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Dynamic effective property of piezoelectric composites with coated piezoelectric nano-fibers
Accepted Manuscript Dynamic effective property of piezoelectric composites with coated piezoelectric nano-fibers Xue-Qian Fang, Ming-Juan Huang, Jin-Xi Liu, Wen-Jie Feng PII: DOI: Reference:
S0266-3538(14)00134-1 http://dx.doi.org/10.1016/j.compscitech.2014.04.017 CSTE 5795
To appear in:
Composites Science and Technology
Received Date: Revised Date: Accepted Date:
29 August 2013 2 April 2014 20 April 2014
Please cite this article as: Fang, X-Q., Huang, M-J., Liu, J-X., Feng, W-J., Dynamic effective property of piezoelectric composites with coated piezoelectric nano-fibers, Composites Science and Technology (2014), doi: http://dx.doi.org/10.1016/j.compscitech.2014.04.017
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Dynamic effective property of piezoelectric composites with coated piezoelectric nano-fibers Xue-Qian Fang* ,
Ming-Juan Huang, Jin-Xi Liu, Wen-Jie Feng
Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China Abstract: This paper addresses the size-dependent dynamic effective electro-elastic properties of piezoelectric composites embedded with coated piezoelectric nano-fibers. By using the Effective Field Method of electric-elastic coupling wave, the randomly distributed coated nano-fibers in the piezoelectric media are reduced into a typical piezoelectric coated nano-fiber. To analyze the size-dependent effect, the coupling surface/interface theory is introduced to analyze the interface effect around the coating layers. Wave function expansion method is used to express the coupling wave field in the effective coupling field. The piezoelectrically stiffened elastic modulus and effective coupling wave number are obtained. Through analysis, it is found that the coupling surface/interface effect on the piezoelectrically stiffened elastic modulus show significant variation with the material properties of coating layer and the incident wave frequency. The effect of piezoelectric property of surfaces/interfaces on the effective shear modulus is also examined. Comparison with the existing results is given to validate this dynamic electro-elastic model. Keywords: A. Nano composites; A. Coating; B. Interface; C. Elastic properties; D. Ultrasonics 1. Introduction The nanosize characteristic, the excellent piezoelectricity performances, and the unique coupling between piezoelectric and semiconducting properties of piezoelectric *Corresponding author. Tel.: +86 311 87936542 E-mail address: [email protected] (X.Q. Fang)
nanocomposites make them attractive for applications such as transducers, sensors, actuators and energy harvesters in the nano-electro-mechanical systems. Recently, the theory and experimental investigations on piezoelectric nano-structures have received considerable attention because of their excellent piezoelectric and micro properties over conventional composites [1-3]. Compared
with
the
macroscopic
piezoelectric
composites,
piezoelectric
nanocomposites exhibit size-dependent mechanical and physical properties due to their large ratio of surface area to volume. In recent years, many experimental investigations and atomistic simulations have shown that the elastic constants or the piezoelectric coefficients of some piezoelectric nanocomposites increase significantly with the decrease of the material size to the nanoscale [4,5]. Determining the effective behavior of nanocomposites embedded with inclusions is of considerable importance in a wide variety of physical problems. In traditional continuum mechanics, surface free energy around the inclusions is typically neglected because it is associated with only a few layers of atoms near the surface, and the ratio of the volume occupied by the surface atoms and the total volume of material of interest are extremely small. However, due to a large ratio of the surface/interface region to the bulk, the effective properties of nano-composites are inevitably be affected by surfaces/interfaces. In the past years, the investigation of surface/interface effect on the static [6-7] and dynamic [8-10] effective properties of nanocomposites is attracting more and more interests. With the advent of piezoelectric nanocomposites in recent years, some attempts have been made to reveal the size-dependent behavior of piezoelectric nano-structures. Within Nb2O5 nanotubes forming a unique class of heterostructure nanotubes combining ferroelectric and semiconducting nanomaterials, multiple NaNbO3 nanoplates were created by Yan et al., and the local piezoelectric
-2-
properties were studied [11]. Both finite element method (FEM) simulations and experiments were carried out to investigate the stress distribution and the interfacial effects between the matrix and piezoelectric NWs by altering the slip conditions at the interface and changing the mechanical properties of the matrix [12]. Fakhzan and Muthalif presented an analytical estimation of voltage production of piezoelectric cantilever beam when subjected to base excitation, with and without attached proof masses [13]. In the above mentioned papers, it can be concluded that the electro-elastic properties of the interface-region become very important, and should be given due consideration while formulating their overall properties. In recent years, the electro-elastic coupling surface/interface model has been proposed to characterize the behavior of piezoelectric nanostructures. In this model, the interfacial properties are defined by introducing the concept of a dividing surface. The surface/interface between the piezoelectric inclusion and the matrix is assumed to be coherent, and no atomic bonds are broken along the surface/interface [14,15]. In the work of Fang et al., the coupling surface/interface theory was used to predict the dynamic electro-elastic properties of piezoelectric nanocomposites with nano-fibers [16]. To obtain a better performance of piezoelectric nano-structures, coating layers around the reinforced inclusions are often introduced to reduce the internal stresses and enhance the piezoelectric behavior. In these nano-structures, a larger ratio of surface/interface area to volume exists, and expresses greater effect on the performance of piezoelectric nano-structures. For conventional nanocomposites, Ozmusul and Picu [17] studied the elastic moduli of composites containing an anisotropic coating layer and nano-particles that are either of finite stiffness or rigid. In addition, Boutaleb et al. [18] proposed a micromechanical analytical model to
-3-
predict the stiffness and yield strength of silica/polymer nano-composites, and it is shown that the interphase is a dominant parameter for controlling the overall nano-composite behavior. However, up to now, no investigations on the prediction of piezoelectric composites with coating nano-fibers are available. To optimally design the piezoelectric nanostructures, this paper addresses the dynamic effective electro-elastic properties of piezoelectric composites embedded with coated piezoelectric nano-fibers. The multiple scattering of electro-elastic waves among the randomly distributed piezoelectric nano-fibers is transformed to the typical piezoelectric nano-fiber problem by using Effective Field Method. The fundamental framework of coupling surface/interface model is generalized to take into account the surface/interface effect around the typical piezoelectric nano-fiber on the effective elastic modulus. The scattered far-field method and the iterative process are employed to obtain the effective propagating wave number and the effective elastic modulus. Through the numerical examples, the effects of surface/interface around the coating layer on the dynamic shear modulus are analyzed in detail. 2. Problem formulation and dynamic effective field method An unbounded piezoelectric matrix containing randomly distributed coated nano-fibers is considered. A microscopic Representative Volume Element (RVE) at the microscopic scale is proposed. The piezoelectric composite is considered to be homogeneous and isotropic at the macro-continuum level; however it consists of a spatially uniform distribution of embedded coated nano-fibers, as shown in the schematic of Fig. 1. The volume fraction of the coated nano-fibers is n0 .The coating layers correspond to the volume defined by the narrow region sandwiched between the nano-fiber and matrix with different properties, while the interface refers to the surface areas between the coating layers and the fibers, and those between the fibers
-4-
and the matrix. This transition of the properties from surface/interface to the bulk value may take place over a few layers of atoms [16]. In the following, the matrix occupies a region is denoted by ‘M’, the coating layers are denoted by ‘C’, and the nano-fibers occupying a region are ‘F’. ‘S’ represents the matrix-coating and coating-fiber interface. The outer surface/interface is denoted as ‘S1’, and the inner one is ‘S2’. The matrix, coating layer, and fiber are both assumed to be transversely isotropic, and are polarized in the z-direction. It is assumed that an electro-elastic wave resulting from the randomly distributed coated nano-fibers propagates in the matrix material. The interaction between the coated nano-fibers gives rise to the dispersion relations for electro-elastic waves, and the propagating coupling wave number denoted as effective wave number will change. The distribution of the piezoelectric nano-fiber in the transverse piezoelectric matrix is assumed to be homogeneous and isotropic. If the incident field is a plane electro-elastic coupling wave, the mean fields are also plane electro-elastic waves in many important cases. To construct these electro-elastic coupling fields, it is very difficult to appropriately descript the interaction between the random coated nano-fibers in the piezoelectric composite media. Strictly speaking, in order to construct these fields, we have to find the detailed wave fields for every realization of the random set of coated nano-fibers and then to average the results over an ensemble of realizations of this set. These principal difficulties of this complex phenomenon enable us only to find the approximate solution. Self-consistent scheme named as Effective Field Method, which is powerful tool, is used to obtain such solutions in this paper, and mainly based on the following two hypotheses. The first hypothesis is expressed as: Every coated nano-fiber in the composites behaves as an isolated one embedded in the original matrix by the action of local
-5-
effective coupling fields (displacement u * (r ) and electric potential J * (r ) ). The local effective fields denote the sum of the incident fields applied to the medium and the scattered fields on all the surrounding coated nano-fibers. Therefore, the local effective fields can be written in the forms N
u * (r ) u in (r ) £ unsc (r ) ,
(1)
n 1
N
J * (r ) J in (r ) £ Jnsc (r ) .
