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Size-dependent bending of an electro-elastic bilayer nanobeam due to flexoelectricity and strain gradient elastic effect
Accepted Manuscript Size-dependent bending of an Electro-elastic bilayer nanobeam due to flexoelectricity and strain gradient elastic effect Lu Qi, Shenjie Zhou, Anqing Li PII: DOI: Reference:
S0263-8223(15)00856-9 http://dx.doi.org/10.1016/j.compstruct.2015.09.020 COST 6865
To appear in:
Composite Structures
Please cite this article as: Qi, L., Zhou, S., Li, A., Size-dependent bending of an Electro-elastic bilayer nanobeam due to flexoelectricity and strain gradient elastic effect, Composite Structures (2015), doi: http://dx.doi.org/10.1016/ j.compstruct.2015.09.020
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Size-dependent bending of an Electro-elastic bilayer nanobeam due to flexoelectricity and strain gradient elastic effect Lu Qia, Shenjie Zhoua,*, Anqing Lia a School of Mechanical Engineering, Shandong University, Jinan City, Shandong 250061, People’s Republic of China
*Corresponding author: Key Laboratory of High Efficiency and Clean Mechanical Manufacture, Shandong University, Jinan City, Shandong 250061, People’s Republic of China. Tel.: +86 531 88396708; fax: +86 531 88392700 E-mail address: [email protected] (S. Zhou).
Abstract: A
size-dependent bending model of an electro-elastic bilayer nanobeam
including an isotropic dielectric layer and an elastic layer is established based on the flexoelectricity theory and the strain gradient theory. The governing equations and boundary conditions are derived from electric enthalpy variation principle with consideration of electrostatic force. The static bending problems of bilayer cantilever under closed and open circuit conditions are solved to show the size-dependency of the flexoelectric effect in both direct and converse flexoelectric processes. Numerical results demonstrate that both the strain gradient elastic effect and the flexoelectric effect significantly influence the deflection of the bending cantilever when the beam thickness is comparable to the material length scale parameters. Due to the flexoelectricity, sharp gradients of the electric field and polarization field arise near the surfaces, which differs greatly from the uniform field in the classical theory. In addition, the electric potential generated in the direct flexoelectric process and the deflection generated in the converse flexoelectric process exhibit obvious size-dependency. Keywords: Bilayer nanobeam, Flexoelectricity, Strain gradient elastic effect, Size-dependency
1. Introduction Flexoelectricity, the coupling between polarization and strain gradients, is a universal effect allowed by symmetry in all dielectrics [1, 2]. Since strain gradient is closely related to the characteristic scale (thickness, radius, etc.) of structures, flexoelectricity has increasing influence on the electrical and mechanical properties of dielectrics with their structural size decreasing to nanometers. In literatures, for example, flexoelectricity was found to play an important role in the physical characteristics of
ferroelectrics. Catalan et.al [3] found that the flexoelectricity causes an order-of–magnitude decrease in the dielectric constant of thin films. Lubomirsky et.al [4] reported that the poling of quasi-amorphous BaTiO3 upon cooling is assisted by flexoelectricity. Based on these observations, flexoelectricity is believed to have a good application potential in the engineering field. Recently, flexoelectricity has stimulated a surge of scientific interests in both experimental and theoretical investigations. Some experiments have been successfully performed to estimate the flexocoupling coefficients. For example, Cross and coworkers [5-7] used the cantilever bending method and the pyramid-compression method to measure the flexocoupling coefficient in some certain perovskite ceramics. Zubko et al. [8-9] employed the three-point bending method to measure the flexocoupling coefficient in nonpiezoelectric SrTiO3 single crystals of different crystallographic orientations. In these experiments, giant flexocoupling coefficients are found in some titanate material whose dielectric constants are very high. Besides, some theoretical studies have also been done to interpret flexoelectricity in dielectrics. Kogan [10] formulated the first phenomenological theory of flexoelectricity in 1964 and estimated the value range of flexoelectric coefficients. Mindlin [11] introduced first gradients of the polarization into the conventional linear electromechanical coupling theory based on the long-wavelength limit of the shell-model of lattice dynamics. Tagantsev [12] developed a microscopic theory of ionic flexoelectricity based on a rigid-ion model. The author verified that there are four mechanisms contributing to the flexoelectric response: dynamic bulk flexoelectricity, static bulk flexoelectricity, surface flexoelectricity and surface piezoelectricity. The first steps towards a microscopic description of the electronic contribution to flexoelectricity were made by Resta [13]. Sharma et al. [14-15] developed a theory considering first gradients of the strain and the polarization and analyzed the size-dependent mechanical and electrical behaviors of piezoelectric and nonpiezoelectric nanostructures based on a combination of theoretical and atomistic approaches. Then a more comprehensive theory with consideration of surface stress, surface polarization, bulk flexoelectric effect and electrostatic force for elastic dielectrics was proposed by Shen and Hu [16]. These theories have already been used to interpret or predict some size-dependent mechanical and electrical behaviors of nanosized dielectric structures. In literatures, flexoelectric effects on the mechanical and electrical properties of specific nanosized structures have been analyzed based on these theories. Liang et al. [17] established a Bernoulli-Euler dielectric nanobeam model with the strain gradient elastic effect and the flexoelectric effect. In their paper the size-dependent mechanical behaviors of pure elastic, piezoelectric and nonpiezoelectric dielectric cantilever beams were discussed. Influence of the flexoelectric effect on the electroelastic and dynamic responses of bending piezoelectric nanobeams under different boundary conditions was also investigated by Yan et al. [18] based on Bernoulli-Euler beam model and
Timoshenko beam model. Zhang et al. [19] proved that flexoelectricity has significant influence on the electroelastic responses and vibrational behaviors of a piezoelectric nanoplate. These researches mainly targeted at monolayer simple structures such as nanobeam and nanoplate. Moreover, a bilayer or a multilayer structure is also widely used in mems-electro-mechanical system (MEMS) and nano-electro-mechanical system (NEMS) [20-23], such as a piezoelectric bi-morph sensor [24]. Several analyses about the effect of flexoelectricity on bilayer or multilayer dielectric structures could be found. Sharma [25] groups proved the possibility of creating apparently piezoelectric multilayer thin films by only symmetrically stacking various nonpiezoelectric dielectric materials with different dielectric constant. When the multilayer structure is subjected to a pressure, the large strain gradients induced near the interfaces will generate a net polarization. Chen and Soh [26] investigated the flexoelectric effects on the distributions of polarization in multiferroic electro-magnetic thin bilayer films. They found that the flexoelectric induced polarization became more and more dominant with the film thickness decreases. Li and Zhou [27] analyzed a three-layer beam structure including an isotropic dielectric layer, an electrode layer and an elastic substrate layer based on the extended linear piezoelectricity theory proposed by Hadjesfandiari [28]. Their analysis demonstrated that the size-dependent flexoelectricity significantly affects the static and dynamic behaviors of the three-layer bending beam. Besides, for bilayer or multilayer nanosized structures, the strain gradient elastic effect may play a key role in its mechanical properties. Such an effect on the composite structures has been widely investigated especially in recent years [29-33]. Analogous to the bilayer piezoelectric intelligent structure, a bilayer flexoelectric dielectric structure (including an isotropic dielectric layer and an elastic layer) may also possess exciting electrical and mechanical properties and have the possibility of the application in engineering. However, the investigation and application of such an electro-elastic bilayer structure is still absent. Further exploration of the flexoelectricity and strain gradient effect in such bilayer structures is necessary. This paper aims at performing some theoretical analyses of the flexoelectricity and strain gradient effect in nanosized bilayer flexoelectric beams. In the present analysis, a size-dependent bending model of a bilayer flexoelectric dielectric nanobeam is proposed based on the Bernoulli-Euler beam model and the flexoelectricity theory with consideration of the effects of strain gradients and polarization gradients. The details are as follows: in section 2, some basic equations in flexoelectric theory are given; in section 3, the governing equations and boundary conditions of a bilayer flexoelectric dielectric nanobeam are derived; in section 4, static bending problems of the bilayer cantilever under closed and open circuit conditions are solved. The new model can be used to illustrate the size-dependent flexoelectricity of a bilayer dielectric cantilever; in section 5, numerical results are given to discuss the influence of strain gradient and flexoelectric effects on the mechanical and electrical
properties. Finally, main conclusions for this paper are summarized in section 6.
2. Basic equations in flexoelectric theory
For an extended linear theory of centrosymmetric dielectrics, the most general expression for the internal energy density U incorporating first gradients of the strain and the polarization can be written as [34] 1 1 1 1 (1) U = akl Pk Pl + cijkl ε ij ε kl + bijkl Pi , j Pk ,l + fijkl Piη jkl + d ijkl Pi , j ε kl + gijklmnηijkηlmn , 2 2 2 2 where a and c are the second-order reciprocal dielectric susceptibility and fourth-order elastic constant tensors, respectively. d and f are the flexocoupling coefficient tensors, and it was justified that d = −f [16, 35]. The tensor g represents strain gradient elastic effect. The summation convention is used in this paper, and the comma in the subscript indicates differentiation with respect to the spatial variables. P denotes the polarization vector. ε and η are the strain and strain gradient tensors, respectively, which are defined as, 1 1 ε ij = (ui , j + u j ,i ) , ηijk = ε ij ,k = (ui , jk + u j ,ik ) . 2 2 Here u represents the displacement vector. The electric enthalpy density is defined by Toupin [36] as 1 H = U − ε 0ϕ,iϕ,i + ϕ,i Pi , 2
(2)
(3)
where ε 0 is the permittivity of vacuum and ϕ is the electric potential of the Maxwell self-field (MS) defined by EiMS = −ϕ ,i .
