4
1.6
1.4
1.2
1.2
1.0 0.0
0.2
0.4
0.6
1.0 0.0
0.8
0.2
w max
(b) 1.6
(b)
0.8
0.6
0.8
C-H: 1.5
V 0 = 0.1 V ( ωl = 0.5707) V 0 = 0.05 V ( ωl = 0.5858) V0 = 0 V ( ωl = 0.6006) V 0 = -0.05 V ( ωl = 0.6149) V 0 = -0.1 V ( ωl = 0.6289)
1.4
ωnl/ ωl
ωnl / ω l
0.6
1.6
C-H:
1.4
0.4
w*max
*
1.2
L / h = 6 (ωl = 0.6006) L / h = 8 (ωl = 0.4806) L / h = 12 (ωl = 0.3378) L / h = 20 (ωl = 0.2088)
1.3 1.2 1.1
1.0 0.0
0.2
0.4
0.6
1.0 0.0
0.8
0.2
*
w max Fig. 4. The effect of the external electric voltage V0 on nonlinear frequency ratio versus amplitude curves of the piezoelectric nanobeams with l = 0.1 and DT = 0 °C: (a) H–H; and (b) C–H.
Table 2 Dimensionless linear and nonlinear frequencies of the H–H and C–H piezoelectric nanobeams with varying total numbers of the nodes N (wmax ¼ 0:5; DT ¼ 0 C; V 0 ¼ 0 and l = 0.1). N
5 6 8 10 14 16
H–H
0.4
w*max
C–H
xl
xnl
xl
xnl
0.41974 0.42447 0.42629 0.42626 0.42626 0.42626
0.47030 0.58486 0.58903 0.58909 0.58905 0.58905
0.59911 0.59891 0.60060 0.60056 0.60057 0.60057
– 0.65786 0.73393 0.73865 0.73852 0.73855
nonlocal Euler beam model and Timoshenko beam model with wmax ¼ 0:5; h ¼ 2 nm; DT ¼ 0 C; V 0 ¼ 0V and l = 0.1. The governing equations and boundary conditions of the piezoelectric
Fig. 5. The effect of the length-to-thickness ratioL/h on nonlinear frequency ratio versus amplitude curves of the piezoelectric nanobeams with l = 0.1, V0 = 0, DT = 0 and h = 2 nm: (a) H–H; and (b) C–H.
nanobeams based on the Euler beam theory and nonlocal theory are given in Appendix A. The difference between the results obtained from the Euler beam model and Timoshenko beam model is observed to be quite small when L/h P 16 but relatively large when L/h < 16. This is because the transverse shear deformation which tends to be more significant for the beams with smaller slenderness ratio is included in the Timoshenko beam theory but neglected in the Euler beam theory. 5.2. Nonlinear vibration analysis Fig. 2 plots the effect of the nonlocal parameter l on the nonlinear frequency ratio versus the amplitude curves of the piezoelectric nanobeams with DT = 0 °C and V0 = 0. Note that the nonlocal parameter l = 0 corresponds to the classical naobeams without the nonlocal effect. Obviously, the nonlinear frequency ratio increases as the vibration amplitude increases for both H–H and
Table 3 Comparison of the dimensionless linear and nonlinear frequencies obtained from Euler beam model and Timoshenko beam model with wmax ¼ 0:5; h ¼ 2 nm; DT ¼ 0 C; V 0 ¼ 0V and l = 0.1. L/h
Euler beam (H–H)
xl 6 8 10 16 20 30
0.4570 0.3428 0.2742 0.1714 0.1371 0.0914
Timoshenko beam (H–H)
Euler beam (C–H)
Timoshenko beam (C–H)
xnl
xl
xnl
xl
xnl
xl
xnl
0.6140 0.4608 0.3687 0.2305 0.1844 0.1229
0.4263 0.3290 0.2670 0.1696 0.1362 0.0911
0.5891 0.4495 0.3628 0.2290 0.1836 0.1227
0.7077 0.5310 0.4250 0.2658 0.2127 0.1420
0.8456 0.6354 0.5088 0.3184 0.2548 0.1700
0.6006 0.4806 0.3976 0.2585 0.2088 0.1406
0.7385 0.5833 0.4800 0.3105 0.2505 0.1685
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L.-L. Ke et al. / Composite Structures 94 (2012) 2038–2047
(a)
(a)
0.5
0.00006
0.4 0.00004
ϕ*
w*
0.3
H-H: μ = 0.00 μ = 0.05 μ = 0.10 μ = 0.15
0.2
0.1
0.0 0.0
0.2
0.4
0.6
H-H: μ = 0.00 μ = 0.05 μ = 0.10 μ = 0.15
0.00002
0.8
0.00000 0.0
1.0
0.2
0.4
x/L
(b)
(b)
0.5
0.6
0.8
1.0
x/L
0.4
0.00009
0.00006
ϕ*
w*
0.3
C-H: μ = 0.00 μ = 0.05 μ = 0.10 μ = 0.15
0.2
0.1
0.0 0.0
0.2
0.4
0.6
C-H:
0.00003
μ = 0.00 μ = 0.05 μ = 0.10 μ = 0.15
0.00000
0.8
1.0
-0.00003 0.0
C–H piezoelectric nanobeams. The nonlocal parameter has a significant effect on the nonlinear vibration behavior. At a given vibration amplitude, an increase in the nonlocal parameter leads to both smaller linear and nonlinear frequencies but a higher nonlinear frequency ratio. The reason is that the presence of the nonlocal effect tends to decrease the stiffness of the nanobeams and hence decreases the values of the linear and nonlinear frequencies. The C–H piezoelectric nanobeam has the higher linear and nonlinear frequencies than the H–H one since the end support is the stronger for the C–H piezoelectric nanobeam than for the H–H piezoelectric nanobeam. Fig. 3 shows the effect of the temperature change DT(°C) on the curves of the nonlinear frequency ratio versus the amplitude for