(2)
n 1
This Hypothesis reduces the problem of interaction between the randomly distributed piezoelectric nano-fibers to the typical piezoelectric nano-fiber problem in the effective electro-elastic medium, as depicted in Fig. 1. The second hypothesis is: Random local effective fields u * (r ) and electric potential J * (r ) acting on an arbitrary fiber are statistically independent on the properties and the geometrical characteristics (containing the presences and locations) of this fiber. According to this hypothesis, one can obtain that
u * (r ) u (r ) | r , J * (r ) J (r ) | r , where symbol
(3)
u | r denotes the averaging under the condition that the point r
belongs to the region occupied by the cross-sections of coated nano-fibers. Based on the above hypotheses and following the work of Levin et al. [19], the governing equations of average electro-elastic medium can be expressed as * c44 (k * )$ u (r ) e15* (k * )$ J (r ) R * (k * )W 2 u (r ) 0 ,
e15* (k * )$ u (r ) C11* (k * )$ J (r ) 0 ,
(4) (5)
* * * where c44 , e15 and C11 are, respectively, the elastic stiffness, piezoelectric constant
and dielectric constant of effective coupling fields, they are all related with the
-6-
effective coupling wave number k * , M F M (1 n0 ) ( C11F C11M )c44 ) C11M (e11F e11M )e11M ¶ (e11F e11M )e11M ¶ § (1 n0 ) (c44
c44 1 · ¨ · M M M M C11 (e11M ) 2 C11 (e11M ) 2 2 c44 2 c44 © ¸© ¸ (1 n0 ) 2 F M M
[(e F e M ) C M ( C11F C11M )e15M ][(c44 )e15M (e15F e15M )c44 ] , (6)
c44 M M C11 (e11M ) 2 ]2 15 15 11 4[c44
§
$ ¨1
k * is the effective wave number, and *
u (r ) U *ei( k r Wt ) ,
J (r )
e15* * 11
C
*
U *ei( k r Wt ) .
(7)
The dispersion relation for effective coupling wave number can be written as ª * § (e15* ) 2 ¶ ¹ * 2 * 2 «c44 ¨ * · º (k ) R W 0 . ¬ © C11 ¸ »
(8)
The relation between the piezoelectrically stiffened elastic modulus and the effective wave number is expressed as 2
* M c44 c44 ( R * / R M ) §© Re(k M / k * ) ¶¸ ,
M in which k M W M M / R M with M M c44
(e15M ) 2
C11M
(9)
. For a small density contrast, the
simple rule of mixture is used in the numerical examples, i.e.,
R * n0 R f (1 n0 ) R M ,
(10)
where R M LR M (1 L ) R C with L b / a . When the multiple scattering among the piezoelectric nano-fibers is reduced to the typical piezoelectric one-fiber problem in the effective piezoelectric medium (see Fig. 1), the scattering of effective electro-elastic waves resulting from the typical nano-fiber comes into being. The coupling wave fields around the typical nano-fiber can be described in Section 4. 3. Solving the typical coated nano-fiber in effective piezoelectric media 3.1 Governing equations in the piezoelectric media -7-
In the electro-elastic medium, an anti-plane electro-elastic wave with frequency W propagates in the positive x-direction. The mechanically and electrically coupled constitutive equations can be written as
S xz c44
tu tJ , e15 tx tx
S yz c44
tu tJ , e15 ty ty
(11)
Dx e15
tu tJ ,
C11 tx tx
Dy e15
tu tJ ,
C11 ty ty
(12)
where S jz , u , J , and D j ( j x, y ) are the shear stress, anti-plane displacement, electric potential, and in-plane electric displacement, respectively. The following governing equations can be expressed as c44 2u e15 2J R
t 2u , tt 2
e15 2u C11 2J 0 .
(13) (14)
Here 2 t 2 / tx 2 t 2 / ty 2 is the two-dimensional Laplace operator in the variables x and y. To decouple Eqs. (13) and (14), another electro-elastic field Y is introduced,
Y J Hu ,
(15)
where H e15 / C11 . Substituting Eq. (15) into Eqs. (13)-(14), one can obtain the following equations 2u
1 t 2u , 2 tt 2 cSH
2Y 0 , where cSH M e / R with M e c44
e152
C11
(16)
(17)
being the wave speed of electro-elastic
waves in the piezoelectric media. 3.2. Wave fields around the typical coated nano-fiber -8-
a. Incident electro-elastic coupling wave field In the local coordinate system (r * , Q * ) of the real cavity, the incident waves can be expanded as c
*
u *(in ) u0 eik x u0 £ E ni n J n (k *r ) cos(nQ ) ,
(18)
n 0
where
ª1, n 0 , ¬2, n 1, 2,3,!
