Length effect on the ME coupling behavior of a simply supported finite composite cylinder

International Journal of Mechanical Sciences 76 (2013) 158–165

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Length effect on the ME coupling behavior of a simply supported finite composite cylinder Yang Huang n, De Zhong Liu Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 16 January 2013 Received in revised form 9 August 2013 Accepted 10 September 2013 Available online 2 October 2013

This work presents a three-dimensional exact solution to axisymmetric vibrations of a simply-supported piezoelectric/piezomagnetic composite cylinder. Both the static and harmonic driving magnetoelectric (ME) effects of the composite structure are investigated. It is shown in the numerical results that the length of the composite cylinder has significant effect on the ME effect and cannot be simply neglected. The increase of the length of the cylinder can enhance the static ME effect, while in the frequency driving case the increase of the length can decrease the resonance frequencies and enlarge the first few resonance peaks. The distributions of the electric potential in the piezoelectric layer are plotted to present the magnitude of the electric potential for each length. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Axisymmetric vibration Simply supported Magnetoelectric effect Composite cylinder

1. Introduction The magnetoelectric (ME) effect is characterized by the ME voltage coefficient α, which is defined by the ratio of electric field output over the magnetic field input or vice versa [1,2]. Compared with the single phase materials, the ME effect is largely enhanced in piezoelectric/piezomagnetic layered systems [1,3,4]. Recently, due to its potential applications in various devices and systems such as energy harvesters, sensors, ME transformers and radio frequency devices, the ME effect of composite structures has attracted broad research attention [5,6]. Layered composite plates have been well studied both theoretically and experimentally [3,4,7,8]. Investigations on ME effect of structures with curved shapes such as disks, cylinders and shells have also been reported [9–11]. The ME effect in layered structures is a product property produced by the mechanical coupling between reverse piezomagnetic and piezoelectric effect [1]. However, most PM/PE composites are coupled through interfacial shear stress which may lead to a lower ME coefficient. Thus, lots of studies have paid attention to special shapes such as disk-ring and cylindrical composite structures, in which normal stresses at the interface play an important role in the mechanical coupling of PE and PM phases. Wang et al. investigated the influences of curvature on the ME effect of a composite PE/PM cylindrical ring and found that a larger radius of the ring enhances the resonance peak of ME voltage coefficient [2]. They also discussed the magnetoelectric

n Correspondence to: Department of Engineering Mechanics, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, PR China. Tel.: þ 86 571 8795 2396; fax: þ 86 571 8795 2570. E-mail addresses: [email protected], [email protected] (Y. Huang).

0020-7403/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2013.09.007

effect of the cylindrical composite PM/PE ring with an imperfect interface [6]. Wu et al. studied resonance magnetoelectric effects in a disk-ring PE/PM composite structure and obtained the enhanced coupling and ME effect [12]. The study presented in [2,12] neglected the anti-plane strain and can be regarded as a plane strain problem. This is because in practical applications, a cylinder with its length much longer than its radius are always treated as a plane strain model, where the radial vibration mode dominates. However, in many situations the length and the radius of the cylinder are comparable, or in other words, the ratio of the length over the radius is not sufficiently large, then the longitudinal vibration mode cannot be totally neglected compared with the radial vibration mode. For arbitrary boundary conditions, the solution of a cylinder with finite length can be acquired in an approximate sense [13–16]. But for a simply supported finite cylinder, the boundary conditions at the two ends can be automatically satisfied and the remaining boundary conditions on the curved surfaces can be used to determine the amplitudes [17,18]. In this study, simply supported boundary conditions are adopted and only the first mode along the axial direction is numerically discussed. The ME effect in the static case is first investigated. The curve of the ME effect vs harmonic driving frequency is plotted. Influences of the cylinder's length and curvature on the resonance peaks are analyzed. Distributions of the electric potential are presented at last.

2. Analytical model and general solutions A piezoelectric/piezomagnetic composite cylinder with finite length as shown in Fig. 1 is considered. The inner layer is PM

Y. Huang, De Zhong Liu / International Journal of Mechanical Sciences 76 (2013) 158–165

159

where Z 0 and Z 1 represent zero- and first-order Bessel functions of the first or second kind. Substituting Eq. (5) into (4) yields:  8 m Cm ρm ω2 C m > 2 44 13 þ C 44 > > <  n þ C m11 k2  C m11 A  C m11 nB ¼ 0   ð6Þ m m m m 2 > > >  C 13Cþm C 44 nA þ  n2 þ ρ mω2  CC 33 B¼0 m : C 44 k

44

44

For nontrivial solutions of amplitudes A and B, the determinant of Eq. (6) is required to be zero, and we get a fourth-order equation quadratic in n2 . Two roots n1 2 , n2 2 can thus be obtained. Ratios of amplitudes are determined as follows: B A

λs ¼ ¼ Fig. 1. Sketch of a piezoelectric/piezomagnetic composite cylindrical structure.