the piezoelectric nanobeams with l = 0.1 and V0 = 0. With the increase of the temperature change, the linear and nonlinear frequencies decrease for both H–H and C–H piezoelectric nanobeams. The reason is that a larger temperature change results in more reduction in the nanobeam stiffness, and hence leads to lower linear and nonlinear frequencies of the nanobeams. At a given vibration amplitude, the nonlinear frequency ratio increases as the temperature change increases. The effect of the external electric voltage V0 on the curves of the nonlinear frequency ratio versus the amplitude for the piezoelectric nanobeams is depicted in Fig. 4 with l = 0.1 and DT = 0°C. It is observed that the positive/negative voltage decreases/increases the linear and nonlinear frequencies of the nanobeams. This is due to the fact that the axial compressive and tensile forces are generated in the nanobeams by the applied positive and negative voltages, respectively. At a given vibration amplitude, it is found that a change in the external electric voltage from 0.1V to 0.1V leads to the decrease of the nonlinear frequency ratio. Comparing
0.4
0.6
0.8
1.0
x/L
x/L Fig. 6. The effect of nonlocal parameter on the nonlinear mode shapes (deflection w⁄) of the piezoelectric nanobeams with DT = 0 °C and V0 = 0: (a) H–H; and (b) C–H.
0.2
Fig. 7. The effect of nonlocal parameter on the nonlinear mode shapes (electrical potential /⁄) of the piezoelectric nanobeams with DT = 0 °C and V0 = 0: (a) H–H; and (b) C–H.
the effects of the temperature change and external electric voltage on the nonlinear vibration behaviors of the piezoelectric nanobeams in Figs. 3 and 4, we find that the external electric voltage has the greater effect than the temperature change. This phenomenon can be explained from the expressions of NT and NE in which the coefficients of NT and NE have the relation of j k1hj < j2e31j. Fig. 5 plots the effect of the length-to-thickness ratio L/h on the curves of the nonlinear frequency ratio versus the amplitude for the piezoelectric nanobeams with l = 0.1, V0 = 0, DT = 0 and h = 2 nm. Both linear and nonlinear frequency ratios decrease as the length-to-thickness ratio increases. The effect of slenderness ratio on the nonlinear frequency ratio tends to be very weak for the long beams (L/h P 12). The effect of the nonlocal parameter on the nonlinear mode shapes (the dimensionless displacement w⁄ and electric potential u⁄) of the piezoelectric nanobeams are plotted in Figs. 6 and 7 with DT = 0 °C and V0 = 0. The maximum displacement and electric potential occur at the midpoint of the H–H piezoelectric nanobeams, buts slightly deviate from the center of the C–H piezoelectric nanobeams. For the H–H piezoelectric nanobeam, the nonlocal parameter nearly has no effect on the mode shape w⁄, while has a slight effect on the mode shape u⁄. However, the nonlocal parameter has a relatively large effect on the mode shapes (w⁄ and u⁄) for the C–H piezoelectric nanobeam, especially for the mode shape of the electric potential. 6. Conclusions This paper investigates the nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory and Timoshenko beam theory. The nonlinear governing equations and boundary
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L.-L. Ke et al. / Composite Structures 94 (2012) 2038–2047
conditions are discretized by using the differential quadrature (DQ) method. A direct iterative method is employed to determine the nonlinear frequencies and mode shapes of the piezoelectric nanobeams. The influences of the nonlocal parameter, temperature change and external electric voltage on the nonlinear vibration characteristics of the piezoelectric nanobeams are discussed in detail. Numerical results show that: (1) An increase in the nonlocal parameter leads to both smaller linear and nonlinear frequencies but a higher nonlinear frequency ratio; (2) The nonlinear frequency ratio increases with the increase of the temperature change; (3) A change in the external electric voltage from a positive value to a negative value leads to the decrease of the nonlinear frequency ratio; (4) The nonlocal parameter has a strong effect on the mode shapes for the C–H piezoelectric nanobeam than those for the H–H piezoelectric nanobeam.