En «
is the amplitude of the incident waves,
u0
k * W / M * / R * is the wave number of the propagating waves, and J n () is the nth
Bessel function of the first kind. Similarly, the incident field J (i ) is expressed as
J *(in ) J0 e
i k * x Wt
c
J0 £ E ni n J n k *r cos(nQ ) ,
(19)
n 0
where J0
e15
C11
u0 .
b. Scattered wave field Using wave function expansion method, the general solution of scattered field resulting from the coated nano-fiber can be expressed, as c
u *( sc ) u0 £ an H n (k *r ) cos(nQ ) ,
(20)
n 0
J *( sc )
u0 e15
C11
c
£a H n
n 0
c
n * cos(nQ ) , n ( k r ) cos( nQ ) £ bn r
(21)
n 0
where an and bn are the mode coefficients of the electro-elastic waves, and can be determined by satisfying the boundary conditions at the surface/interface. H n < is the nth Hankel function of the first kind, and denotes the outgoing wave. c. Wave field in the piezoelectric coating layer of the typical nano-fiber The wave field in the piezoelectric coating layer may be described by the sum of -9-
the two components (outgoing and ingoing waves), and are expressed in the following form c §c ¶ u C u0 ¨ £ cn H n (k C r ) cos(nQ ) £ d nYn (k C r ) cos(nQ ) · , n 0 © n 0 ¸
JC
e15C
C11C
c
c
n 0
n 0
u0 £ §©cn H n k C r d nYn k C r ¶¸ cos(nQ ) £ §©e1n r n e2 n r n ¶¸ cos(nQ ) ,
(22)
(23)
where k C W / M C / R C , Yn < are the nth Hankel functions of the second kind, and denote the ingoing waves. cn , d n , e1n and e2n are the mode coefficients in the coating layer. d. Refracted wave field The refracted waves confined inside the piezoelectric nano-fiber are standing electro-elastic waves, and represented by c
u F u0 £ f n J n k F r cos(nQ ) ,
(24)
n 0
JF
e15F
C11F
c
c
n 0
n 0
u0 £ f n J n k F r cos(nQ ) £ g n r n cos(nQ ) ,
(25)
where k F W / M F / R F is the wave number inside the coated nano-fiber with
M F c44F
(e15F ) 2
C11F
, f n and g n are the mode coefficients of the refracted waves, and
the cylindrical Bessel functions of the first kind are used to obtain the standing waves. d. Total wave field in the effective piezoelectric medium The total coupling electro-elastic field in the effective piezoelectric medium are the superposition of the incident electro-elastic waves and the scattered coupling waves resulting from the typical coated nano-fiber, i.e., u * u *(in ) u *( sc ) ,
J * J *(in ) J *( sc ) .
- 10 -
(26)
3.3 Boundary conditions around the nano-fiber involving surface/interface effect The surface near the piezoelectric coating layer usually penetrates several atomic layers into the matrix and fibers. The surface/interface is usually modeled as a medium with material properties different from those of the matrix and fibers. As stated in our previous work [15,16], the coupling surface/interface model can be used to analyze the surface/interface effect qualitatively. The boundary conditions at the two interfaces are expressed as follows: a. At the inner interface, i.e., r a ,
JC
u C |r a u F |r a ,
S rzC
r a
S rzF
r a
tS QSz2 r tQ
JF
r a
DrC
,
r a
r a
,
DrF
(27)
r a
r a
tDQS 2 r tQ
.
(28)
r a
b. At the outer interface, i.e., r b ,
JM
u M |r b u C |r b ,
S rzM
r b
S rzC
r b
tS QSz1 r tQ
,
DrM
r b
r b
JC
DrC
r b
It should be noted that the surface stress tensor S QSz
r b
r b
,
tDQS 1 r tQ
(29)
.
(30)
r b
is charge dependent, and the
surface electric displacement DQS is deformation dependent, i.e.,
S QSz c44S EQSz e15S EQS ,
DQS DQ 0 e15S EQSz C11S EQS ,
(31)
S where c44 , C11S , and e15S are the elastic stiffness, dielectric constant and piezoelectric
constant of interface. Determination of them often requires extensive atomistic simulations. In the following formulation, the surface elastic stiffness, dielectric constant and piezoelectric constant are considered as known quantities. In this study, a coherent interface is considered. The interfacial strain and electric potential are equal to the associated tangential strain and electric potential in the
- 11 -
abutting bulk materials, respectively. They can be expressed as
EQSz EQMz EQFz ,
EQS EQM EQF .
(32)
4. Effective propagating wave number in the effective piezoelectric medium Substituting the wave field in the effective piezoelectric medium into the boundary conditions around the coating layer, the scattered electro-elastic field due to a single piezoelectric nano-fiber can be solved. Then, by using the scattered far-field method, the phase velocities and attenuations of the effective propagating wave number of electro-elastic waves in piezoelectric composites can be easily calculated by means of an iterative process based on the following equation
k
( j 1)
2
2
2 2 2 / k ( j ) §¨1 2P n0 = ( j ) (0 )/ k ( j ) ¶· §¨ 2P n0 = ( j ) (P ) / k ( j ) ¶· , © ¸ © ¸
(33)
where c
= ( j ) (Q ) £ ( i) n n 0
an( j ) inQ e . u0
(34)
The functions = ( j ) (Q ) (Q 0, P ) are the forward and backward scattering amplitudes, respectively. To start the iteration, the initial effective wave number k (0) with k (0) k M is introduced. By using the initial effective wave number, the forward and backward scattered amplitudes can be calculated. Next, the obtained effective wave numbers are used as the new effective wave numbers in the typical nano-fiber problem, and Eq. (33) are repeated until the convergence is obtained. During the iterative process, the convergence criterion | (k ( j 1) k ( j ) ) / k ( j ) |a 10 4 is applied. 5. Numerical examples and analyses Effective shear modulus plays a significant role in the determining the strength of piezoelectric nanocomposites under different loadings. Specially designed coating
- 12 -
layer and the surfaces/interfaces can improve the effective shear modulus of piezoelectric nanocomposites. In order to illustrate the surfaces/interfaces effect at the piezoelectric coating layer on the effective shear modulus, some numerical examples are given in the following. In the following numerical analyses, it is convenient to make the variables dimensionless. To this end, a characteristic length a , where a 1nm is the inner radius of the coated nano-fiber, is introduced. The following dimensionless variables and quantities have been chosen for computation: the incident wave number is k k M a 1.2 8.0 , the ratio of the outer radius to the inner radius is
L b / a 1.2 2.0 . The elastic stiffness ratio, dielectric constant ratio, and F M piezoelectric constant ratio of nano-fiber to matrix are c44F c44 / c44 0.1 25.0 ,
e15F e15F / e15M 0.1 20.0 , and C11F E11F / E11M 0.1 30.0 , respectively. Those ratios of C M coating-fiber to matrix are c44C c44 / c44 0.1 25.0 , e15C e15C / e15M 0.1 20.0 , and
C11C C11C / C11M 0.1 30.0 , respectively. Those of surface/interface to matrix are c44S c44S /(c44M a) 0.1 12.0 , e15S e15S /(e15M a) 0.1 10.0 , and C11S C11S /( C11M a) 0.1 10.0 ,
respectively. The density ratio is R R F / R M 1.0 3.0 . To validate this dynamic model, comparison with the existing numerical results is given in Figs. 2 and 3. Fig. 2 shows the comparison with the experimental results of the static effective shear modulus in Ref. [20]. In Ref. [20], the experimental data of shear modulus of a carbon/epoxy composite is presented. If the wave number is k l 0 , the dynamic effective shear modulus reduces to the static one. The dynamic
effective shear modulus as a function of the dimensionless wave number is shown in Fig. 3. When the value of c is 1.0, the coating layer will not exist. If the material properties are equal to zero, the nano-size effect cannot be considered. Then, the
- 13 -
coated piezoelectric nano-fibers are reduced to the piezoelectric fibers, which is studied in Refs. [16,19]. Excellent agreement with Refs. [16,19] can be seen in Fig. 3. With the increase of wave frequency, the dynamic effective shear modulus decreases, and tends to be stable. Fig. 4 illustrates the dynamic effective shear modulus under different elastic modulus of surfaces/interfaces. In this figure, the nano-fiber is stiffer than the matrix, and the coating layer is softer than the nano-fiber. It can be seen that if the dimensionless effective wave numbers are greater than 3.5, the effective shear modulus nearly shows no variation with surface/interface properties. By comparing with the results in Fig. 3, it can be found that the coating layer and surface/interface result in the increase of dynamic effective shear modulus. The elastic modulus of the outer interface shows greater effect on the dynamic effective shear modulus than that of the inner interface. If the outer interface is stiffer than the fiber, the inner interface effect is greater than that of softer interface. Please note that this conclusion is different from that in Ref. [16]. The reason is that the material properties of surface/interface in Ref. [16] are not properly selected, and there is an error in selecting the parameters. Fig. 5 illustrates the dynamic effective shear modulus with the elastic constants. In this figure, the nano-fiber and coating layer are both stiffer than the matrix, and the coating layer is softer than the nano-fiber. It can be seen that the dynamic effective shear modulus decreases if the coating layer is softer. By comparing with the results in Fig. 4, it is clear that the effect of the inner interface on the effective shear modulus increases if the coating layer is softer than the fiber and matrix. To analyze the effect of thickness of coating layer on the dynamic effective shear modulus, Fig. 6 and Fig. 7 are given. The coating layer in Fig.6 is stiff, and that in Fig.