(piezomagnetic) and the outer is PE (piezoelectric). Both layers are polarized in the axial direction and have the same material symmetry as the 6 mm crystal. The PE layer is coated with electrodes on the interface, outer surface and two end surfaces. The radii of the inner surface, interface, outer surface and length of the cylinder are denoted by a, b, c and L respectively. For axisymmetric problems, the mechanical displacements should satisfy uz ¼ uz ðr; z; tÞ, ur ¼ ur ðr; z; tÞ, uθ ¼ 0, and electric potential is ϕ ¼ ϕðr; z; tÞ. The constitutive equations and governing equations for the piezomagnetic layer are [5,12,19] 8 m ∂um ∂ψ m um m ∂um r r z T ¼ Cm > 11 ∂r þ C 12 r þ C 13 ∂z þ h31 ∂z > > rr > m m m > ∂u ∂u u m m m m z r r > < T θθ ¼ C 12 ∂r þ C 11 r þ C 13 ∂z þ h31 ∂∂zψ ∂ψ m r m r m z Tm > zz ¼ C 13 ∂r þ C 13 r þ C 33 ∂z þ h33 ∂z > >
m  > m > ∂u ∂u ∂ ψ m > Tm ¼ C z : þ r þh ∂um

44

rz

∂um

um

∂r

15 ∂r

∂z

:

∂r

þ

∂z

þ

Tm rz r

¼ρ

ð2Þ

2 m m ∂ uz 2

∂t

where the superscript ‘m’ denotes the piezomagnetic phase and m m represent stresses, elastic modulus, piezoTm ij , C ij , hij , and ρ magnetic coefficients, and density separately. ψ expðiωtÞ is the time harmonic magnetic potential exerted on the structure. For a specific mode in the axial direction, ψ can be assumed as ψ ¼  H0 r sin ðkzÞ, for which form ψ ¼ 0 can be satisfied at the two boundaries. The magnetic field can thus be derived as ! H ¼ ðH r ; 0; H z Þ ¼

 

∂ψ ∂ψ ; 0;  ∂r ∂z

 ¼ ðH 0 sin ðkzÞ; 0; H 0 rk cos ðkzÞÞ

44

0

s¼1

Substituting (9) again into (4) and requiring 8 > > > > <

J 1 ðkn1 rÞ

Y 1 ðkn1 rÞ

J 1 ðkn2 rÞ

Y 1 ðkn2 rÞ

kn1 J 0 ðkn1 rÞ

kn1 Y 0 ðkn1 rÞ

kn2 J 0 ðkn2 rÞ

kn2 Y 0 ðkn2 rÞ

A′1 ðrÞ

p ′ A1 ðrÞ

1

0

C C C C 0 C A h15 H 0 h33 H 0 rk2  m m C r C h15 þ h31 H0 k Cm 11

44

44

′

′

2

h m p m um r ¼ ur þ ur ¼ ∑ ½As J 1 ðkns rÞ þAs Y 1 ðkns rÞ cos ðkzÞ s¼1

"Z

s¼1

r

a

p ′ As ð

ξÞdξ U J 1 ðkns rÞ þ

Z

r a

# p ′ As ð

ξÞdξ UY 1 ðkns rÞ cos ðkzÞ ð11Þ

2

h m p m um z ¼ uz þ uz ¼ ∑ ½As λs J 0 ðkns rÞ þAs λs Y 0 ðkns rÞ sin ðkzÞ

"Z

þ ∑

s¼1

a

s¼1

r

p ′ As ð

ξÞdξ U λs J 0 ðkns rÞ þ

ð4Þ

Z a

r

# p ′ As ð

ξÞdξ U λs Y 0 ðkns rÞ sin ðkzÞ ð12Þ

# m Cm 11 C 12 J 1 ðkns rÞ þkC m λ J ðkn rÞ As s 13 s 0 r s¼1 # " m 2 Cm m 11  C 12 Y kn Y ðkn rÞ  ðkn rÞ þ kC λ Y ðkn rÞ As þ ∑ Cm s 0 s s s 1 11 13 s 0 r s¼1 (

ð5Þ

ð10Þ

one can get coefficients p A′1 ðrÞ, p A1 ðrÞ, p A′2 ðrÞ, p A2 ðrÞ, where the prime denotes the derivation. The general solutions for the piezomagnetic layer are

2

The solutions for the homogeneous equations (when H0 equals zero) can be expressed as the following form [20]:

9 > > > > =

λ1 Y 0 ðkn1 rÞ λ2 J 0 ðkn2 rÞ λ2 Y 0 ðkn2 rÞ > > > >  λ1 kn1 Y 1 ðkn1 rÞ  λ2 kn2 J 1 ðkn2 rÞ  λ2 kn2 Y 1 ðkn2 rÞ ;

0

þ ∑

44

h m ðh um r ; uz Þ ¼ ½AZ 1 ðknrÞ cos ðkzÞ; BZ 0 ðknrÞ sin ðkzÞ

1

Bp ′ C B B A ðrÞ C B B 1 C B Bp ′ C¼B B A1 ðrÞ C B @ A @

ð3Þ

8  
ρm ω2 C m k2  m C m þ C m ∂um h þ h ∂ 1∂ m >  C44m ur þ 13C m 44 k ∂rz ¼ 15C m 31 H 0 k cos ðkzÞ > > ∂r r ∂r ðrur Þ þ Cm > 11 11 11 11 >
m
m 2 < 2 Cm k Cm þ Cm ∂uz ρ ω 1 ∂  k 13C m 44 1r ∂r∂ þ C m  C33m um z r ∂r r ∂r 44 44 44 > > > 2 > h15 H 0 h33 H0 rk > : ðrum sin ðkzÞ r Þ ¼ C m r sin kz  Cm