The constitutive equations are expressed in terms of the internal energy as ∂U σ ij = = cijkl ε kl + dijkl Pi , j , ∂ε ij
(4)
(5)
τ ijk =
∂U = fijkl Pi + gijklmnηlmn , ∂ηijk
(6)
Ei =
∂U = aij Pj + fijklη jkl , ∂Pi
(7)
X ij =
∂U = bijkl Pk ,l + dijklε kl . ∂Pi , j
(8)
In the above equations, σ is the classical Cauchy Stress tensor, τ is the high-order stress tensor, E is the effective local electric field vector, X is the
high-order local electric field. For an isotropic dielectric, the symmetries of the material constants introduced in Eq. (1) are as follows [34], aij = aδ ij , cijkl = c12δ kl + c44 (δ ikδ jl + δilδ jk ) bijkl = b12δ kl + b44 (δikδ jl + δ ilδ jk ) + b77 (δikδ jl − δ ilδ jk ) . (9) dijkl = d12δ kl + d 44 (δ ikδ jl + δilδ jk ) f = f δ + f (δ δ + δ δ ) 12 kl 44 ik jl il jk ijkl The coefficient tensor g corresponds to the adopted strain gradient elastic theory. The differences/similarities of various strain gradient theories have been investigated by Zhang and Sharma [37]. In this paper, the theory proposed by Kleinert [38] is used. In this theory, for an isotropic material, the elements of the coefficient tensor g can be expressed as g ijklmn = (c12 + 2c44 )l12δ ijδ knδ lm + c44 l2 2 (δ il δ jk δ mn − δ ijδ knδ lm ) ,
(10)
where l1 and l2 are two material length scale parameters having the dimensions of length. For an isotropic elastic material, the internal energy density is expressed as 1 1 U elas = cijkl ε ijε kl + g ijklmnηijkηlmn . 2 2 The constitutive equations are expressed as ∂U σ ij = = cijkl ε kl , ∂ε ij
τ ijk =
∂U = gijklmnηlmn . ∂ηijk
(11)
(12) (13)
3. Size-dependent model of a bilayer nanobeam As shown in Fig. 1, an isotropic dielectric layer, i.e. the upper layer, is bonded to an elastic layer, which is normally a metal layer. The whole thickness of the bilayer beam is H , the length is L and the width is B . The beam is subjected to a lateral distributed force q ( x) and a voltage v on the upper surface. A Cartesian coordinate system is used in the beam with the x-axis coincident with the interface and the z-axis along the thickness direction. The thicknesses of the upper layer and the lower layer are h1 and
h2 , respectively. For the Bernoulli-Euler beam model, the displacement components of the bilayer beam model are taken as [39] dw( x) , v = 0 , w = w( x) , u = u0 ( x ) − z dx
(14)
where u , v , w , are the x -, y -, z -components of the displacement vector, u0 ( x)
is the axial displacement at z = 0 and w( x ) is the deflection of the zero-strain axis. If
d ( x) is z coordinates of the zero-strain axis across the length, the axial displacement can be expressed as u0 ( x) = − d ( x)
dw( x) . The only non-zero strain and strain gradients dx
are obtained from Eq. (2) as
ε xx =
du0 ( x) d 2 w( x) , −z dx dx 2
d 2u0 ( x ) d 3 w( x) d 2 w( x ) − z , η = − . (15) xxz dx 2 dx3 dx 2 For a slender beam with large ratio of length to thickness, i.e. H << L , the electric field is assumed to exist and vary only in the z direction. Moreover, for a bending beam the axial displacement is so small compared to the transverse displacement that the
η xxx =
strain gradient η xxx can be neglected compared to ηxxz [18]. Thus, only the strain gradient ηxxz is considered in the following analysis. For the flexoelectric dielectric layer, the electric enthalpy density in Eq. (3) is then expressed with the non-trivial variables as 1 1 1 a33 Pz Pz + c11(1)ε xxε xx + b33 Pz,z Pz,z + f13 Pzη xxz 2 2 2 . (16) 1 1 − f13 Pz , z ε xx + g13(1)η xxzη xxz − ε 0ϕ, zϕ, z + ϕ, z Pz 2 2 Here, the subscripts 1, 2, 3 represent the x, y, z directions, respectively, and the subscripts (1) and (2) is used to represent the upper layer and the lower layer, respectively. For simplicity, the contracted notation for the subscripts of the material H1 =
property tensors is adopted, including c11 = c1111 , b33 = b3333 , f13 = f3113 , g13 = g113113 . For the elastic layer, the internal energy density is expressed as 1 1 (17) U 2 = c11(2)ε xxε xx + g13(2)η xxzη xxz . 2 2 The summation of the electric enthalpy in the upper layer and the internal energy in the lower layer is
∫
V
H total dV = B ∫
L
0
∫
h1
− h2
H total dzdx = B ∫
L
0
∫
h1
0
H1dzdx + B ∫
L
0
∫
0
− h1
U 2 dzdx .
(18)
Without volume charge density, the virtual work done by the external distributed force q ( x) , the boundary shear force Q and the boundary moment M is L
δ ∫ WdV = ∫ q(x)δ w(x)dx + Qδ w(x) |0L + M δ w′(x) |0L . V
0
(19)
Thus, for a static problem of the established bilayer beam model, the variation principle takes
δ ∫ ( − H total +W ) dV = 0 . V
(20)
To consider the effect of electrostatic force, the Reynolds’s transport theorem [16]
is adopted in the enthalpy variation, i.e.
δ (B ∫
L
0
∫
h1
0
H1dzdx) = B ∫
L
0
∫
h1
0
δ H1dzdx + B ∫
L
0
∫
h1
0
H1δ ui ,i dzdx .
(21)
Substituting Eqs. (14), (16- 19) and (21) into Eq. (20) yields
δ ( − ∫ H total dV + ∫ W dV ) = V
V
d ( M M + M E ) + q (x) δ w(x) dx 2 0 dx L d +∫ ( F M + F E )δ u0 (x) dx 0 dx
∫
2
L
+ B∫
L
−B∫
L
0
0
∫
h1
∫
h1
0
0
( Ez − X zz , z + ϕ, z )δ Pdzdx ( −ε 0ϕ, zz + Pz , z )δϕdzdx
d + − ( M M + M E ) + Q δ w(x) |0L dx M E + ( M + M + M )δ w′(x) |0L − ( F M + F E )δ u0 (x) |0L + ( −ε 0ϕ, z + Pz )δϕ |0h1 − X zzδ Pz |h01 = 0
,
(22)
where M M + M E , F M + F E are the total bending moment and total axial force on the beam cross-section, respectively. M M is the mechanical bending moment on the cross-section of the cantilever. M E is the electrostatic bending moment. Analogously, F M is the axial mechanical force on the cross-section of the cantilever. F E is the axial electrostatic force. They are defined, respectively, as M M = B∫
h1
− h2
h1
(σ 11 z + τ 113 )dz , M E = B ∫ σ 11E zdz , 0
h1
h1
− h2
0
F M = B ∫ σ 11dz , F E = B ∫ σ 11E dz ,
(23)
where σ 11 and τ113 are the axial stress and axial high order stress both having different expressions in each layer defined as Eqs. (5), (6) and Eqs. (12), (13), respectively. σ 11E is the axial electrostatic stress [40] defined as 1 1 1 (24) Ez Pz − ε 0ϕ, zϕ, z + ϕ, z Pz + X zz Pz , z , 2 2 2 in which the former three terms represent the generalized Maxwell Stress and the last term represents the generalized electrostatic stress corresponding to the flexoelectric effect. The governing equations and the boundary conditions are naturally obtained from Eq. (22) as follows, the governing equations,
σE = 11
d2 M E 2 ( M + M ) + q (x) = 0 dx d (F M + F E ) = 0 dx
∀x ∈ (0, L) ,
(25)
in V1 ,
(26)
Ez − Vzz , z + ϕ, z = 0 −ε 0ϕ, zz + Pz , z = 0 the boundary conditions, ( M M + M E + M )δ w′(x) |0L = 0 − d ( M M + M E ) + Q δ w(x) |L = 0 0 dx . −( F M + F E )δ u (x) |L = 0 0 0 (−ε 0ϕ, z + Pz )δϕ |0h1 = 0 Vzzδ Pz |h01 = 0
(27)
The two equations in Eqs. (25) are the moment equilibrium and the axial force equilibrium equations of the cross section, respectively. The two equations in Eqs. (26) are the so-called intramolecular force equilibrium [36] and the conventional Gauss equation in the dielectric layer, respectively. When the electric field effect is ignored, the governing equations reduce to that of pure strain gradient bilayer beam model. Furthermore, if the material length scale parameters equal to zero, only the two governing equations are coincident with the classical bilayer Bernoulli-Euler beam theory. When the thickness of the elastic layer equals to zero, the present model is a special case of the theory proposed in [16] without consideration of surface effect. Furthermore, when the pure strain gradient effect and the electrostatic force are ignored, the governing equations and the boundary conditions are the same as the ones derived in [25].
4. Static bending of a bilayer cantilever
In this section, the static bending problems of a bilayer cantilever under the closed circuit and the open circuit conditions are solved.