- 14 -
7 is soft. It can be seen that the dynamic effective shear modulus increases with the increase of thickness of coating layer. In the region of very high frequency, the effect vanishes. The effect of thickness of coating layer is significant if the coating layer is soft. To find the effect of the piezoelectric properties of surfaces/interfaces on the dynamic shear modulus, Figs. 8 is given. In Fig. 8, the piezoelectric and dielectric constants of the coating layer are greater than the matrix. It can be clearly seen that the piezoelectric constants of interfaces have greatly influence on the effective shear modulus. The effective shear modulus increases due to the existence of piezoelectric constants. It is clear that the surfaces/interfaces effects increase with the increases of the piezoelectric constants of the coating layer. The effect of the outer surface/interface is greater than that of the inner surface/interface. By comparing with the results in Figs. 3-6, it can be observed that the effect of piezoelectric properties at the surfaces/interfaces is less than that of the elastic properties. 6. Conclusion In this work, the effective elastic modulus of piezoelectric nanocomposites with piezoelectric coated nano-fibers is characterized by combing Effective Field Method and electro-elastic surface/interface model. The effect of interfacial properties around the coating layers on the effective shear modulus under different material and geometrical parameters is analyzed in detail. Comparison with the previous results validates this nano-scale dynamic model. It has been found that the effect of surface/interface elasticity on the effective shear modulus is significantly related to the coating layer. The main findings of this work are as follows: a. The coating layer shows significant effect on the dynamic effective shear modulus. The surface/interface effect increases because of the existence of soft
- 15 -
coating layer. b. The dynamic effective shear modulus decreases with the increase of the thickness of coating layer. The influence becomes great in the case of soft coating layer. c. The surfaces/interfaces result in the increase of effective shear modulus. The effect of the outer interface is greater than that of the inner interface. The surface/interface effect on the effective properties increases if the coating layer is softer than the fiber and matrix. d. The effective shear modulus increases due to the existence of piezoelectric constants of surfaces/interfaces. The surfaces/interfaces effects increase with the increases of the piezoelectric constants of the coating layer. Acknowledgements The paper is supported by National Natural Science Foundation of China (Nos. 11172185; 11272222), National Key Basic Research Program of China (No. 2012CB723300) , and the Training Program for Leading Talent in University Innovative Research Team in Hebei Province (No.LJRC006). References [1] Zhang D, Tian P, Chen X, Lu J, Zhou Z, Fan X, Huang R. Fullerene C60-induced growth of hollow piezoelectric nanowire arrays of poly (vinylidene fluoride) at high pressure. Compos Sci Technol 2013;77(22): 29–36. [2] Qi Y, Jafferis NT, Lyons K, Lee CM, Ahmad H, McAlpine MC. Piezoelectric ribbons printed onto rubber for flexible energy conversion. Nano Lett 2010;10(2):524–8. [3] Feng X, Yang BD, Liu YM, Wang Y, Dagdeviren C, Liu ZJ, Carlson A, Li JY, Huang YG, Rogers JA. Stretchable ferroelectric nanoribbons with wavy
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Figures and figure captions:
Fig. 1. Schematic drawing of scale transition, the surface/interface model and the
coating layer from atomistic simulation, and the Effective Field Method employed to compute the dynamic effective properties
1.40
Present paper Experiment data
Shear modulus (GPa)
1.35 1.30 1.25 1.20 1.15 1.10 1.05 1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Volume fraction of fibers Fig. 2. Comparison with the experimental results of carbon/epoxy case in Ref. [20]
- 19 -
Obtained from this paper Obtained from Ref.[16] Obtained from Ref.[19]
1.30 1.25 1.20
c*44
1.15 1.10 1.05 1.00 0.95 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
@k F
Fig. 3. Comparison with the existing literatures (Refs. [16,19]) ( L 1.0 , n0 0.15 , c44 10.0 , C
S 1
S 2
e15F * 5.0 , C11F * 5.0 , c44 4.0 , e15C* 2.0 , C11C* 2.0 c44 0 , c44 0 , e15S 1* 0 , e15S 2* 0 ,
c*44
C11S 1 0 , C11S 2 0 , R * 1.0 )
1.50 cS1
=0.1,cS2
=0.1 1.45 44 44 S1
S2
c44 =0.1,c44 =9.0 1.40 1.35 cS1
=7.0,cS2
=0.1 44 44 S2
1.30 cS1
=7.0,c =9.0 44 44 1.25 1.20 1.15 1.10 1.05 1.00 0.95 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
@k Fig. 4. Dynamic effective shear modulus as a function of dimensionless wave numbers ( L 1.3 , F
C
n0 0.15 , c44 10.0 , e15F * 5.0 , C11F * 5.0 , c44 4.0 ,
e15C* 2.0 , C11C* 2.0 , e15S 1* 4.0 , e15S 2* 8.0 , C11S 1 13.0 , C11S 2 15.0 , R * 1.0 )
- 20 -
1.30 cS1
=0.1,cS2
=0.1 44 44
1.25
cS1
=0.1,cS2
=9.0 44 44 cS1
=7.0,cS2
=0.1 44 44
1.20
cS1
=7.0,cS2
=9.0 44 44
c*44
1.15 1.10 1.05 1.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
@k Fig. 5. Dynamic effective shear modulus as a function of dimensionless wave numbers F
C
( L 1.3 , n0 0.15 , c44 10.0 , e15F * 5.0 , C11F * 5.0 , c44 0.5 ,
e15C* 2.0 , C11C* 3.0 , e15S 1* 4.0 , e15S 2* 8.0 , C11S 1 13.0 , C11S 2 15.0 , R * 1.0 )
1.8
L L L
1.7 1.6
c*44
1.5 1.4 1.3 1.2 1.1 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
@k Fig. 6. Dynamic effective shear modulus as a function of dimensionless wave numbers F
F* F* C
C* C* ( n0 0.15 , c44 10.0 , e15 5.0 , C11 5.0 , c44 4.0 , e15 2.0 , C11 2.0 , S 1
S 2
c44 7.0 , c44 9.0 , e15S 1* 4.0 , e15S 2* 8.0 , C11S 1 13.0 , C11S 2 15.0 , R * 1.0 )
- 21 -
1.40 @c =1.15 @c =1.30 @c =1.50
1.35 1.30
c*44
1.25 1.20 1.15 1.10 1.05 1.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
@k Fig. 7. Dynamic effective shear modulus as a function of dimensionless wave numbers F
F* F* C
C* C* ( n0 0.15 , c44 10.0 , e15 5.0 , C11 5.0 , c44 0.5 , e15 2.0 , C11 5.0 ,
c*44
S 1
S 2
c44 7.0 , c44 9.0 , e15S 1* 4.0 , e15S 2* 8.0 , C11S 1 13.0 , C11S 2 15.0 , R * 1.0 )
1.50 eS1
=0.1,eS2
=0.1 1.45 15 15 S2
1.40 eS1
=0.1,e =8.0 15 15 S1
S2
1.35 e15 =4.0,e15 =0.1 1.30 eS1
=4.0,eS2
=8.0 15 15 1.25 1.20 1.15 1.10 1.05 1.00 0.95 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
@k Fig. 8. Dynamic effective shear modulus as a function of dimensionless wave numbers F
F* F* C
( c 1.3 , n0 0.15 , c44 10.0 , e15 5.0 , C11 5.0 , c44 4.0 , S 1
S 2
e15C* 2.0 , C11C* 2.0 , c44 7.0 , c44 9.0 , C11S 1 13.0 , C11S 2 15.0 , R * 1.0 )
- 22 -
S0266-3538(14)00134-1 http://dx.doi.org/10.1016/j.compscitech.2014.04.017 CSTE 5795
To appear in:
Composites Science and Technology
Received Date: Revised Date: Accepted Date:
29 August 2013 2 April 2014 20 April 2014