ð7Þ

0

s¼1

2

The time harmonic factor expðiωtÞ is suppressed here. Combining the constitutive equations with the governing equations leads to the following nonhomogeneous equations:

ðs ¼ 1 or 2Þ

where J and Y are the first and second Bessel functions. The particular solutions for the nonhomogeneous equations are supposed in the following forms based on the method of variation of constants: 8 2 > p p > > p um r ¼ ∑ ½ As ðrÞJ 1 ðkns rÞ þ As ðrÞY 1 ðkns rÞ cos ðkzÞ < s¼1 ð9Þ 2 > > > : p um ¼ ∑ ½λs p As ðrÞJ ðkns rÞ þ λs p As ðrÞY 0 ðkns rÞ sin ðkzÞ:

0p

∂t

∂T m zz

z

λ1 J 0 ðkn1 rÞ > > > > :  λ1 kn1 J ðkn1 rÞ 1

and 8 ∂T m ∂T m T m  T m 2 m < ∂rrr þ ∂zrz þ rr r θθ ¼ ρm ∂ u2r ∂T m rz

2

The solutions for the homogeneous equations are thus written as follows: 8 2 > h m > > < ur ¼ ∑ ½As J 1 ðkns rÞ þ As Y 1 ðkns rÞ cos ðkzÞ s¼1 ð8Þ 2 > > > h um ¼ ∑ ½λs As J ðkns rÞ þ λs As Y 0 ðkns rÞ sin ðkzÞ: :

z

ð1Þ

m m  ns 2 þ ρm ω2 =C m 11 k  C 44 =C 11 m m m C 13 þ C 44 =C 11 ns

Tm rr ¼

2

∑

"

Cm 11 kns J 0 ðkns rÞ 

160

Y. Huang, De Zhong Liu / International Journal of Mechanical Sciences 76 (2013) 158–165

þ Cm 11

"Z

2

∑

s¼1

r a

ξÞdξ Ukns J 0 ðkns rÞ þ

Z

r a

"

2

# p ′ As ð

s¼1

#

m  ðC m 11  C 12 Þ=r

"Z

2

∑

s¼1

2

"Z

∑

s¼1

r a

a

p ′ As ð

r

p ′ As ð

ξÞdξ U J 1 ðkns rÞ þ

ξÞdξ U λs J 0 ðkns rÞ þ

Z

r a

Z

r a

Tm rz ¼

# p ′ As ð

Substituting Eq. (17) into Eqs. (15) and (16) gives the following equations: 8 e e e e e ω2 2 > > > ðρ k2  C 11 m  C 44 ÞC  ðC 13 þ C 44 ÞmD ðe31 þ e15 ÞmE ¼ 0 < 2  ðC e13 þ C e44 ÞmC þ ðρe ω2 C e44 m2  C e33 ÞD  ðe33 þ e15 m2 ÞE ¼ 0 k > > > :  ðe þ e ÞmC  ðe þ e m2 ÞD þ ðε þ ε m2 ÞE ¼ 0 31

15

33

15

33

11

ð18Þ

ξÞdξ U Y 1 ðkns rÞ #

p ′ As ð

 h13 H 0 rkg cos ðkzÞ 

ξÞdξ Ukns Y 0 ðkns rÞ

′ p ′ As ðrÞJ 1 ðkns rÞ þ p As ðrÞY 1 ðkns rÞ

þ Cm 11 ∑

þ Cm 13 k

p ′ As ð

ξÞdξ U λs Y 0 ðkns rÞ ð13Þ

For nontrivial solutions of amplitudes C, D and E, the determinant of Eq. (18) is required to be zero. One can obtain a sixth-order equation quadratic in m2 . Three roots m1 2 , m2 2 , m3 2 can thus be obtained. Ratios of amplitudes are determined as follows:

αj ¼

D C


 2 C e11 mj 2 þ C e44  ρe ω2 ðe33 þ e15 mj 2 Þ  ðe31 þ e15 ÞðC e13 þ C e44 Þmj 2 k
 ¼ 2 ðC e13 þ C e44 Þðe33 þ e15 mj 2 Þmj  ðe31 þ e15 Þ C e44 m2 þ C e33  ρe ω2 mj

2

 Cm 44 ∑ ½λs kns J 1 ðkns rÞ þkJ 1 ðkns rÞAs s¼1

2

 Cm 44 ∑ ½λs kns Y 1 ðkns rÞ þ kY 1 ðkns rÞAs

k

ð19Þ

s¼1 2

′

p ′ p þ Cm 44 ∑ ½ As ðrÞλs J 0 ðkns rÞ þ As ðrÞλs Y 0 ðkns rÞ

βj ¼

s¼1

 Cm 44 Z

r

þ a

"Z

2

∑

s¼1

a

 Cm 44 k ∑ r a



 2 2 ðC e13 þ C e44 Þ2 mj 2  C e11 mj 2 þ C e44  ρe ω2 C e44 mj 2 þ C e33  ρe ω2 k k
 : ¼ 2 ðC e13 þ C e44 Þðe33 þ e15 mj 2 Þmj  ðe31 þ e15 Þ C e44 m2 þ C e33  ρe ω2 mj