4.1 Closed circuit condition
As shown in Fig.2, a bilayer cantilever is subjected to a shear force Q at the free
end and a voltage V on the upper surface. Under closed circuit condition, the governing equations are obtained from Eqs. (25) and (26) as d2 M E dx 2 ( M + M ) = 0 d (F M + F E ) = 0 dx
∀x ∈ (0, L) ,
(28)
in V1 ,
(29)
E z − X zz , z + ϕ, z = 0 −ε 0ϕ, zz + Pz , z = 0
and the boundary conditions are obtained from Eqs. (27) as
w′(x) | x =0 = 0,( M M + M E ) | x = L = 0 w(x) | = 0, − d ( M M + M E ) + Q x=0 dx M E u0 (x) |x =0 = 0, ( F + F ) | x = L = 0 ϕ |z = 0 = 0, ϕ |z = h = V 1 X zz |z =0 = 0, X zz | z =h1 = 0
x=L
=0 .
(30)
By solving the two electrical governing equations in Eqs. (29) with the last four electrical boundary conditions in Eqs. (30), the expressions of the electrical variables in terms of w( x) and u0 ( x) are obtained as Pz = C1eλ z + C2 e− λ z +
f13 (1 + 2a33ε 0 ) d 2 w(x) V − , 2 a33 (1 + a33ε 0 ) dx a33 h1
Ez = a33C1eλ z + a33C2 e− λ z +
ϕ=
C1
ε 0λ
eλ z −
C2
ε 0λ
e −λ z +
f13 a33ε 0 d 2 w(x) V − , h1 1 + a33ε 0 dx 2
f13 d 2 w(x) f13 du0 ( x) V , z+ z− 2 h1 1 + a33ε 0 dx 1 + a33ε 0 dx
(31) (32) (33)
where
λ=
1 + a33ε 0 , b33ε 0
f13 h1 C1 =
f13 h1 C2 =
(34)
du ( x) −λ h1 d 2 w(x) + f13 0 (e − 1) 2 dx dx , b33λ (e− λh1 − eλ h1 )
(35)
du ( x) λ h1 d 2 w(x) + f13 0 (e − 1) dx 2 dx . b33λ (e− λ h1 − e λh1 )
(36)
Without consideration of flexoelectric effect ( f = 0 ), the expressions of the polarization, electric field and electrical potential will reduce to the classical forms
Pz = −
V V V , Ez = − , ϕ = z . a33h1 h1 h1
By substituting Eqs. (31-33) into (24), the axial electrostatic stress is expressed as f13V (1 + a33ε 0 ) eλ z + e− λ z f13V d 2 w(x) V 2 (1 + a33ε 0 ) + − − λ h1 − eλ h1 2a33 h1 dx 2 2a33 h12 a33ε 0b33λ e
σ E = − 11
f V (1 + a33ε 0 ) ( e− λ h1 − 1) eλ z + (eλ h1 − 1)e− λ z du0 ( x) − 13 e− λ h1 − eλ h1 a33ε 0 h1b33 λ dx
.
(37)
Here, the quadratic nonlinear terms are ignored since the distribution of these terms are checked and found that they are negligible [19] in comparison with the existing terms in Eq. (37). Furthermore, if the flexoelectric effect is neglected, only the term −
V 2 (1 + a33ε 0 ) is retained, which makes the electrostatic stress a constant. 2a33h12
By substituting Eqs. (5), (6) and (31-33) into (23), the total moment and the total axial force are obtained as, du ( x) d 2 w(x) + β1 0 + γ1, 2 dx dx
(38)
du ( x) d 2 w(x) + β2 0 +γ2 , 2 dx dx
(39)
M M + M E = α1
F M + F E = α2
in which the coefficients are expressed as
α1 = −( I (1) c11(1) + I (2) c11(2) + A(1) g13(1) + A(2) g13(2) ) +
3 f13Vh1B f13VB e− λ h1 + eλ h1 − 2 + 4a33 a33λ e− λ h1 − eλh1
f132 h1B f 2 h 2 B e− λ h1 + eλh1 + − 13 1 a33 (1 + a33ε 0 ) b33λ e −λ h1 − eλ h1
β1 = S(1)c11(1) + S (2) c11(2) +
γ1 = −
f132 h1B e− λ h1 + eλ h1 − 2 f13VB 2 f13VB e− λ h1 + eλ h1 − 2 , − − b33λ e− λ h1 − eλ h1 a33 a33λ h1 e− λ h1 − eλ h1
f13VB V 2 (1 + a33ε 0 ) B , − a33 4a33
α 2 = −( S (1) c11(1) + S(2) c11(2) ) +
f13VB f132 h1B e− λ h1 + eλ h1 − 2 − , 2a33 b33λ eh1 − eλ h1
,
(40)
(41)
(42)
(43)
β 2 = A(1) c11(1) + A(2) c11(2) +
γ2 = −
2 f132 B e− λ h1 + eλ h1 − 2 , b33λ e − λh1 − eλ h1
V 2 (1 + a33ε 0 ) B. 2a33h1
(44)
(45)
Here, we define the volume, cross-section area, static moment and the second moment of inertia for the model with respect to x-axis as V , A , S , I , respectively. Then, the governing equations in Eqs. (28) and the first six boundary conditions in Eqs. (30) are expressed as d 4 w(x) d 3u 0 ( x ) + =0 α β 1 1 dx 4 dx3 2 3 α d w(x) + β d u0 ( x) = 0 2 2 dx3 dx 2
∀x ∈ (0, L) ,
dw( x) d 2 w(x) du ( x ) | x =0 = 0, α1 + β1 0 + γ1 =0 2 dx dx x=L dx d 3 w(x) d 2u0 ( x ) . α β w ( x ) | = 0, + =Q x=0 1 1 3 2 dx dx x=L 2 u ( x) | = 0, α d w(x) + β du0 ( x) + γ =0 x =0 2 2 2 dx 2 0 dx x= L Solutions of Eqs. (46) with the boundary conditions are easily obtained as
(46)
(47)
w(x) =
1 Qβ 2 1 γ β − γ β − QLβ 2 2 x3 + 2 1 1 2 x , 6 α1β 2 − α 2 β1 2 α1β 2 − α 2 β1
(48)
u0 ( x) =
γ α − γ α − QLα 2 1 Qα 2 x2 + 2 1 1 2 x. 2 α 2 β1 − α1β 2 α 2 β1 − α1β 2
(49)
Then the analytical solutions of the variables Pz , Ez , ϕ can be obtained by substituting Eqs. (48) and (49) into their expressions. When the external shear force Q is zero, polarization-induced bending under the closed circuit condition is the so called converse flexoelectric process.
4.2 Open circuit condition
As shown in Fig. 3, the bilayer cantilever is subjected to a shear force Q at the
free end. The electric potential of the lower surface of the dielectric layer is set to zero but no voltage is applied on the upper surface of the cantilever. Under the open circuit condition, the governing equations are obtained from Eqs. (25) and (26) as d2 M E dx 2 ( M + M ) = 0 d (F M + F E ) = 0 dx
∀x ∈ (0, L) ,
(50)
in V1 ,
(51)
Ez − X zz , z + ϕ, z = 0 −ε 0ϕ, zz + Pz , z = 0
and the boundary conditions are obtained from Eqs. (27) as w′(x) |x =0 = 0, (M M + M E ) |x = L = 0 w(x) | = 0, − d ( M M + M E ) + Q | = 0 x =0 dx x = L . M E u (x) | = 0, ( F + F ) | = 0 x =0 x= L 0 ϕ |z =0 = 0, (−ε 0ϕ, z + Pz ) |z = h = 0 1 X zz | z =0 = 0, X zz |z =h1 = 0
(52)
By solving the two electrical governing equations in Eqs. (51) with the last four electrical boundary conditions in Eqs. (52), the expressions of the electrical variables in terms of w( x) and u0 ( x) are obtained as, Pz = C1eλ z + C2 e− λ z +
f13ε 0 d 2 w(x) , 1 + a33ε 0 dx 2
Ez = a33C1eλ z + a33C2e − λ z +
ϕ=
C1
ε 0λ
eλ z −
C2
ε 0λ
e− λ z +
f13 (1 − a33ε 0 ) d 2 w(x) , dx 2 1 + a33ε 0
2 f13 d 2 w(x) f13 du0 ( x) z− . 2 1 + a33ε 0 dx 1 + a33ε 0 dx
(53)
(54)
(55)
where,
λ=
1 + a33ε 0 , b33ε 0 f13h1
C1 =
du ( x) d 2 w(x) + f13 0 (e−λ h1 − 1) dx 2 dx , − λ h1 b33λ (e − eλ h1 )
(56)
(57)
f13h1 C2 =
du ( x) λ h1 d 2 w(x) + f13 0 (e − 1) 2 dx dx . b33λ (e− λ h1 − e λh1 )
(58)
Under the open circuit condition, the axial electrostatic stress σ 11E includes only the quadratic nonlinear terms, which makes it negligible compared to the axial classical Cauchy Stress σ 11 . Thus, the axial electrostatic stress σ 11E and the electrostatic moment M E can be set to zero. By substituting Eqs. (5), (6) and (53-55) into (23), the total moment and the total axial force are obtained as M M = α1
du ( x) d 2 w(x) , + β1 0 2 dx dx
(59)
du ( x) d 2 w(x) + β2 0 , 2 dx dx in which the coefficients are expressed as F M = α2
α1 = −( I (1) c11(1) + I (2) c11(2) + A(1) g13(1) + A(2) g13(2) ) −
β1 = S(1)c11(1) + S (2) c11(2) +
f132 h12 B e− λ h1 + eλ h1 , b33λ e − λh1 − eλh1
f132 h1B e− λh1 + eλh1 − 2 , b33λ e− λ h1 − eλh1
α 2 = −( S(1) c11(1) + S(2)c11( 2) ) −
β 2 = A(1) c11(1) + A(2) c11(2) −
(60)
f132 h1 B e− λ h1 + eλ h1 − 2 , b33λ e− λ h1 − eλh1
2 f132 B e− λ h1 + eλ h1 − 2 . b33λ e −λ h1 − eλ h1
(61)
(62)
(63)
(64)
Then, the governing equations in Eqs. (50) and the first six boundary conditions in Eqs. (52) are expressed as d 4 w(x) d 3 u0 ( x ) + =0 β 1 α1 dx 4 dx3 ∀x ∈ (0, L) , 2 3 α d w(x) + β d u0 ( x) = 0 2 2 dx3 dx 2
(65)
dw( x) d 2 w(x) du ( x) |x =0 = 0, α1 + β1 0 =0 2 dx dx x = L dx d 3 w(x) d 2u 0 ( x ) + =Q. β w( x) |x =0 = 0, α1 1 dx3 dx 2 x = L 2 u ( x) | = 0, α d w(x) + β du0 ( x) =0 x=0 2 2 dx 2 0 dx x= L Solutions of Eqs. (65) with the boundary conditions are obtained as
(66)
w(x) =
1 Qβ 2 1 QLβ 2 x3 − x2 , 6 α1β 2 − α 2 β1 2 α1β 2 − α 2 β1
(67)
u0 ( x) =
1 Qα 2 QLα 2 x2 − x. α 2 β1 − α 1β 2 2 α 2 β1 − α1β 2
(68)
The analytical solutions of the variables Pz , Ez , ϕ can be obtained by substituting Eqs. (67) and (68) into their expressions. Bending-induced polarization under the open circuit condition is the so called direct flexoelectric process.