Please cite this article as: Fang, X-Q., Huang, M-J., Liu, J-X., Feng, W-J., Dynamic effective property of piezoelectric composites with coated piezoelectric nano-fibers, Composites Science and Technology (2014), doi: http://dx.doi.org/10.1016/j.compscitech.2014.04.017
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Dynamic effective property of piezoelectric composites with coated piezoelectric nano-fibers Xue-Qian Fang* ,
Ming-Juan Huang, Jin-Xi Liu, Wen-Jie Feng
Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China Abstract: This paper addresses the size-dependent dynamic effective electro-elastic properties of piezoelectric composites embedded with coated piezoelectric nano-fibers. By using the Effective Field Method of electric-elastic coupling wave, the randomly distributed coated nano-fibers in the piezoelectric media are reduced into a typical piezoelectric coated nano-fiber. To analyze the size-dependent effect, the coupling surface/interface theory is introduced to analyze the interface effect around the coating layers. Wave function expansion method is used to express the coupling wave field in the effective coupling field. The piezoelectrically stiffened elastic modulus and effective coupling wave number are obtained. Through analysis, it is found that the coupling surface/interface effect on the piezoelectrically stiffened elastic modulus show significant variation with the material properties of coating layer and the incident wave frequency. The effect of piezoelectric property of surfaces/interfaces on the effective shear modulus is also examined. Comparison with the existing results is given to validate this dynamic electro-elastic model. Keywords: A. Nano composites; A. Coating; B. Interface; C. Elastic properties; D. Ultrasonics 1. Introduction The nanosize characteristic, the excellent piezoelectricity performances, and the unique coupling between piezoelectric and semiconducting properties of piezoelectric *Corresponding author. Tel.: +86 311 87936542 E-mail address: [email protected] (X.Q. Fang)
nanocomposites make them attractive for applications such as transducers, sensors, actuators and energy harvesters in the nano-electro-mechanical systems. Recently, the theory and experimental investigations on piezoelectric nano-structures have received considerable attention because of their excellent piezoelectric and micro properties over conventional composites [1-3]. Compared
with
the
macroscopic
piezoelectric
composites,
piezoelectric
nanocomposites exhibit size-dependent mechanical and physical properties due to their large ratio of surface area to volume. In recent years, many experimental investigations and atomistic simulations have shown that the elastic constants or the piezoelectric coefficients of some piezoelectric nanocomposites increase significantly with the decrease of the material size to the nanoscale [4,5]. Determining the effective behavior of nanocomposites embedded with inclusions is of considerable importance in a wide variety of physical problems. In traditional continuum mechanics, surface free energy around the inclusions is typically neglected because it is associated with only a few layers of atoms near the surface, and the ratio of the volume occupied by the surface atoms and the total volume of material of interest are extremely small. However, due to a large ratio of the surface/interface region to the bulk, the effective properties of nano-composites are inevitably be affected by surfaces/interfaces. In the past years, the investigation of surface/interface effect on the static [6-7] and dynamic [8-10] effective properties of nanocomposites is attracting more and more interests. With the advent of piezoelectric nanocomposites in recent years, some attempts have been made to reveal the size-dependent behavior of piezoelectric nano-structures. Within Nb2O5 nanotubes forming a unique class of heterostructure nanotubes combining ferroelectric and semiconducting nanomaterials, multiple NaNbO3 nanoplates were created by Yan et al., and the local piezoelectric
-2-
properties were studied [11]. Both finite element method (FEM) simulations and experiments were carried out to investigate the stress distribution and the interfacial effects between the matrix and piezoelectric NWs by altering the slip conditions at the interface and changing the mechanical properties of the matrix [12]. Fakhzan and Muthalif presented an analytical estimation of voltage production of piezoelectric cantilever beam when subjected to base excitation, with and without attached proof masses [13]. In the above mentioned papers, it can be concluded that the electro-elastic properties of the interface-region become very important, and should be given due consideration while formulating their overall properties. In recent years, the electro-elastic coupling surface/interface model has been proposed to characterize the behavior of piezoelectric nanostructures. In this model, the interfacial properties are defined by introducing the concept of a dividing surface. The surface/interface between the piezoelectric inclusion and the matrix is assumed to be coherent, and no atomic bonds are broken along the surface/interface [14,15]. In the work of Fang et al., the coupling surface/interface theory was used to predict the dynamic electro-elastic properties of piezoelectric nanocomposites with nano-fibers [16]. To obtain a better performance of piezoelectric nano-structures, coating layers around the reinforced inclusions are often introduced to reduce the internal stresses and enhance the piezoelectric behavior. In these nano-structures, a larger ratio of surface/interface area to volume exists, and expresses greater effect on the performance of piezoelectric nano-structures. For conventional nanocomposites, Ozmusul and Picu [17] studied the elastic moduli of composites containing an anisotropic coating layer and nano-particles that are either of finite stiffness or rigid. In addition, Boutaleb et al. [18] proposed a micromechanical analytical model to
-3-