ξÞdξ U λs kns J 1 ðkns rÞ

k

General solutions of the piezoelectric layer are as follows:

"Z

r a

p ′ As ð

3

ξÞdξ UJ 1 ðkns rÞ #

p ′ As ð

ð20Þ

ξÞdξ U λs kns Y 1 ðkns rÞ

s¼1

Z

p ′ As ð

# p ′ As ð

2

þ

r

ξÞdξ U Y 1 ðkns rÞ h15 H0

uer ¼ ∑ ½C j J 1 ðkmj rÞ þ C j Y 1 ðkmj rÞ cos ðkzÞ

ð21Þ

j¼1

) sin ðkzÞ

ð14Þ

3

uez ¼ ∑ ½αj C j J 0 ðkmj rÞ þ αj C j Y 0 ðkmj rÞ sin ðkzÞ

ð22Þ

j¼1

General solutions for the piezoelectric layer can be acquired in a very similar way. The constitutive and governing equations for the piezoelectric layer are [5,12,19]  ∂ue ue ∂ue ∂ϕ T err ¼ C e11 r þ C e12 r þ C e13 z þ e31 ∂r r ∂z ∂z T eθθ ¼ C e12

E C

∂uer ue ∂ue ∂ϕ þ C e11 r þ C e13 z þe31 ∂r r ∂z ∂z

3

ϕ ¼ ∑ ½βj C j J 0 ðkmj rÞ þ βj C j Y 0 ðkmj rÞ sin ðkzÞ ( T err

j¼1

2 e ∂T erz ∂T ezz T erz e ∂ uz ∂r þ ∂z þ r ¼ ∂t 2 > > > : ∂Dz þ 1 ∂ ðrD Þ ¼ 0 r ∂z r ∂r

ρ

C e11  C e12 J 1 ðkmj rÞ þ kC e13 αj J 0 ðkmj rÞ r 3

þ ke31 βj J 0 ðkmj rÞBj þ ∑ ½C e11 kmj Y 0 ðkmj rÞ

∂uer ue ∂ue ∂ϕ þ C e13 r þ C e33 z þe33 ∂r r ∂z ∂z  e  e ∂uz ∂ur ∂ϕ þ þ e15 T erz ¼ C e44 ∂r ∂z ∂r  e  e ∂uz ∂ur ∂ϕ þ  ε11 Dr ¼ e15 ∂r ∂z ∂r

and 8 e T err  T eθθ ∂T erz ∂T rr ∂ 2 ue > ¼ ρe ∂t 2r > ∂r þ ∂z þ r > <

3

∑ ½C e11 kmj J 0 ðkmj rÞ 

¼

j¼1

T ezz ¼ C e13

1 ∂ ∂ue ∂ϕ ðrur Þ þ e33 z  ε33 Dz ¼ e31 r ∂r ∂z ∂z

ð23Þ

j¼1

C e  C e12 Y 1 ðkmj rÞ þ kC e13 αj Y 0 ðkmj rÞ  11 r ) þ ke31 βj Y 0 ðkmj rÞBj ( ð15Þ

T erz

cos ðkzÞ:

ð24Þ

3

 ∑ ½C e44 αj kmj J 1 ðkmj rÞ þ C e44 kJ 1 ðkmj rÞ þ e15 β j kmj J 1 ðkmj rÞBj

¼

j¼1 3

 ∑ ½C e44 αj kmj Y 1 ðkmj rÞ þC e44 kY 1 ðkmj rÞ j¼1

ð16Þ

respectively, where the superscript ‘e’ denotes the piezoelectric layer and T eij , C eij , eij , εij , Di , and ρe represents stresses, elastic modulus, piezoelectric coefficients, dielectric constants, electric displacements and density respectively. ϕ is the electric potential. The solutions can be supposed to be ðuer ; uez ; ϕÞ ¼ ½CZ 1 ðkmrÞ cos ðkzÞ; DZ 0 ðkmrÞ sin ðkzÞ; EZ 0 ðkmrÞ sin ðkzÞ ð17Þ

þ e15 β j kmj Y 1 ðkmj rÞBj ( Dr ¼

) sin ðkzÞ

ð25Þ

3

 ∑ ½e15 αj kmj J 1 ðkmj rÞ þe15 kJ 1 ðkmj rÞ  ε11 βj kmj J 1 ðkmj rÞBj j¼1

3

)