5. Numerical results and discussion
In this section, some numerical results and discussions are given based on the solutions obtained in the last section. For a slender rectangular bilayer cantilever, the length and width of the beam are set as L = 20 H and B = H , where H = 20nm is the total thickness of the bilayer cantilever. To ensure validation of the infinitesimal strain assumption, the applied shear force Q at the free end is 0.02 nN under both conditions and the voltage applied on the upper surface under closed circuit is 0.2 V . For the dielectric layer, the material parameters are taken according to the Ref. [34] as a33 = 8.767 ×1010 Nm 2 /C 2 , b33 = 1.921×10−9 Nm 4 /C 2 , c12(1) = 0.538 × 1011 Nm 2 /C , c44(1) = 0.325 × 1011 Nm 2 /C
,
and
c11(1) = c12(1) + 2c44(1) = 1.188 × 1011 Nm 2 /C
,
g13(1) = (c12(1) + 2c44(1) )l1(1) 2 − c44l2(1) 2 . Here, the two internal material length scale parameters are taken as the same i.e. l1(1) = l2(1) = l = 2.25nm . And f13 = 10V is adopted according to Kogan’s [10] estimate of the flexocoupling coefficient f ≈ 1 − 10V . For the elastic layer, the material parameters are taken as c11(2) = 1.5c11(1) , l1(2) = l2( 2) = 1.5l . The dimensionless height ratio h1 / h2 is adopted and set as 1 / 3 . All
the data listed above will be used in the following numerical calculations unless
otherwise stated. Effects of the strain gradient elasticity and flexoelectricity on the deflection of the bending cantilever under closed circuit condition are investigated. Fig. 4 shows the deflection across the length direction under different thicknesses and applied voltages. First, the applied voltage is set to be zero to observe the strain gradient elastic effect on the bending cantilever. Under this condition, the only effect makes the deflection deviate from that of the classical theory is strain gradient elastic effect. It is obviously observed that the strain gradient elastic effect dramatically decreases the deflection when the thickness is 20nm. In addition, from Fig. 4, the applied voltage also deviates the deflection due to the flexoelectric effect and the asymmetrical electrostatic stress on the cross section (in this numerical analysis the electrostatic stress and electrostatic moment can be neglected due to the small dielectric constant). It is clear that the flexoelectric effect decreases the deflection when the voltage is 0.2V while increases the deflection when the voltage is − 0.2V , which reveals that whether the flexoelectric effect stiffens or softens the cantilever depends on the applied voltage. Fig. 5 and Fig. 6 are presented to investigate the size-dependency of the strain gradient and flexoelectric effects. In Fig. 5, relation between the deflection at the free end of the cantilever and the thickness H is shown. With the increasing thickness H , the deflections under different applied voltages are gradually coincident with that of the classical theory. The curve at v = 0 gets close to the one of classical theory more rapidly than the other two curves. Meanwhile, variation of the deflection with the dimensionless thickness H / l at H = 20 nm is plotted in Fig.6. It is obvious that flexoelectricity enhances with higher f value, which indicate that the material with high f value may possess more opportunities for application. From the curves in Fig. 6, it is not difficult to infer that the strain gradient effect is dependent on the beam dimensionless thickness H / l , whereas the flexoelectric effect is almost independent of the material length scale parameters. It is indicated that for a bilayer flexoelectric cantilever both the strain gradient elastic effect and the flexoelectric effect significantly affect the deflection of the cantilever in nanoscale. Furthermore, both effects exhibit size-dependency and the strain gradient elastic effect weakens more rapidly than the flexoelectric effect with increasing thickness. Next, the effects of flexoelectricity on the electric field distribution and the polarization field distribution across the thickness under closed circuit condition are investigated. The curves plotted in Fig. 7 clearly manifest that the electric field across the thickness is different from that of the classical theory due to the flexoelectric effect. When the flexoelectric effect is concerned ( f = 10V ), the electric field can still be assumed as uniform across the thickness except near the upper and lower surfaces of the dielectric layer. Comparing Fig. 7(a) and Fig. 7(b), we can see that the effect of
flexoelectricity on the electric field distribution is independent of the direction of the applied voltage, which is also demonstrated in Eq. (32). Furthermore, from comparison of Fig. 7(b) and Fig. 7(c), the intensity and influencing scale of the flexoelectric effect on the electric field distribution decrease with the increasing thickness. The similar trend is found in the polarization field distribution across the thickness, which is not shown here. In the direct flexoelectric process, the electric potential across the thickness at x = L / 2 under different shear forces, thicknesses and f values are plotted in Fig. 8. It appears a smooth and steady increase of the induced electric potential in the central section, while sharp variation near the surfaces. This kind of electric potential distribution means large electric field near the surfaces while small uniform one in the central part. The induced electric potential can be explained as follows: the shear force induces the strain gradient along the thickness direction, and then polarization will be generated through the coupling between strain gradient and polarization. The sharp variation of the electric field near the surfaces with consideration of the flexoelectricity is a typical boundary behavior for a gradient theory [19]. Comparing the four curves in the figure, it is clearly observed that larger shear force and f value generates larger electric potential due to the larger strain gradient. In addition, as the total thickness increases from 20nm to 50nm , the generated electric potential decreases rapidly, which demonstrates obvious size-dependency. The deflection in the converse flexoelectric process is shown in Fig. 9. Due to the boundary conditions, the applied voltage will generate a non-uniform electric field and then a non-uniform polarization field near the boundaries. A bending moment is then generated through the coupling between strain and polarization gradient to make the cantilever bend. It is noted that the voltage applied in the opposite direction may induce a reverse bending in the cantilever. In addition, it is obviously to see that the induced electrical potential is size-dependent.
6. Conclusions In this work, a size-dependent bending model of a bilayer flexoelectric dielectric nanobeam, with consideration of the strain gradient elastic effect and the flexoelectric effect, is established based on the Bernoulli-Euler beam model and the theory for flexoelectric dielectrics. The solutions of static bending problems of a bilayer cantilever under open circuit and closed circuit conditions are obtained. Numerical results show that the strain gradient elastic effect decreases the deflection of the bilayer bending cantilever while the flexoelectric effect decreases or increases the deflection depends on whether the applied voltage is positive or negative. Both effects exhibit obvious size-dependency that the strain gradient effect is dependent on the beam dimensionless
thickness H / l , whereas the flexoelectric effect is only dependent on the beam thickness. Flexoelectricity also changes the electrical and polarization field distributions in the beam that sharp field gradients arise near the surfaces. Futhermore, in the direct flexoelectric process, the induced electric potential increases with the decreasing thickness and increasing f value. In the converse flexoelectric process, the applied voltage can generate bending and the deflection decreases as the thickness increases. The presented bilayer nanobeam model is helpful in understanding the mechanical and electrical properties in bilayer flexoelectric dielectric nanobeam.
Acknowledgments The work reported here is funded by the National Natural Science Foundation of China (11272186), Specialized Research Fund for the Doctoral Program of Higher Education of China (20120131110045) and the Natural Science Fund of Shandong Province of China (ZR2012AM014).