predict the stiffness and yield strength of silica/polymer nano-composites, and it is shown that the interphase is a dominant parameter for controlling the overall nano-composite behavior. However, up to now, no investigations on the prediction of piezoelectric composites with coating nano-fibers are available. To optimally design the piezoelectric nanostructures, this paper addresses the dynamic effective electro-elastic properties of piezoelectric composites embedded with coated piezoelectric nano-fibers. The multiple scattering of electro-elastic waves among the randomly distributed piezoelectric nano-fibers is transformed to the typical piezoelectric nano-fiber problem by using Effective Field Method. The fundamental framework of coupling surface/interface model is generalized to take into account the surface/interface effect around the typical piezoelectric nano-fiber on the effective elastic modulus. The scattered far-field method and the iterative process are employed to obtain the effective propagating wave number and the effective elastic modulus. Through the numerical examples, the effects of surface/interface around the coating layer on the dynamic shear modulus are analyzed in detail. 2. Problem formulation and dynamic effective field method An unbounded piezoelectric matrix containing randomly distributed coated nano-fibers is considered. A microscopic Representative Volume Element (RVE) at the microscopic scale is proposed. The piezoelectric composite is considered to be homogeneous and isotropic at the macro-continuum level; however it consists of a spatially uniform distribution of embedded coated nano-fibers, as shown in the schematic of Fig. 1. The volume fraction of the coated nano-fibers is n0 .The coating layers correspond to the volume defined by the narrow region sandwiched between the nano-fiber and matrix with different properties, while the interface refers to the surface areas between the coating layers and the fibers, and those between the fibers
-4-
and the matrix. This transition of the properties from surface/interface to the bulk value may take place over a few layers of atoms [16]. In the following, the matrix occupies a region is denoted by ‘M’, the coating layers are denoted by ‘C’, and the nano-fibers occupying a region are ‘F’. ‘S’ represents the matrix-coating and coating-fiber interface. The outer surface/interface is denoted as ‘S1’, and the inner one is ‘S2’. The matrix, coating layer, and fiber are both assumed to be transversely isotropic, and are polarized in the z-direction. It is assumed that an electro-elastic wave resulting from the randomly distributed coated nano-fibers propagates in the matrix material. The interaction between the coated nano-fibers gives rise to the dispersion relations for electro-elastic waves, and the propagating coupling wave number denoted as effective wave number will change. The distribution of the piezoelectric nano-fiber in the transverse piezoelectric matrix is assumed to be homogeneous and isotropic. If the incident field is a plane electro-elastic coupling wave, the mean fields are also plane electro-elastic waves in many important cases. To construct these electro-elastic coupling fields, it is very difficult to appropriately descript the interaction between the random coated nano-fibers in the piezoelectric composite media. Strictly speaking, in order to construct these fields, we have to find the detailed wave fields for every realization of the random set of coated nano-fibers and then to average the results over an ensemble of realizations of this set. These principal difficulties of this complex phenomenon enable us only to find the approximate solution. Self-consistent scheme named as Effective Field Method, which is powerful tool, is used to obtain such solutions in this paper, and mainly based on the following two hypotheses. The first hypothesis is expressed as: Every coated nano-fiber in the composites behaves as an isolated one embedded in the original matrix by the action of local
-5-
effective coupling fields (displacement u * (r ) and electric potential J * (r ) ). The local effective fields denote the sum of the incident fields applied to the medium and the scattered fields on all the surrounding coated nano-fibers. Therefore, the local effective fields can be written in the forms N
u * (r ) u in (r ) £ unsc (r ) ,
(1)
n 1
N
J * (r ) J in (r ) £ Jnsc (r ) .
(2)
n 1
This Hypothesis reduces the problem of interaction between the randomly distributed piezoelectric nano-fibers to the typical piezoelectric nano-fiber problem in the effective electro-elastic medium, as depicted in Fig. 1. The second hypothesis is: Random local effective fields u * (r ) and electric potential J * (r ) acting on an arbitrary fiber are statistically independent on the properties and the geometrical characteristics (containing the presences and locations) of this fiber. According to this hypothesis, one can obtain that
u * (r ) u (r ) | r , J * (r ) J (r ) | r , where symbol
(3)
u | r denotes the averaging under the condition that the point r
belongs to the region occupied by the cross-sections of coated nano-fibers. Based on the above hypotheses and following the work of Levin et al. [19], the governing equations of average electro-elastic medium can be expressed as * c44 (k * )$ u (r ) e15* (k * )$ J (r ) R * (k * )W 2 u (r ) 0 ,
e15* (k * )$ u (r ) C11* (k * )$ J (r ) 0 ,
(4) (5)
* * * where c44 , e15 and C11 are, respectively, the elastic stiffness, piezoelectric constant
and dielectric constant of effective coupling fields, they are all related with the
-6-
effective coupling wave number k * , M F M (1 n0 ) ( C11F C11M )c44 ) C11M (e11F e11M )e11M ¶ (e11F e11M )e11M ¶ § (1 n0 ) (c44
c44 1 · ¨ · M M M M C11 (e11M ) 2 C11 (e11M ) 2 2 c44 2 c44 © ¸© ¸ (1 n0 ) 2 F M M
[(e F e M ) C M ( C11F C11M )e15M ][(c44 )e15M (e15F e15M )c44 ] , (6)
c44 M M C11 (e11M ) 2 ]2 15 15 11 4[c44
§
$ ¨1
k * is the effective wave number, and *
u (r ) U *ei( k r Wt ) ,
J (r )
e15* * 11
C
*
U *ei( k r Wt ) .
(7)
The dispersion relation for effective coupling wave number can be written as ª * § (e15* ) 2 ¶ ¹ * 2 * 2 «c44 ¨ * · º (k ) R W 0 . ¬ © C11 ¸ »
(8)
The relation between the piezoelectrically stiffened elastic modulus and the effective wave number is expressed as 2
* M c44 c44 ( R * / R M ) §© Re(k M / k * ) ¶¸ ,
M in which k M W M M / R M with M M c44
(e15M ) 2
C11M
(9)
. For a small density contrast, the
simple rule of mixture is used in the numerical examples, i.e.,
R * n0 R f (1 n0 ) R M ,
(10)
where R M LR M (1 L ) R C with L b / a . When the multiple scattering among the piezoelectric nano-fibers is reduced to the typical piezoelectric one-fiber problem in the effective piezoelectric medium (see Fig. 1), the scattering of effective electro-elastic waves resulting from the typical nano-fiber comes into being. The coupling wave fields around the typical nano-fiber can be described in Section 4. 3. Solving the typical coated nano-fiber in effective piezoelectric media 3.1 Governing equations in the piezoelectric media -7-
In the electro-elastic medium, an anti-plane electro-elastic wave with frequency W propagates in the positive x-direction. The mechanically and electrically coupled constitutive equations can be written as
S xz c44
tu tJ , e15 tx tx
S yz c44
tu tJ , e15 ty ty
(11)
Dx e15
tu tJ ,
C11 tx tx
Dy e15
tu tJ ,
C11 ty ty
(12)
where S jz , u , J , and D j ( j x, y ) are the shear stress, anti-plane displacement, electric potential, and in-plane electric displacement, respectively. The following governing equations can be expressed as c44 2u e15 2J R
t 2u , tt 2
e15 2u C11 2J 0 .