 ∑ ½e15 αj kmj Y 1 ðkmj rÞ þ e15 kY 1 ðkmj rÞ  ε11 β j kmj Y 1 ðkmj rÞBj j¼1

sin ðkzÞ

ð26Þ

Y. Huang, De Zhong Liu / International Journal of Mechanical Sciences 76 (2013) 158–165

161

3. Boundary conditions and vibration equations The end surfaces of the piezoelectric cylinder are electrically grounded, that is, ϕðr; 0Þ ¼ ϕðr; LÞ ¼ 0, as shown in Fig. 1. This condition can be satisfied according to Eq. (23) if k ¼ nLπ ðn ¼ 0; 7 1; 7 2; :::Þ. The composite cylinder is simplysupported in such a way that the supports allow inplane tangential displacements ur, which means T rz ðr; 0Þ ¼ T rz ðr; LÞ ¼ 0 and prevent displacements normal to the boundary, that is, uz ðr; 0Þ ¼ uz ðr; LÞ ¼ 0. These conditions can also be automatically satisfied if k ¼ nLπ ðn ¼ 0; 7 1; 7 2; :::Þ. The inner and outer surfaces of the composite cylinder are clamped: um r ða; zÞ ¼ 0

ð27Þ

um z ða; zÞ ¼ 0

ð28Þ

uer ðc; zÞ ¼ 0

ð29Þ

uez ðc; zÞ ¼ 0

ð30Þ

The interface is grounded whereas the outer surface is electrically open, and one has

ϕðb; zÞ ¼ 0

ð31Þ

Dr ðc; zÞ ¼ 0

ð32Þ

Fig. 2. Variation of ME effect vs thickness ratio for the static case. The unit of α is (V/m)(A/m)  1.

Both the displacements and stresses are continuous at the interface, and one gets e um r ðb; zÞ þ ur ðb; zÞ ¼ 0

ð33Þ

e um z ðb; zÞ þ uz ðb; zÞ ¼ 0

ð34Þ

e T m rr ðb; zÞ þ T rr ðb; zÞ ¼ 0

ð35Þ

e T m rz ðb; zÞ þ T rz ðb; zÞ ¼ 0

ð36Þ

Combining Eqs. (27)–(36) together leads to the following vibration equations: TAT ¼ F T

ð37Þ

where A ¼½A1 ; A1 ; A2 ; A2 ; C 1 ; C 1 ; C 2 ; C 2 ; C 3 ; C 3 . Components of the 10  10 matrix T, F are listed in the appendix and amplitudes A can thus be determined. The electric potential used to define the ME effect is taken from the middle point z ¼ L=2 of the interface and outer surface. Thus, we define the ME effect as ϕðc; L=2Þ  ϕðb; L=2Þ ϕðc; L=2Þ ¼ α ¼ ð38Þ ðc  aÞH 0 ðc  aÞH 0

4. Numerical results and discussions Numerical results of axisymmetric vibrations of the PE/PM composite cylinder are obtained and discussed. CoFe2O4 and BaTiO3 are used here as piezomagnetic and piezoelectric materials respectively. The values for the different material properties are given in [21]. In the calculation, the first mode is used, where k ¼ 1L π . Two quantities are introduced in the discussions: one is the thickness ratio m defined as m ¼ ðc  bÞ=ðc  aÞ, which is the ratio of the thickness of PE layer over the total thickness. The other is the ratio of inner radius over the total thickness: R ¼ a=ðc  aÞ. In this calculation, the elastic constants of PE and PM layers are multiplied by a complex factor (1 þ 0.01i) for viscous damping to avoid singularity at the resonance peaks. We at first investigate the ME effect for static or low-frequency (ω  0) case. The total thickness of the composite cylinder is 10 mm. Fig. 2 shows ME effect vs thickness ratio m, where R¼ 0.5 is used. It can be concluded from Fig. 2 that (1) for every

Fig. 3. Variation of the static ME effect vs dimensionless radius R. The unit of α is (V/m)(A/m)  1.

length L, the ME effect reaches a maximum value at a specific value m. (2) The longer the length of the composite ring is, the larger the specific value m is. (3) The increase of the length L not only increases m, but also enlarges the maximum α. Fig. 3 is the variation of static ME effect vs the dimensionless radius R of the composite structure, where m ¼0.5 is used. Some features can be observed in Fig. 3: (1) the ME effect increases with the increase of the dimensionless radius R (decrease of the curvature of the cylinder). (2) The longer the cylinder is, the larger the ME effect is. The ME effect vs the length of the cylinder is given in Fig. 4 to help us understand the length effect more directly. For the length in the region (5–50 mm) comparable to the radius (thickness) of the cylinder, the ME effect changes sharply with the increase of the length. But when the length in the region (50–200 mm) is much larger than the radius, the ME effect tend to slowly increase with the length and finally approaches a constant value when the length becomes sufficiently large. This phenomenon can be understood as that when the length is large enough compared with the radius, the structure behaves in a plane-strain state and the radial deformation dominates. Thus, changing the length does not affect the ME effect very much. Fig. 5 shows the variation of ME effect vs the driving circular frequency with different lengths, where c a¼10 mm, R¼0.5 and

162

Y. Huang, De Zhong Liu / International Journal of Mechanical Sciences 76 (2013) 158–165

Fig. 4. The ME effect vs the length of the cylinder in the static case. The unit of α is (V/m)(A/m)  1.

Fig. 6. ME effect vs harmonic driving frequency with large lengths of the composite cylinder. The unit of α is (V/m)(A/m)  1.

Fig. 5. Influences of the length of the composite cylinder on the frequency dependence of the ME effect. The unit of α is (V/m)(A/m)  1.