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Fig 1. Schematic of a bilayer nanobeam and its coordinate system Fig 2. Schematic of a bilayer cantilever under closed circuit condition Fig 3. Schematic of a bilayer cantilever under open circuit condition Fig 4. Variation of the deflection of the bilayer cantilever with dimensionless length Fig 5. Variation of the generated maximum deflection with beam thickness under different voltages Fig 6. Variation of the generated maximum deflection with the dimensionless thickness at H=20nm Fig 7. Electric potential distribution with the dimensionless thickness at x=L/2 (a) H=20nm, v=0.2V; (b) H=20nm, v=-0.2V; (c) H=50nm, v=-0.2V Fig 8. Electric potential distribution along the thickness direction at x=L/2 under different shear forces and thicknesses (direct flexoelectric process) Fig 9. Deflection of the cantilever with the dimensionless distance under different applied voltages and thicknesses (converse flexoelectric process)
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7(a)
Figure 7(b)
Figure 7(c)
Figure 8
Figure 9
S0263-8223(15)00856-9 http://dx.doi.org/10.1016/j.compstruct.2015.09.020 COST 6865
To appear in:
Composite Structures
Please cite this article as: Qi, L., Zhou, S., Li, A., Size-dependent bending of an Electro-elastic bilayer nanobeam due to flexoelectricity and strain gradient elastic effect, Composite Structures (2015), doi: http://dx.doi.org/10.1016/ j.compstruct.2015.09.020
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Size-dependent bending of an Electro-elastic bilayer nanobeam due to flexoelectricity and strain gradient elastic effect Lu Qia, Shenjie Zhoua,*, Anqing Lia a School of Mechanical Engineering, Shandong University, Jinan City, Shandong 250061, People’s Republic of China
*Corresponding author: Key Laboratory of High Efficiency and Clean Mechanical Manufacture, Shandong University, Jinan City, Shandong 250061, People’s Republic of China. Tel.: +86 531 88396708; fax: +86 531 88392700 E-mail address: [email protected] (S. Zhou).
Abstract: A
size-dependent bending model of an electro-elastic bilayer nanobeam
including an isotropic dielectric layer and an elastic layer is established based on the flexoelectricity theory and the strain gradient theory. The governing equations and boundary conditions are derived from electric enthalpy variation principle with consideration of electrostatic force. The static bending problems of bilayer cantilever under closed and open circuit conditions are solved to show the size-dependency of the flexoelectric effect in both direct and converse flexoelectric processes. Numerical results demonstrate that both the strain gradient elastic effect and the flexoelectric effect significantly influence the deflection of the bending cantilever when the beam thickness is comparable to the material length scale parameters. Due to the flexoelectricity, sharp gradients of the electric field and polarization field arise near the surfaces, which differs greatly from the uniform field in the classical theory. In addition, the electric potential generated in the direct flexoelectric process and the deflection generated in the converse flexoelectric process exhibit obvious size-dependency. Keywords: Bilayer nanobeam, Flexoelectricity, Strain gradient elastic effect, Size-dependency
1. Introduction Flexoelectricity, the coupling between polarization and strain gradients, is a universal effect allowed by symmetry in all dielectrics [1, 2]. Since strain gradient is closely related to the characteristic scale (thickness, radius, etc.) of structures, flexoelectricity has increasing influence on the electrical and mechanical properties of dielectrics with their structural size decreasing to nanometers. In literatures, for example, flexoelectricity was found to play an important role in the physical characteristics of
ferroelectrics. Catalan et.al [3] found that the flexoelectricity causes an order-of–magnitude decrease in the dielectric constant of thin films. Lubomirsky et.al [4] reported that the poling of quasi-amorphous BaTiO3 upon cooling is assisted by flexoelectricity. Based on these observations, flexoelectricity is believed to have a good application potential in the engineering field. Recently, flexoelectricity has stimulated a surge of scientific interests in both experimental and theoretical investigations. Some experiments have been successfully performed to estimate the flexocoupling coefficients. For example, Cross and coworkers [5-7] used the cantilever bending method and the pyramid-compression method to measure the flexocoupling coefficient in some certain perovskite ceramics. Zubko et al. [8-9] employed the three-point bending method to measure the flexocoupling coefficient in nonpiezoelectric SrTiO3 single crystals of different crystallographic orientations. In these experiments, giant flexocoupling coefficients are found in some titanate material whose dielectric constants are very high. Besides, some theoretical studies have also been done to interpret flexoelectricity in dielectrics. Kogan [10] formulated the first phenomenological theory of flexoelectricity in 1964 and estimated the value range of flexoelectric coefficients. Mindlin [11] introduced first gradients of the polarization into the conventional linear electromechanical coupling theory based on the long-wavelength limit of the shell-model of lattice dynamics. Tagantsev [12] developed a microscopic theory of ionic flexoelectricity based on a rigid-ion model. The author verified that there are four mechanisms contributing to the flexoelectric response: dynamic bulk flexoelectricity, static bulk flexoelectricity, surface flexoelectricity and surface piezoelectricity. The first steps towards a microscopic description of the electronic contribution to flexoelectricity were made by Resta [13]. Sharma et al. [14-15] developed a theory considering first gradients of the strain and the polarization and analyzed the size-dependent mechanical and electrical behaviors of piezoelectric and nonpiezoelectric nanostructures based on a combination of theoretical and atomistic approaches. Then a more comprehensive theory with consideration of surface stress, surface polarization, bulk flexoelectric effect and electrostatic force for elastic dielectrics was proposed by Shen and Hu [16]. These theories have already been used to interpret or predict some size-dependent mechanical and electrical behaviors of nanosized dielectric structures. In literatures, flexoelectric effects on the mechanical and electrical properties of specific nanosized structures have been analyzed based on these theories. Liang et al. [17] established a Bernoulli-Euler dielectric nanobeam model with the strain gradient elastic effect and the flexoelectric effect. In their paper the size-dependent mechanical behaviors of pure elastic, piezoelectric and nonpiezoelectric dielectric cantilever beams were discussed. Influence of the flexoelectric effect on the electroelastic and dynamic responses of bending piezoelectric nanobeams under different boundary conditions was also investigated by Yan et al. [18] based on Bernoulli-Euler beam model and
Timoshenko beam model. Zhang et al. [19] proved that flexoelectricity has significant influence on the electroelastic responses and vibrational behaviors of a piezoelectric nanoplate. These researches mainly targeted at monolayer simple structures such as nanobeam and nanoplate. Moreover, a bilayer or a multilayer structure is also widely used in mems-electro-mechanical system (MEMS) and nano-electro-mechanical system (NEMS) [20-23], such as a piezoelectric bi-morph sensor [24]. Several analyses about the effect of flexoelectricity on bilayer or multilayer dielectric structures could be found. Sharma [25] groups proved the possibility of creating apparently piezoelectric multilayer thin films by only symmetrically stacking various nonpiezoelectric dielectric materials with different dielectric constant. When the multilayer structure is subjected to a pressure, the large strain gradients induced near the interfaces will generate a net polarization. Chen and Soh [26] investigated the flexoelectric effects on the distributions of polarization in multiferroic electro-magnetic thin bilayer films. They found that the flexoelectric induced polarization became more and more dominant with the film thickness decreases. Li and Zhou [27] analyzed a three-layer beam structure including an isotropic dielectric layer, an electrode layer and an elastic substrate layer based on the extended linear piezoelectricity theory proposed by Hadjesfandiari [28]. Their analysis demonstrated that the size-dependent flexoelectricity significantly affects the static and dynamic behaviors of the three-layer bending beam. Besides, for bilayer or multilayer nanosized structures, the strain gradient elastic effect may play a key role in its mechanical properties. Such an effect on the composite structures has been widely investigated especially in recent years [29-33]. Analogous to the bilayer piezoelectric intelligent structure, a bilayer flexoelectric dielectric structure (including an isotropic dielectric layer and an elastic layer) may also possess exciting electrical and mechanical properties and have the possibility of the application in engineering. However, the investigation and application of such an electro-elastic bilayer structure is still absent. Further exploration of the flexoelectricity and strain gradient effect in such bilayer structures is necessary. This paper aims at performing some theoretical analyses of the flexoelectricity and strain gradient effect in nanosized bilayer flexoelectric beams. In the present analysis, a size-dependent bending model of a bilayer flexoelectric dielectric nanobeam is proposed based on the Bernoulli-Euler beam model and the flexoelectricity theory with consideration of the effects of strain gradients and polarization gradients. The details are as follows: in section 2, some basic equations in flexoelectric theory are given; in section 3, the governing equations and boundary conditions of a bilayer flexoelectric dielectric nanobeam are derived; in section 4, static bending problems of the bilayer cantilever under closed and open circuit conditions are solved. The new model can be used to illustrate the size-dependent flexoelectricity of a bilayer dielectric cantilever; in section 5, numerical results are given to discuss the influence of strain gradient and flexoelectric effects on the mechanical and electrical
properties. Finally, main conclusions for this paper are summarized in section 6.
2. Basic equations in flexoelectric theory
For an extended linear theory of centrosymmetric dielectrics, the most general expression for the internal energy density U incorporating first gradients of the strain and the polarization can be written as [34] 1 1 1 1 (1) U = akl Pk Pl + cijkl ε ij ε kl + bijkl Pi , j Pk ,l + fijkl Piη jkl + d ijkl Pi , j ε kl + gijklmnηijkηlmn , 2 2 2 2 where a and c are the second-order reciprocal dielectric susceptibility and fourth-order elastic constant tensors, respectively. d and f are the flexocoupling coefficient tensors, and it was justified that d = −f [16, 35]. The tensor g represents strain gradient elastic effect. The summation convention is used in this paper, and the comma in the subscript indicates differentiation with respect to the spatial variables. P denotes the polarization vector. ε and η are the strain and strain gradient tensors, respectively, which are defined as, 1 1 ε ij = (ui , j + u j ,i ) , ηijk = ε ij ,k = (ui , jk + u j ,ik ) . 2 2 Here u represents the displacement vector. The electric enthalpy density is defined by Toupin [36] as 1 H = U − ε 0ϕ,iϕ,i + ϕ,i Pi , 2
(2)
(3)
where ε 0 is the permittivity of vacuum and ϕ is the electric potential of the Maxwell self-field (MS) defined by EiMS = −ϕ ,i .