nonlinear governing equations (A3)–(A5) for the piezoelectric nanobeams based on the nonlocal Euler beam model can be written as
A11
F 31
@ 2 U @W @ 2 W þ @x2 @x @x2
Appendix A Based on the Euler beam theory, the displacements of an arbitrary point in the beam along the x- and z-axes, denoted by ux (x, z, t) and uz(x, z, t) respectively, are
uz ðx; z; tÞ ¼ Wðx; tÞ:
ðA1Þ
The distribution of the electric potential for the piezoelectric nanobeam is given in Eq. (8). The von Kármán type nonlinear strain–displacement relationship is written as
ex ¼
" # 2 @2 2@ U ; U ðe aÞ 0 @x2 @t 2
ðA10Þ
@4W þ Z 1 ðe0 aÞ2 Z 2 @x4 " # 2 @2 2@ W ¼ I1 2 W ðe0 aÞ ; @x2 @t ðe0 aÞ2 ðNT þ N E Þ
F 31
@2W @2/ þ X 33 / X 11 2 ¼ 0: @x2 @x
ðA11Þ
ðA12Þ
The associated boundary conditions can be written as
@W ¼ / ¼ 0; @x
ðA13Þ
for a clamped end and
The work described in this paper was supported by National Natural Science Foundation of China (No. 11002019), Ph.D. Programs Foundation of Ministry of Education of China (No. 20100009120018) and Fundamental Research Funds for the Central Universities (No. 2009JBM073).
@W ; ux ðx; z; tÞ ¼ Uðx; tÞ z @x
¼ I1
@2/ @2W @2W D11 þ ðNT þ N E Þ 2 2 2 @x @x @x
U¼W ¼ Acknowledgements
!
2 @U @ 2 W 1 @W z 2 þ : @x @x 2 @x
ðA2Þ
By using the Hamilton’s principle, the equations of motion of the Euler nanobeam are derived as
@Nx @2U ¼ I1 2 @x @t @ 2 Mx @ @W @2W Nx ¼ I1 þ ; dW : 2 @x @x @x @t 2 Z h=2 @Dx d/ : cosðbzÞ þ b sinðbzÞDz dz ¼ 0: @x h=2
dU :
ðA3Þ ðA4Þ ðA5Þ
The corresponding boundary conditions at beam ends (x = 0, L) require
U ¼ 0 or Nx ¼ 0; x ¼ 0; L; @M x @W W ¼ 0 or ¼ 0; x ¼ 0; L; þ Nx @x @x @W ¼ 0 or M x ¼ 0; x ¼ 0; L; @x Z h=2 Dx cosðbzÞdz ¼ 0; x ¼ 0; L: / ¼ 0 or
ðA6Þ ðA7Þ ðA8Þ ðA9Þ
h=2
Similar with the derivation in the nonlocal Timoshenko beam model, we can re-express Eqs. (A3)–(A5) in the form of U, W and /. Then, the
" # @2W @2W @ @W 2 Nx ¼ 0; Mx ¼ F 31 / D11 þ ðe0 aÞ I1 @x2 @x @x @t 2 U ¼ W ¼ / ¼ 0;
ðA14Þ
for a hinged end. Then, The DQ method is used to solve the nonlinear equations (A10)–(A12) and the associated boundary of the piezoelectric nanobeams.
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