Fig. 7. Influences of the curvature of the composite cylinder on the frequency dependence of the ME effect. The unit of α is (V/m)(A/m)  1.

m¼0.5 are used. The ME effect is largely enhanced when the composite structure is driven at the resonance frequencies, as was observed by other studies [2,12]. It can be seen obviously that the larger the length is, the smaller the resonance frequencies are. Moreover, the length of the cylinder is relevant with the magnitudes of the resonance peaks. When the length is 6 mm, the highest peak is the fifth one and the frequency for this highest peak is 4400 kHz. When the length increases to 10 mm, the highest peak becomes the third one and the frequency for it decreases to 3000 kHz. When the length rises again to 14 mm, the first peak becomes the highest and the resonance frequency is 1500 kHz. The variation of the magnitude of resonance peaks for different lengths may be caused by the coupling between the radial and longitudinal vibration modes. When the radius and the length of the cylinder are comparable, the ME effects induced by the radial and longitudinal vibrations may cancel with or enhance each other, leading to the somewhat complex distribution behavior of the resonance peak. The ME effect vs the frequency with a large length is plotted in Fig. 6 separately from Fig. 5 for clearness. We can see that the shapes of the curves L¼ 50 mm and L¼ 200 mm are almost the same except for the first resonance frequencies and the associated magnitudes which show a little difference. More calculations for L larger than 200 mm have been done and the curves all coincide with the one for L¼200 mm. For simplicity and clearness, they are not presented in Fig. 6. This

observation comes to the conclusion similar to that of Fig. 4. When the length gets large enough, the radial vibration mode dominates in the coupled vibration. In the dynamic case, the resonance frequencies of a long composite cylinder are thus largely determined by the radial size and are hardly influenced by the longitudinal size. Influences of the curvature of the composite cylinder on resonance peaks are plotted in Fig. 7, where ca¼10 mm, L¼ 14 mm and m¼0.5 are used. We notice that the radius of the composite structure has significant effect on resonance peaks. When R is 0.2, the first resonance peak is 25 (V/m)(A/m)  1. But the magnitude of this peak increases to more than four times larger when R increases to 2. The enhancement of the ME effect when increasing the radius of the cylinder is mainly caused by the stimulation of the radial vibration mode. A larger radius makes the radial vibration more violent and thus contributes more to the ME effect. Further, the interval of the first and second resonance peaks is shortened when the dimensionless radius R of the cylinder is enlarged. Distributions of the electric potential in the piezoelectric layer induced by unit magnetic field are presented in Fig. 8(a)–(d), which show ϕ vs radial coordinate r under the static case, ω ¼ 1500 kHz, ω ¼ 3000 kHz and ω ¼ 4000 kHz respectively. Here R¼0.5, m ¼0.5 and c a ¼10 mm are used. Fig. 8(a) indicates that without the harmonic driving frequency, the magnitudes of electric potentials increase with the increase of the length.

Y. Huang, De Zhong Liu / International Journal of Mechanical Sciences 76 (2013) 158–165

163

Fig. 8. Distributions of the electric potential in the piezoelectric layer, where (a)–(d) correspond to frequencies ω ¼0 kHz, ω¼ 1500 kHz, ω¼ 3000 kHz and ω¼ 4000 kHz respectively.

In Fig. 8(b) where the driving frequency is 1500 kHz, the L¼14 mm structure is at the resonance frequency, which agrees with that in Fig. 5 and the magnitude of ϕ is much larger than others. Similarly, Fig. 8(c) illustrates that the L¼ 10 mm is at the resonance frequency, which is also in accordance with Fig. 5.

5. Conclusions Three-dimensional exact solutions for axisymmetric vibrations of a PE/PM composite cylinder are obtained. The cylinder is simply supported and electrically grounded at the two boundaries. The first mode along the axial direction is discussed in the numerical results. The static ME effect is analyzed, which shows that the ME effect can be enhanced by increasing the length of the composite cylinder. Frequency dependence of the ME effect is analyzed and one can conclude that the length of the composite cylinder not only affects the resonance frequencies but also has effect on the

magnitudes of the resonance peaks. That is, the larger the length is, the smaller the resonance frequencies are and the higher the first few peaks are. Results also show that when the length of the cylinder becomes much larger than the radius, the ME effect in either the static case or the dynamic case is determined only by the radial size while does not vary with the changing of the longitudinal size significantly. Effect of the curvature on the ME effect is also graphically presented. The decrease of the curvature is observed to increase the resonance peaks. Distributions of the electric potential in the PE layer are plotted and the magnitudes of ϕ under different lengths are seen clearly in Fig. 8(a)–(d) and the length for which the cylinder is under resonance can be inferred, too.

Acknowledgments This work was supported by the National Science Foundation of China (Grant no. 11090333).