The constitutive equations are expressed in terms of the internal energy as ∂U σ ij = = cijkl ε kl + dijkl Pi , j , ∂ε ij
(4)
(5)
τ ijk =
∂U = fijkl Pi + gijklmnηlmn , ∂ηijk
(6)
Ei =
∂U = aij Pj + fijklη jkl , ∂Pi
(7)
X ij =
∂U = bijkl Pk ,l + dijklε kl . ∂Pi , j
(8)
In the above equations, σ is the classical Cauchy Stress tensor, τ is the high-order stress tensor, E is the effective local electric field vector, X is the
high-order local electric field. For an isotropic dielectric, the symmetries of the material constants introduced in Eq. (1) are as follows [34], aij = aδ ij , cijkl = c12δ kl + c44 (δ ikδ jl + δilδ jk ) bijkl = b12δ kl + b44 (δikδ jl + δ ilδ jk ) + b77 (δikδ jl − δ ilδ jk ) . (9) dijkl = d12δ kl + d 44 (δ ikδ jl + δilδ jk ) f = f δ + f (δ δ + δ δ ) 12 kl 44 ik jl il jk ijkl The coefficient tensor g corresponds to the adopted strain gradient elastic theory. The differences/similarities of various strain gradient theories have been investigated by Zhang and Sharma [37]. In this paper, the theory proposed by Kleinert [38] is used. In this theory, for an isotropic material, the elements of the coefficient tensor g can be expressed as g ijklmn = (c12 + 2c44 )l12δ ijδ knδ lm + c44 l2 2 (δ il δ jk δ mn − δ ijδ knδ lm ) ,
(10)
where l1 and l2 are two material length scale parameters having the dimensions of length. For an isotropic elastic material, the internal energy density is expressed as 1 1 U elas = cijkl ε ijε kl + g ijklmnηijkηlmn . 2 2 The constitutive equations are expressed as ∂U σ ij = = cijkl ε kl , ∂ε ij
τ ijk =
∂U = gijklmnηlmn . ∂ηijk
(11)
(12) (13)
3. Size-dependent model of a bilayer nanobeam As shown in Fig. 1, an isotropic dielectric layer, i.e. the upper layer, is bonded to an elastic layer, which is normally a metal layer. The whole thickness of the bilayer beam is H , the length is L and the width is B . The beam is subjected to a lateral distributed force q ( x) and a voltage v on the upper surface. A Cartesian coordinate system is used in the beam with the x-axis coincident with the interface and the z-axis along the thickness direction. The thicknesses of the upper layer and the lower layer are h1 and
h2 , respectively. For the Bernoulli-Euler beam model, the displacement components of the bilayer beam model are taken as [39] dw( x) , v = 0 , w = w( x) , u = u0 ( x ) − z dx
(14)
where u , v , w , are the x -, y -, z -components of the displacement vector, u0 ( x)
is the axial displacement at z = 0 and w( x ) is the deflection of the zero-strain axis. If
d ( x) is z coordinates of the zero-strain axis across the length, the axial displacement can be expressed as u0 ( x) = − d ( x)
dw( x) . The only non-zero strain and strain gradients dx
are obtained from Eq. (2) as
ε xx =
du0 ( x) d 2 w( x) , −z dx dx 2
d 2u0 ( x ) d 3 w( x) d 2 w( x ) − z , η = − . (15) xxz dx 2 dx3 dx 2 For a slender beam with large ratio of length to thickness, i.e. H << L , the electric field is assumed to exist and vary only in the z direction. Moreover, for a bending beam the axial displacement is so small compared to the transverse displacement that the
η xxx =
strain gradient η xxx can be neglected compared to ηxxz [18]. Thus, only the strain gradient ηxxz is considered in the following analysis. For the flexoelectric dielectric layer, the electric enthalpy density in Eq. (3) is then expressed with the non-trivial variables as 1 1 1 a33 Pz Pz + c11(1)ε xxε xx + b33 Pz,z Pz,z + f13 Pzη xxz 2 2 2 . (16) 1 1 − f13 Pz , z ε xx + g13(1)η xxzη xxz − ε 0ϕ, zϕ, z + ϕ, z Pz 2 2 Here, the subscripts 1, 2, 3 represent the x, y, z directions, respectively, and the subscripts (1) and (2) is used to represent the upper layer and the lower layer, respectively. For simplicity, the contracted notation for the subscripts of the material H1 =
property tensors is adopted, including c11 = c1111 , b33 = b3333 , f13 = f3113 , g13 = g113113 . For the elastic layer, the internal energy density is expressed as 1 1 (17) U 2 = c11(2)ε xxε xx + g13(2)η xxzη xxz . 2 2 The summation of the electric enthalpy in the upper layer and the internal energy in the lower layer is
∫
V
H total dV = B ∫
L
0
∫
h1
− h2
H total dzdx = B ∫
L
0
∫
h1
0
H1dzdx + B ∫
L
0
∫
0
− h1
U 2 dzdx .
(18)
Without volume charge density, the virtual work done by the external distributed force q ( x) , the boundary shear force Q and the boundary moment M is L
δ ∫ WdV = ∫ q(x)δ w(x)dx + Qδ w(x) |0L + M δ w′(x) |0L . V
0
(19)
Thus, for a static problem of the established bilayer beam model, the variation principle takes
δ ∫ ( − H total +W ) dV = 0 . V
(20)
To consider the effect of electrostatic force, the Reynolds’s transport theorem [16]
is adopted in the enthalpy variation, i.e.
δ (B ∫
L
0
∫
h1
0
H1dzdx) = B ∫
L
0
∫
h1
0
δ H1dzdx + B ∫
L
0
∫
h1
0
H1δ ui ,i dzdx .
(21)
Substituting Eqs. (14), (16- 19) and (21) into Eq. (20) yields
δ ( − ∫ H total dV + ∫ W dV ) = V
V
d ( M M + M E ) + q (x) δ w(x) dx 2 0 dx L d +∫ ( F M + F E )δ u0 (x) dx 0 dx
∫
2
L
+ B∫
L
−B∫
L
0
0
∫
h1
∫
h1
0
0
( Ez − X zz , z + ϕ, z )δ Pdzdx ( −ε 0ϕ, zz + Pz , z )δϕdzdx
d + − ( M M + M E ) + Q δ w(x) |0L dx M E + ( M + M + M )δ w′(x) |0L − ( F M + F E )δ u0 (x) |0L + ( −ε 0ϕ, z + Pz )δϕ |0h1 − X zzδ Pz |h01 = 0
,
(22)
where M M + M E , F M + F E are the total bending moment and total axial force on the beam cross-section, respectively. M M is the mechanical bending moment on the cross-section of the cantilever. M E is the electrostatic bending moment. Analogously, F M is the axial mechanical force on the cross-section of the cantilever. F E is the axial electrostatic force. They are defined, respectively, as M M = B∫
h1
− h2
h1
(σ 11 z + τ 113 )dz , M E = B ∫ σ 11E zdz , 0
h1
h1
− h2
0
F M = B ∫ σ 11dz , F E = B ∫ σ 11E dz ,
(23)
where σ 11 and τ113 are the axial stress and axial high order stress both having different expressions in each layer defined as Eqs. (5), (6) and Eqs. (12), (13), respectively. σ 11E is the axial electrostatic stress [40] defined as 1 1 1 (24) Ez Pz − ε 0ϕ, zϕ, z + ϕ, z Pz + X zz Pz , z , 2 2 2 in which the former three terms represent the generalized Maxwell Stress and the last term represents the generalized electrostatic stress corresponding to the flexoelectric effect. The governing equations and the boundary conditions are naturally obtained from Eq. (22) as follows, the governing equations,
σE = 11
d2 M E 2 ( M + M ) + q (x) = 0 dx d (F M + F E ) = 0 dx
∀x ∈ (0, L) ,
(25)
in V1 ,
(26)
Ez − Vzz , z + ϕ, z = 0 −ε 0ϕ, zz + Pz , z = 0 the boundary conditions, ( M M + M E + M )δ w′(x) |0L = 0 − d ( M M + M E ) + Q δ w(x) |L = 0 0 dx . −( F M + F E )δ u (x) |L = 0 0 0 (−ε 0ϕ, z + Pz )δϕ |0h1 = 0 Vzzδ Pz |h01 = 0
(27)
The two equations in Eqs. (25) are the moment equilibrium and the axial force equilibrium equations of the cross section, respectively. The two equations in Eqs. (26) are the so-called intramolecular force equilibrium [36] and the conventional Gauss equation in the dielectric layer, respectively. When the electric field effect is ignored, the governing equations reduce to that of pure strain gradient bilayer beam model. Furthermore, if the material length scale parameters equal to zero, only the two governing equations are coincident with the classical bilayer Bernoulli-Euler beam theory. When the thickness of the elastic layer equals to zero, the present model is a special case of the theory proposed in [16] without consideration of surface effect. Furthermore, when the pure strain gradient effect and the electrostatic force are ignored, the governing equations and the boundary conditions are the same as the ones derived in [25].
4. Static bending of a bilayer cantilever
In this section, the static bending problems of a bilayer cantilever under the closed circuit and the open circuit conditions are solved.