(13) (14)
Here 2 t 2 / tx 2 t 2 / ty 2 is the two-dimensional Laplace operator in the variables x and y. To decouple Eqs. (13) and (14), another electro-elastic field Y is introduced,
Y J Hu ,
(15)
where H e15 / C11 . Substituting Eq. (15) into Eqs. (13)-(14), one can obtain the following equations 2u
1 t 2u , 2 tt 2 cSH
2Y 0 , where cSH M e / R with M e c44
e152
C11
(16)
(17)
being the wave speed of electro-elastic
waves in the piezoelectric media. 3.2. Wave fields around the typical coated nano-fiber -8-
a. Incident electro-elastic coupling wave field In the local coordinate system (r * , Q * ) of the real cavity, the incident waves can be expanded as c
*
u *(in ) u0 eik x u0 £ E ni n J n (k *r ) cos(nQ ) ,
(18)
n 0
where
ª1, n 0 , ¬2, n 1, 2,3,!
En «
is the amplitude of the incident waves,
u0
k * W / M * / R * is the wave number of the propagating waves, and J n () is the nth
Bessel function of the first kind. Similarly, the incident field J (i ) is expressed as
J *(in ) J0 e
i k * x Wt
c
J0 £ E ni n J n k *r cos(nQ ) ,
(19)
n 0
where J0
e15
C11
u0 .
b. Scattered wave field Using wave function expansion method, the general solution of scattered field resulting from the coated nano-fiber can be expressed, as c
u *( sc ) u0 £ an H n (k *r ) cos(nQ ) ,
(20)
n 0
J *( sc )
u0 e15
C11
c
£a H n
n 0
c
n * cos(nQ ) , n ( k r ) cos( nQ ) £ bn r
(21)
n 0
where an and bn are the mode coefficients of the electro-elastic waves, and can be determined by satisfying the boundary conditions at the surface/interface. H n < is the nth Hankel function of the first kind, and denotes the outgoing wave. c. Wave field in the piezoelectric coating layer of the typical nano-fiber The wave field in the piezoelectric coating layer may be described by the sum of -9-
the two components (outgoing and ingoing waves), and are expressed in the following form c §c ¶ u C u0 ¨ £ cn H n (k C r ) cos(nQ ) £ d nYn (k C r ) cos(nQ ) · , n 0 © n 0 ¸
JC
e15C
C11C
c
c
n 0
n 0
u0 £ §©cn H n k C r d nYn k C r ¶¸ cos(nQ ) £ §©e1n r n e2 n r n ¶¸ cos(nQ ) ,
(22)
(23)
where k C W / M C / R C , Yn < are the nth Hankel functions of the second kind, and denote the ingoing waves. cn , d n , e1n and e2n are the mode coefficients in the coating layer. d. Refracted wave field The refracted waves confined inside the piezoelectric nano-fiber are standing electro-elastic waves, and represented by c
u F u0 £ f n J n k F r cos(nQ ) ,
(24)
n 0
JF
e15F
C11F
c
c
n 0
n 0
u0 £ f n J n k F r cos(nQ ) £ g n r n cos(nQ ) ,
(25)
where k F W / M F / R F is the wave number inside the coated nano-fiber with
M F c44F
(e15F ) 2
C11F
, f n and g n are the mode coefficients of the refracted waves, and
the cylindrical Bessel functions of the first kind are used to obtain the standing waves. d. Total wave field in the effective piezoelectric medium The total coupling electro-elastic field in the effective piezoelectric medium are the superposition of the incident electro-elastic waves and the scattered coupling waves resulting from the typical coated nano-fiber, i.e., u * u *(in ) u *( sc ) ,
J * J *(in ) J *( sc ) .
- 10 -
(26)
3.3 Boundary conditions around the nano-fiber involving surface/interface effect The surface near the piezoelectric coating layer usually penetrates several atomic layers into the matrix and fibers. The surface/interface is usually modeled as a medium with material properties different from those of the matrix and fibers. As stated in our previous work [15,16], the coupling surface/interface model can be used to analyze the surface/interface effect qualitatively. The boundary conditions at the two interfaces are expressed as follows: a. At the inner interface, i.e., r a ,
JC
u C |r a u F |r a ,
S rzC
r a
S rzF
r a
tS QSz2 r tQ
JF
r a
DrC
,
r a
r a
,
DrF
(27)
r a
r a
tDQS 2 r tQ
.
(28)
r a
b. At the outer interface, i.e., r b ,
JM
u M |r b u C |r b ,
S rzM
r b
S rzC
r b
tS QSz1 r tQ
,
DrM
r b
r b
JC
DrC
r b
It should be noted that the surface stress tensor S QSz
r b
r b
,
tDQS 1 r tQ
(29)
.
(30)
r b
is charge dependent, and the
surface electric displacement DQS is deformation dependent, i.e.,
S QSz c44S EQSz e15S EQS ,
DQS DQ 0 e15S EQSz C11S EQS ,
(31)
S where c44 , C11S , and e15S are the elastic stiffness, dielectric constant and piezoelectric
constant of interface. Determination of them often requires extensive atomistic simulations. In the following formulation, the surface elastic stiffness, dielectric constant and piezoelectric constant are considered as known quantities. In this study, a coherent interface is considered. The interfacial strain and electric potential are equal to the associated tangential strain and electric potential in the
- 11 -
abutting bulk materials, respectively. They can be expressed as
EQSz EQMz EQFz ,
EQS EQM EQF .