Appendix T 11 ¼  J 1 ðkn1 aÞ; T 12 ¼  Y 1 ðkn1 aÞ; T 13 ¼  J 1 ðkn2 aÞ; T 13 ¼  Y 1 ðkn2 aÞ; T 21 ¼  λ1 J 0 ðkn1 aÞ; T 22 ¼  λ1 Y 0 ðkn1 aÞ; T 23 ¼  λ2 J 0 ðkn2 aÞ; T 24 ¼  λ2 Y 0 ðkn2 aÞ; T 35 ¼ J 1 ðkm1 cÞ; T 36 ¼ Y 1 ðkm1 cÞ; T 37 ¼ J 1 ðkm2 cÞ; T 38 ¼ Y 1 ðkm2 cÞ; T 39 ¼ J 1 ðkm3 cÞ; T 310 ¼ Y 1 ðkm3 cÞ; T 45 ¼ α1 J 0 ðkm1 cÞ; T 46 ¼ α1 Y 0 ðkm1 cÞ; T 47 ¼ α2 J 0 ðkm2 cÞ; T 48 ¼ α2 Y 0 ðkm2 cÞ;

T 49 ¼ α3 J 0 ðkm3 cÞ; T 410 ¼ α3 Y 0 ðkm3 cÞ;

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T 55 ¼ β1 J 0 ðkm1 bÞ; T 56 ¼ β1 Y 0 ðkm1 bÞ; T 57 ¼ β 2 J 0 ðkm2 bÞ; T 58 ¼ β 2 Y 0 ðkm2 bÞ; T 59 ¼ β3 J 0 ðkm3 bÞ; T 510 ¼ β3 Y 0 ðkm3 bÞ; T 65 ¼  ½e15 α1 km1 J 1 ðkm1 cÞ þ e15 kJ 1 ðkm1 cÞ  ε11 β 1 km1 J 1 ðkm1 cÞ T 66 ¼  ½e15 α1 km1 Y 1 ðkm1 cÞ þ e15 kY 1 ðkm1 cÞ  ε11 β1 km1 Y 1 ðkm1 cÞ T 65 ¼  ½e15 α2 km2 J 1 ðkm2 cÞ þ e15 kJ 1 ðkm2 cÞ  ε11 β 2 km2 J 1 ðkm2 cÞ

T 66 ¼  ½e15 α2 km2 Y 1 ðkm2 cÞ þ e15 kY 1 ðkm2 cÞ  ε11 β2 km2 Y 1 ðkm2 cÞ

T 65 ¼  ½e15 α3 km3 J 1 ðkm3 cÞ þ e15 kJ 1 ðkm3 cÞ  ε11 β 3 km3 J 1 ðkm3 cÞ

T 66 ¼  ½e15 α3 km3 Y 1 ðkm3 cÞ þ e15 kY 1 ðkm3 cÞ  ε11 β3 km3 Y 1 ðkm3 cÞ

T 71 ¼  J 1 ðkn1 bÞ; T 72 ¼  Y 1 ðkn1 bÞ; T 73 ¼ J 1 ðkn2 bÞ; T 74 ¼  Y 1 ðkn2 bÞ; T 75 ¼ J 1 ðkm1 bÞ; T 76 ¼ Y 1 ðkm1 bÞ; T 77 ¼ J 1 ðkm2 bÞ; T 78 ¼ Y 1 ðkm2 bÞ; T 79 ¼ J 1 ðkm3 bÞ; T 710 ¼ Y 1 ðkm3 bÞ; T 81 ¼  λ1 J 0 ðkn1 bÞ; T 82 ¼  λ1 Y 0 ðkn1 bÞ; T 83 ¼  λ2 J 0 ðkn2 bÞ; T 84 ¼  λ2 Y 0 ðkn2 bÞ; T 85 ¼ α1 J 0 ðkm1 bÞ; T 86 ¼ α1 Y 0 ðkm1 bÞ; T 87 ¼ α2 J 0 ðkm2 bÞ; T 88 ¼ α2 Y 0 ðkm2 bÞ; T 89 ¼ α3 J 0 ðkm3 bÞ; T 810 ¼ α3 Y 0 ðkm3 bÞ; m m m T 91 ¼  ½C m 11 kn1 J 0 ðkn1 bÞ  ðC 11  C 12 Þ=b U J 1 ðkn1 bÞ þ kC 13 λ1 J 0 ðkn1 bÞ;

m m m T 92 ¼  ½C m 11 kn1 Y 0 ðkn1 bÞ  ðC 11  C 12 Þ=b U Y 1 ðkn1 bÞ þkC 13 λ1 Y 0 ðkn1 bÞ; m m m T 93 ¼  ½C m 11 kn2 J 0 ðkn2 bÞ  ðC 11  C 12 Þ=b U J 1 ðkn2 bÞ þ kC 13 λ2 J 0 ðkn2 bÞ; m m m T 94 ¼  ½C m 11 kn2 Y 0 ðkn2 bÞ  ðC 11  C 12 Þ=b U Y 1 ðkn2 bÞ þkC 13 λ2 Y 0 ðkn2 bÞ;