4.1 Closed circuit condition
As shown in Fig.2, a bilayer cantilever is subjected to a shear force Q at the free
end and a voltage V on the upper surface. Under closed circuit condition, the governing equations are obtained from Eqs. (25) and (26) as d2 M E dx 2 ( M + M ) = 0 d (F M + F E ) = 0 dx
∀x ∈ (0, L) ,
(28)
in V1 ,
(29)
E z − X zz , z + ϕ, z = 0 −ε 0ϕ, zz + Pz , z = 0
and the boundary conditions are obtained from Eqs. (27) as
w′(x) | x =0 = 0,( M M + M E ) | x = L = 0 w(x) | = 0, − d ( M M + M E ) + Q x=0 dx M E u0 (x) |x =0 = 0, ( F + F ) | x = L = 0 ϕ |z = 0 = 0, ϕ |z = h = V 1 X zz |z =0 = 0, X zz | z =h1 = 0
x=L
=0 .
(30)
By solving the two electrical governing equations in Eqs. (29) with the last four electrical boundary conditions in Eqs. (30), the expressions of the electrical variables in terms of w( x) and u0 ( x) are obtained as Pz = C1eλ z + C2 e− λ z +
f13 (1 + 2a33ε 0 ) d 2 w(x) V − , 2 a33 (1 + a33ε 0 ) dx a33 h1
Ez = a33C1eλ z + a33C2 e− λ z +
ϕ=
C1
ε 0λ
eλ z −
C2
ε 0λ
e −λ z +
f13 a33ε 0 d 2 w(x) V − , h1 1 + a33ε 0 dx 2
f13 d 2 w(x) f13 du0 ( x) V , z+ z− 2 h1 1 + a33ε 0 dx 1 + a33ε 0 dx
(31) (32) (33)
where
λ=
1 + a33ε 0 , b33ε 0
f13 h1 C1 =
f13 h1 C2 =
(34)
du ( x) −λ h1 d 2 w(x) + f13 0 (e − 1) 2 dx dx , b33λ (e− λh1 − eλ h1 )
(35)
du ( x) λ h1 d 2 w(x) + f13 0 (e − 1) dx 2 dx . b33λ (e− λ h1 − e λh1 )
(36)
Without consideration of flexoelectric effect ( f = 0 ), the expressions of the polarization, electric field and electrical potential will reduce to the classical forms
Pz = −
V V V , Ez = − , ϕ = z . a33h1 h1 h1
By substituting Eqs. (31-33) into (24), the axial electrostatic stress is expressed as f13V (1 + a33ε 0 ) eλ z + e− λ z f13V d 2 w(x) V 2 (1 + a33ε 0 ) + − − λ h1 − eλ h1 2a33 h1 dx 2 2a33 h12 a33ε 0b33λ e
σ E = − 11
f V (1 + a33ε 0 ) ( e− λ h1 − 1) eλ z + (eλ h1 − 1)e− λ z du0 ( x) − 13 e− λ h1 − eλ h1 a33ε 0 h1b33 λ dx
.
(37)
Here, the quadratic nonlinear terms are ignored since the distribution of these terms are checked and found that they are negligible [19] in comparison with the existing terms in Eq. (37). Furthermore, if the flexoelectric effect is neglected, only the term −
V 2 (1 + a33ε 0 ) is retained, which makes the electrostatic stress a constant. 2a33h12
By substituting Eqs. (5), (6) and (31-33) into (23), the total moment and the total axial force are obtained as, du ( x) d 2 w(x) + β1 0 + γ1, 2 dx dx
(38)
du ( x) d 2 w(x) + β2 0 +γ2 , 2 dx dx
(39)
M M + M E = α1
F M + F E = α2
in which the coefficients are expressed as
α1 = −( I (1) c11(1) + I (2) c11(2) + A(1) g13(1) + A(2) g13(2) ) +
3 f13Vh1B f13VB e− λ h1 + eλ h1 − 2 + 4a33 a33λ e− λ h1 − eλh1
f132 h1B f 2 h 2 B e− λ h1 + eλh1 + − 13 1 a33 (1 + a33ε 0 ) b33λ e −λ h1 − eλ h1
β1 = S(1)c11(1) + S (2) c11(2) +
γ1 = −
f132 h1B e− λ h1 + eλ h1 − 2 f13VB 2 f13VB e− λ h1 + eλ h1 − 2 , − − b33λ e− λ h1 − eλ h1 a33 a33λ h1 e− λ h1 − eλ h1
f13VB V 2 (1 + a33ε 0 ) B , − a33 4a33
α 2 = −( S (1) c11(1) + S(2) c11(2) ) +
f13VB f132 h1B e− λ h1 + eλ h1 − 2 − , 2a33 b33λ eh1 − eλ h1
,
(40)
(41)
(42)
(43)
β 2 = A(1) c11(1) + A(2) c11(2) +
γ2 = −
2 f132 B e− λ h1 + eλ h1 − 2 , b33λ e − λh1 − eλ h1
V 2 (1 + a33ε 0 ) B. 2a33h1
(44)
(45)
Here, we define the volume, cross-section area, static moment and the second moment of inertia for the model with respect to x-axis as V , A , S , I , respectively. Then, the governing equations in Eqs. (28) and the first six boundary conditions in Eqs. (30) are expressed as d 4 w(x) d 3u 0 ( x ) + =0 α β 1 1 dx 4 dx3 2 3 α d w(x) + β d u0 ( x) = 0 2 2 dx3 dx 2
∀x ∈ (0, L) ,
dw( x) d 2 w(x) du ( x ) | x =0 = 0, α1 + β1 0 + γ1 =0 2 dx dx x=L dx d 3 w(x) d 2u0 ( x ) . α β w ( x ) | = 0, + =Q x=0 1 1 3 2 dx dx x=L 2 u ( x) | = 0, α d w(x) + β du0 ( x) + γ =0 x =0 2 2 2 dx 2 0 dx x= L Solutions of Eqs. (46) with the boundary conditions are easily obtained as
(46)
(47)
w(x) =
1 Qβ 2 1 γ β − γ β − QLβ 2 2 x3 + 2 1 1 2 x , 6 α1β 2 − α 2 β1 2 α1β 2 − α 2 β1
(48)
u0 ( x) =
γ α − γ α − QLα 2 1 Qα 2 x2 + 2 1 1 2 x. 2 α 2 β1 − α1β 2 α 2 β1 − α1β 2
(49)
Then the analytical solutions of the variables Pz , Ez , ϕ can be obtained by substituting Eqs. (48) and (49) into their expressions. When the external shear force Q is zero, polarization-induced bending under the closed circuit condition is the so called converse flexoelectric process.
4.2 Open circuit condition
As shown in Fig. 3, the bilayer cantilever is subjected to a shear force Q at the
free end. The electric potential of the lower surface of the dielectric layer is set to zero but no voltage is applied on the upper surface of the cantilever. Under the open circuit condition, the governing equations are obtained from Eqs. (25) and (26) as d2 M E dx 2 ( M + M ) = 0 d (F M + F E ) = 0 dx
∀x ∈ (0, L) ,
(50)
in V1 ,
(51)
Ez − X zz , z + ϕ, z = 0 −ε 0ϕ, zz + Pz , z = 0
and the boundary conditions are obtained from Eqs. (27) as w′(x) |x =0 = 0, (M M + M E ) |x = L = 0 w(x) | = 0, − d ( M M + M E ) + Q | = 0 x =0 dx x = L . M E u (x) | = 0, ( F + F ) | = 0 x =0 x= L 0 ϕ |z =0 = 0, (−ε 0ϕ, z + Pz ) |z = h = 0 1 X zz | z =0 = 0, X zz |z =h1 = 0
(52)
By solving the two electrical governing equations in Eqs. (51) with the last four electrical boundary conditions in Eqs. (52), the expressions of the electrical variables in terms of w( x) and u0 ( x) are obtained as, Pz = C1eλ z + C2 e− λ z +
f13ε 0 d 2 w(x) , 1 + a33ε 0 dx 2
Ez = a33C1eλ z + a33C2e − λ z +
ϕ=
C1
ε 0λ
eλ z −
C2
ε 0λ
e− λ z +
f13 (1 − a33ε 0 ) d 2 w(x) , dx 2 1 + a33ε 0
2 f13 d 2 w(x) f13 du0 ( x) z− . 2 1 + a33ε 0 dx 1 + a33ε 0 dx
(53)
(54)
(55)
where,
λ=
1 + a33ε 0 , b33ε 0 f13h1
C1 =
du ( x) d 2 w(x) + f13 0 (e−λ h1 − 1) dx 2 dx , − λ h1 b33λ (e − eλ h1 )
(56)
(57)
f13h1 C2 =
du ( x) λ h1 d 2 w(x) + f13 0 (e − 1) 2 dx dx . b33λ (e− λ h1 − e λh1 )
(58)
Under the open circuit condition, the axial electrostatic stress σ 11E includes only the quadratic nonlinear terms, which makes it negligible compared to the axial classical Cauchy Stress σ 11 . Thus, the axial electrostatic stress σ 11E and the electrostatic moment M E can be set to zero. By substituting Eqs. (5), (6) and (53-55) into (23), the total moment and the total axial force are obtained as M M = α1
du ( x) d 2 w(x) , + β1 0 2 dx dx
(59)
du ( x) d 2 w(x) + β2 0 , 2 dx dx in which the coefficients are expressed as F M = α2
α1 = −( I (1) c11(1) + I (2) c11(2) + A(1) g13(1) + A(2) g13(2) ) −
β1 = S(1)c11(1) + S (2) c11(2) +
f132 h12 B e− λ h1 + eλ h1 , b33λ e − λh1 − eλh1
f132 h1B e− λh1 + eλh1 − 2 , b33λ e− λ h1 − eλh1
α 2 = −( S(1) c11(1) + S(2)c11( 2) ) −
β 2 = A(1) c11(1) + A(2) c11(2) −
(60)
f132 h1 B e− λ h1 + eλ h1 − 2 , b33λ e− λ h1 − eλh1
2 f132 B e− λ h1 + eλ h1 − 2 . b33λ e −λ h1 − eλ h1
(61)
(62)
(63)
(64)
Then, the governing equations in Eqs. (50) and the first six boundary conditions in Eqs. (52) are expressed as d 4 w(x) d 3 u0 ( x ) + =0 β 1 α1 dx 4 dx3 ∀x ∈ (0, L) , 2 3 α d w(x) + β d u0 ( x) = 0 2 2 dx3 dx 2
(65)
dw( x) d 2 w(x) du ( x) |x =0 = 0, α1 + β1 0 =0 2 dx dx x = L dx d 3 w(x) d 2u 0 ( x ) + =Q. β w( x) |x =0 = 0, α1 1 dx3 dx 2 x = L 2 u ( x) | = 0, α d w(x) + β du0 ( x) =0 x=0 2 2 dx 2 0 dx x= L Solutions of Eqs. (65) with the boundary conditions are obtained as
(66)
w(x) =
1 Qβ 2 1 QLβ 2 x3 − x2 , 6 α1β 2 − α 2 β1 2 α1β 2 − α 2 β1
(67)
u0 ( x) =
1 Qα 2 QLα 2 x2 − x. α 2 β1 − α 1β 2 2 α 2 β1 − α1β 2
(68)
The analytical solutions of the variables Pz , Ez , ϕ can be obtained by substituting Eqs. (67) and (68) into their expressions. Bending-induced polarization under the open circuit condition is the so called direct flexoelectric process.
5. Numerical results and discussion
In this section, some numerical results and discussions are given based on the solutions obtained in the last section. For a slender rectangular bilayer cantilever, the length and width of the beam are set as L = 20 H and B = H , where H = 20nm is the total thickness of the bilayer cantilever. To ensure validation of the infinitesimal strain assumption, the applied shear force Q at the free end is 0.02 nN under both conditions and the voltage applied on the upper surface under closed circuit is 0.2 V . For the dielectric layer, the material parameters are taken according to the Ref. [34] as a33 = 8.767 ×1010 Nm 2 /C 2 , b33 = 1.921×10−9 Nm 4 /C 2 , c12(1) = 0.538 × 1011 Nm 2 /C , c44(1) = 0.325 × 1011 Nm 2 /C
,
and
c11(1) = c12(1) + 2c44(1) = 1.188 × 1011 Nm 2 /C
,
g13(1) = (c12(1) + 2c44(1) )l1(1) 2 − c44l2(1) 2 . Here, the two internal material length scale parameters are taken as the same i.e. l1(1) = l2(1) = l = 2.25nm . And f13 = 10V is adopted according to Kogan’s [10] estimate of the flexocoupling coefficient f ≈ 1 − 10V . For the elastic layer, the material parameters are taken as c11(2) = 1.5c11(1) , l1(2) = l2( 2) = 1.5l . The dimensionless height ratio h1 / h2 is adopted and set as 1 / 3 . All
the data listed above will be used in the following numerical calculations unless
otherwise stated. Effects of the strain gradient elasticity and flexoelectricity on the deflection of the bending cantilever under closed circuit condition are investigated. Fig. 4 shows the deflection across the length direction under different thicknesses and applied voltages. First, the applied voltage is set to be zero to observe the strain gradient elastic effect on the bending cantilever. Under this condition, the only effect makes the deflection deviate from that of the classical theory is strain gradient elastic effect. It is obviously observed that the strain gradient elastic effect dramatically decreases the deflection when the thickness is 20nm. In addition, from Fig. 4, the applied voltage also deviates the deflection due to the flexoelectric effect and the asymmetrical electrostatic stress on the cross section (in this numerical analysis the electrostatic stress and electrostatic moment can be neglected due to the small dielectric constant). It is clear that the flexoelectric effect decreases the deflection when the voltage is 0.2V while increases the deflection when the voltage is − 0.2V , which reveals that whether the flexoelectric effect stiffens or softens the cantilever depends on the applied voltage. Fig. 5 and Fig. 6 are presented to investigate the size-dependency of the strain gradient and flexoelectric effects. In Fig. 5, relation between the deflection at the free end of the cantilever and the thickness H is shown. With the increasing thickness H , the deflections under different applied voltages are gradually coincident with that of the classical theory. The curve at v = 0 gets close to the one of classical theory more rapidly than the other two curves. Meanwhile, variation of the deflection with the dimensionless thickness H / l at H = 20 nm is plotted in Fig.6. It is obvious that flexoelectricity enhances with higher f value, which indicate that the material with high f value may possess more opportunities for application. From the curves in Fig. 6, it is not difficult to infer that the strain gradient effect is dependent on the beam dimensionless thickness H / l , whereas the flexoelectric effect is almost independent of the material length scale parameters. It is indicated that for a bilayer flexoelectric cantilever both the strain gradient elastic effect and the flexoelectric effect significantly affect the deflection of the cantilever in nanoscale. Furthermore, both effects exhibit size-dependency and the strain gradient elastic effect weakens more rapidly than the flexoelectric effect with increasing thickness. Next, the effects of flexoelectricity on the electric field distribution and the polarization field distribution across the thickness under closed circuit condition are investigated. The curves plotted in Fig. 7 clearly manifest that the electric field across the thickness is different from that of the classical theory due to the flexoelectric effect. When the flexoelectric effect is concerned ( f = 10V ), the electric field can still be assumed as uniform across the thickness except near the upper and lower surfaces of the dielectric layer. Comparing Fig. 7(a) and Fig. 7(b), we can see that the effect of
flexoelectricity on the electric field distribution is independent of the direction of the applied voltage, which is also demonstrated in Eq. (32). Furthermore, from comparison of Fig. 7(b) and Fig. 7(c), the intensity and influencing scale of the flexoelectric effect on the electric field distribution decrease with the increasing thickness. The similar trend is found in the polarization field distribution across the thickness, which is not shown here. In the direct flexoelectric process, the electric potential across the thickness at x = L / 2 under different shear forces, thicknesses and f values are plotted in Fig. 8. It appears a smooth and steady increase of the induced electric potential in the central section, while sharp variation near the surfaces. This kind of electric potential distribution means large electric field near the surfaces while small uniform one in the central part. The induced electric potential can be explained as follows: the shear force induces the strain gradient along the thickness direction, and then polarization will be generated through the coupling between strain gradient and polarization. The sharp variation of the electric field near the surfaces with consideration of the flexoelectricity is a typical boundary behavior for a gradient theory [19]. Comparing the four curves in the figure, it is clearly observed that larger shear force and f value generates larger electric potential due to the larger strain gradient. In addition, as the total thickness increases from 20nm to 50nm , the generated electric potential decreases rapidly, which demonstrates obvious size-dependency. The deflection in the converse flexoelectric process is shown in Fig. 9. Due to the boundary conditions, the applied voltage will generate a non-uniform electric field and then a non-uniform polarization field near the boundaries. A bending moment is then generated through the coupling between strain and polarization gradient to make the cantilever bend. It is noted that the voltage applied in the opposite direction may induce a reverse bending in the cantilever. In addition, it is obviously to see that the induced electrical potential is size-dependent.
6. Conclusions In this work, a size-dependent bending model of a bilayer flexoelectric dielectric nanobeam, with consideration of the strain gradient elastic effect and the flexoelectric effect, is established based on the Bernoulli-Euler beam model and the theory for flexoelectric dielectrics. The solutions of static bending problems of a bilayer cantilever under open circuit and closed circuit conditions are obtained. Numerical results show that the strain gradient elastic effect decreases the deflection of the bilayer bending cantilever while the flexoelectric effect decreases or increases the deflection depends on whether the applied voltage is positive or negative. Both effects exhibit obvious size-dependency that the strain gradient effect is dependent on the beam dimensionless
thickness H / l , whereas the flexoelectric effect is only dependent on the beam thickness. Flexoelectricity also changes the electrical and polarization field distributions in the beam that sharp field gradients arise near the surfaces. Futhermore, in the direct flexoelectric process, the induced electric potential increases with the decreasing thickness and increasing f value. In the converse flexoelectric process, the applied voltage can generate bending and the deflection decreases as the thickness increases. The presented bilayer nanobeam model is helpful in understanding the mechanical and electrical properties in bilayer flexoelectric dielectric nanobeam.
Acknowledgments The work reported here is funded by the National Natural Science Foundation of China (11272186), Specialized Research Fund for the Doctoral Program of Higher Education of China (20120131110045) and the Natural Science Fund of Shandong Province of China (ZR2012AM014).
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Fig 1. Schematic of a bilayer nanobeam and its coordinate system Fig 2. Schematic of a bilayer cantilever under closed circuit condition Fig 3. Schematic of a bilayer cantilever under open circuit condition Fig 4. Variation of the deflection of the bilayer cantilever with dimensionless length Fig 5. Variation of the generated maximum deflection with beam thickness under different voltages Fig 6. Variation of the generated maximum deflection with the dimensionless thickness at H=20nm Fig 7. Electric potential distribution with the dimensionless thickness at x=L/2 (a) H=20nm, v=0.2V; (b) H=20nm, v=-0.2V; (c) H=50nm, v=-0.2V Fig 8. Electric potential distribution along the thickness direction at x=L/2 under different shear forces and thicknesses (direct flexoelectric process) Fig 9. Deflection of the cantilever with the dimensionless distance under different applied voltages and thicknesses (converse flexoelectric process)
Figure 1
Figure 2
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Figure 7(a)
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