T 95 ¼ C e11 km1 J 0 ðkm1 bÞ ðC e11  C e12 Þ=bU J 1 ðkm1 bÞ þ kC e13 α1 J 0 ðkm1 bÞ þ ke31 β1 J 0 ðkm1 bÞ; T 96 ¼ C e11 km1 Y 0 ðkm1 bÞ  ðC e11  C e12 Þ=b U Y 1 ðkm1 bÞ þ kC e13 α1 Y 0 ðkm1 bÞ þke31 β 1 Y 0 ðkm1 bÞ; T 97 ¼ C e11 km1 J 0 ðkm2 bÞ ðC e11  C e12 Þ=bU J 1 ðkm2 bÞ þ kC e13 α2 J 0 ðkm2 bÞ þ ke31 β2 J 0 ðkm2 bÞ; T 98 ¼ C e11 km1 Y 0 ðkm2 bÞ  ðC e11  C e12 Þ=b U Y 1 ðkm2 bÞ þ kC e13 α2 Y 0 ðkm2 bÞ þke31 β 2 Y 0 ðkm2 bÞ; T 99 ¼ C e11 km1 J 0 ðkm3 bÞ ðC e11  C e12 Þ=bU J 1 ðkm3 bÞ þ kC e13 α3 J 0 ðkm3 bÞ þ ke31 β3 J 0 ðkm3 bÞ; T 910 ¼ C e11 km1 Y 0 ðkm3 bÞ  ðC e11 C e12 Þ=b UY 1 ðkm3 bÞ þkC e13 α3 Y 0 ðkm3 bÞ þ ke31 β3 Y 0 ðkm3 bÞ; m T 101 ¼ C m 44 λ1 kn1 J 1 ðkn1 bÞ þ C 44 kJ 1 ðkn1 bÞ;

m T 102 ¼ C m 44 λ1 kn1 Y 1 ðkn1 bÞ þ C 44 kY 1 ðkn1 bÞ; m T 103 ¼ C m 44 λ2 kn2 J 1 ðkn2 bÞ þ C 44 kJ 1 ðkn2 bÞ; m T 104 ¼ C m 44 λ2 kn2 Y 1 ðkn2 bÞ þ C 44 kY 1 ðkn2 bÞ;

T 105 ¼  ½C e44 α1 km1 J 1 ðkm1 bÞ þC e44 kJ 1 ðkm1 bÞ þ e15 β 1 km1 J 1 ðkm1 bÞ; T 106 ¼  ½C e44 α1 km1 Y 1 ðkm1 bÞ þ C e44 kY 1 ðkm1 bÞ þe15 β1 km1 Y 1 ðkm1 bÞ; T 107 ¼  ½C e44 α2 km2 J 1 ðkm2 bÞ þC e44 kJ 1 ðkm2 bÞ þ e15 β 2 km2 J 1 ðkm2 bÞ; T 108 ¼  ½C e44 α2 km2 Y 1 ðkm2 bÞ þ C e44 kY 1 ðkm2 bÞ þe15 β2 km2 Y 1 ðkm2 bÞ; T 109 ¼  ½C e44 α3 km3 J 1 ðkm3 bÞ þC e44 kJ 1 ðkm3 bÞ þ e15 β 3 km3 J 1 ðkm3 bÞ; T 1010 ¼  ½C e44 α3 km3 Y 1 ðkm3 bÞ þ C e44 kY 1 ðkm3 bÞ þ e15 β 3 km3 Y 1 ðkm3 bÞ; "Z

2

F7 ¼ ∑

s¼1

b a

"Z

2

F8 ¼ ∑

s¼1

b a

p ′ As ð

ξÞdξ UJ 1 ðkns bÞ þ

p ′ As ð

s¼1

b a

p ′ As ð

m  ðC m 11  C 12 Þ=b

p ′ As ð

Z

∑

s¼1

"Z

b a

b

ξÞdξ U Y 1 ðkns bÞ ; #

p

a

ξÞdξ U kns J 0 ðkns bÞ þ 2

2

#

b

a

ξÞdξ U λs J 0 ðkns bÞ þ

"Z

2

F 9 ¼ Cm 11 ∑

Z

Z

A′ðξÞdξ U λs Y 0 ðkns bÞ ; b

a

#

2

p ′ As ð

ξÞdξ U kns Y 0 ðkns bÞ þ C m 11 ∑

p ′ As ð

ξÞdξ U J 1 ðkns bÞ þ

s¼1

Z a

b

# p ′ As ð

"Z

∑

s¼1 2

C m 44 k ∑

s¼1

b a

"Z a

p ′ As ð

ξÞdξ U λs kns J 1 ðkns bÞ þ

b

p ′ As ð

ξÞdξ U J 1 ðkns bÞ þ

Z

b a

Z

b a

p ′ As ð

# p ′ As ð

ξÞdξ U λs kns Y 1 ðkns bÞ #

ξÞdξ UY 1 ðkns bÞ  h15 H 0 ;

The components not appearing here are zeros.

2

s¼1

′

s¼1

2

# ′ p ′ As ðbÞJ 1 ðkns bÞ þ p As ðbÞY 1 ðkns bÞ

ξÞdξ U Y 1 ðkns bÞ þ C m 13 k ∑

p ′ p F 10 ¼ C m 44 ∑ ½ As ðbÞλs J 0 ðkns rÞ þ As ðbÞλs Y 0 ðkns bÞ

C m 44

" "Z

b a

p ′ As ð

ξÞdξ U λs J 0 ðkns bÞ þ

Z

b a

# p ′ As ð

ξÞdξ U λs Y 0 ðkns bÞ  h13 H0 bk

Y. Huang, De Zhong Liu / International Journal of Mechanical Sciences 76 (2013) 158–165

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