(32)
4. Effective propagating wave number in the effective piezoelectric medium Substituting the wave field in the effective piezoelectric medium into the boundary conditions around the coating layer, the scattered electro-elastic field due to a single piezoelectric nano-fiber can be solved. Then, by using the scattered far-field method, the phase velocities and attenuations of the effective propagating wave number of electro-elastic waves in piezoelectric composites can be easily calculated by means of an iterative process based on the following equation
k
( j 1)
2
2
2 2 2 / k ( j ) §¨1 2P n0 = ( j ) (0 )/ k ( j ) ¶· §¨ 2P n0 = ( j ) (P ) / k ( j ) ¶· , © ¸ © ¸
(33)
where c
= ( j ) (Q ) £ ( i) n n 0
an( j ) inQ e . u0
(34)
The functions = ( j ) (Q ) (Q 0, P ) are the forward and backward scattering amplitudes, respectively. To start the iteration, the initial effective wave number k (0) with k (0) k M is introduced. By using the initial effective wave number, the forward and backward scattered amplitudes can be calculated. Next, the obtained effective wave numbers are used as the new effective wave numbers in the typical nano-fiber problem, and Eq. (33) are repeated until the convergence is obtained. During the iterative process, the convergence criterion | (k ( j 1) k ( j ) ) / k ( j ) |a 10 4 is applied. 5. Numerical examples and analyses Effective shear modulus plays a significant role in the determining the strength of piezoelectric nanocomposites under different loadings. Specially designed coating
- 12 -
layer and the surfaces/interfaces can improve the effective shear modulus of piezoelectric nanocomposites. In order to illustrate the surfaces/interfaces effect at the piezoelectric coating layer on the effective shear modulus, some numerical examples are given in the following. In the following numerical analyses, it is convenient to make the variables dimensionless. To this end, a characteristic length a , where a 1nm is the inner radius of the coated nano-fiber, is introduced. The following dimensionless variables and quantities have been chosen for computation: the incident wave number is k k M a 1.2 8.0 , the ratio of the outer radius to the inner radius is
L b / a 1.2 2.0 . The elastic stiffness ratio, dielectric constant ratio, and F M piezoelectric constant ratio of nano-fiber to matrix are c44F c44 / c44 0.1 25.0 ,
e15F e15F / e15M 0.1 20.0 , and C11F E11F / E11M 0.1 30.0 , respectively. Those ratios of C M coating-fiber to matrix are c44C c44 / c44 0.1 25.0 , e15C e15C / e15M 0.1 20.0 , and
C11C C11C / C11M 0.1 30.0 , respectively. Those of surface/interface to matrix are c44S c44S /(c44M a) 0.1 12.0 , e15S e15S /(e15M a) 0.1 10.0 , and C11S C11S /( C11M a) 0.1 10.0 ,
respectively. The density ratio is R R F / R M 1.0 3.0 . To validate this dynamic model, comparison with the existing numerical results is given in Figs. 2 and 3. Fig. 2 shows the comparison with the experimental results of the static effective shear modulus in Ref. [20]. In Ref. [20], the experimental data of shear modulus of a carbon/epoxy composite is presented. If the wave number is k l 0 , the dynamic effective shear modulus reduces to the static one. The dynamic
effective shear modulus as a function of the dimensionless wave number is shown in Fig. 3. When the value of c is 1.0, the coating layer will not exist. If the material properties are equal to zero, the nano-size effect cannot be considered. Then, the
- 13 -
coated piezoelectric nano-fibers are reduced to the piezoelectric fibers, which is studied in Refs. [16,19]. Excellent agreement with Refs. [16,19] can be seen in Fig. 3. With the increase of wave frequency, the dynamic effective shear modulus decreases, and tends to be stable. Fig. 4 illustrates the dynamic effective shear modulus under different elastic modulus of surfaces/interfaces. In this figure, the nano-fiber is stiffer than the matrix, and the coating layer is softer than the nano-fiber. It can be seen that if the dimensionless effective wave numbers are greater than 3.5, the effective shear modulus nearly shows no variation with surface/interface properties. By comparing with the results in Fig. 3, it can be found that the coating layer and surface/interface result in the increase of dynamic effective shear modulus. The elastic modulus of the outer interface shows greater effect on the dynamic effective shear modulus than that of the inner interface. If the outer interface is stiffer than the fiber, the inner interface effect is greater than that of softer interface. Please note that this conclusion is different from that in Ref. [16]. The reason is that the material properties of surface/interface in Ref. [16] are not properly selected, and there is an error in selecting the parameters. Fig. 5 illustrates the dynamic effective shear modulus with the elastic constants. In this figure, the nano-fiber and coating layer are both stiffer than the matrix, and the coating layer is softer than the nano-fiber. It can be seen that the dynamic effective shear modulus decreases if the coating layer is softer. By comparing with the results in Fig. 4, it is clear that the effect of the inner interface on the effective shear modulus increases if the coating layer is softer than the fiber and matrix. To analyze the effect of thickness of coating layer on the dynamic effective shear modulus, Fig. 6 and Fig. 7 are given. The coating layer in Fig.6 is stiff, and that in Fig.
- 14 -
7 is soft. It can be seen that the dynamic effective shear modulus increases with the increase of thickness of coating layer. In the region of very high frequency, the effect vanishes. The effect of thickness of coating layer is significant if the coating layer is soft. To find the effect of the piezoelectric properties of surfaces/interfaces on the dynamic shear modulus, Figs. 8 is given. In Fig. 8, the piezoelectric and dielectric constants of the coating layer are greater than the matrix. It can be clearly seen that the piezoelectric constants of interfaces have greatly influence on the effective shear modulus. The effective shear modulus increases due to the existence of piezoelectric constants. It is clear that the surfaces/interfaces effects increase with the increases of the piezoelectric constants of the coating layer. The effect of the outer surface/interface is greater than that of the inner surface/interface. By comparing with the results in Figs. 3-6, it can be observed that the effect of piezoelectric properties at the surfaces/interfaces is less than that of the elastic properties. 6. Conclusion In this work, the effective elastic modulus of piezoelectric nanocomposites with piezoelectric coated nano-fibers is characterized by combing Effective Field Method and electro-elastic surface/interface model. The effect of interfacial properties around the coating layers on the effective shear modulus under different material and geometrical parameters is analyzed in detail. Comparison with the previous results validates this nano-scale dynamic model. It has been found that the effect of surface/interface elasticity on the effective shear modulus is significantly related to the coating layer. The main findings of this work are as follows: a. The coating layer shows significant effect on the dynamic effective shear modulus. The surface/interface effect increases because of the existence of soft
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coating layer. b. The dynamic effective shear modulus decreases with the increase of the thickness of coating layer. The influence becomes great in the case of soft coating layer. c. The surfaces/interfaces result in the increase of effective shear modulus. The effect of the outer interface is greater than that of the inner interface. The surface/interface effect on the effective properties increases if the coating layer is softer than the fiber and matrix. d. The effective shear modulus increases due to the existence of piezoelectric constants of surfaces/interfaces. The surfaces/interfaces effects increase with the increases of the piezoelectric constants of the coating layer. Acknowledgements The paper is supported by National Natural Science Foundation of China (Nos. 11172185; 11272222), National Key Basic Research Program of China (No. 2012CB723300) , and the Training Program for Leading Talent in University Innovative Research Team in Hebei Province (No.LJRC006). References [1] Zhang D, Tian P, Chen X, Lu J, Zhou Z, Fan X, Huang R. Fullerene C60-induced growth of hollow piezoelectric nanowire arrays of poly (vinylidene fluoride) at high pressure. Compos Sci Technol 2013;77(22): 29–36. [2] Qi Y, Jafferis NT, Lyons K, Lee CM, Ahmad H, McAlpine MC. Piezoelectric ribbons printed onto rubber for flexible energy conversion. Nano Lett 2010;10(2):524–8. [3] Feng X, Yang BD, Liu YM, Wang Y, Dagdeviren C, Liu ZJ, Carlson A, Li JY, Huang YG, Rogers JA. Stretchable ferroelectric nanoribbons with wavy
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configurations on elastomeric substrates. ACS Nano 2011;5(4):3326–32. [4] Stan G, Ciobanu CV, Parthangal PM, Cook RF. Diameter-dependent radial and tangential elastic moduli of ZnO nanowires. Nano Lett 2007;7(12):3691–7. [5] Fang F, Yang W, Zhang MZ, Wang Z. Mechanical response of barium-titanate/polymer 0–3 ferroelectric nano-composite film under uniaxial tension. Compos Sci Technol 2009;69(5): 602–5. [6] Duan H, Wang J, Huang ZP, Karihaloo BL. Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. J Mech Phys Solids 2005;53(7):1574 –96. [7] Paliwal B, Cherkaoui M. Estimation of anisotropic elastic properties of nanocomposites using atomistic-continuum interphase model. Int J Solids Struct 2012;49(18):2424–38. [8] Hasheminejad SM, Avazmohammadi R. Size-dependent effective dynamic properties of unidirectional nanocomposites with interface energy effects. Compos Sci Technol 2009; 69: 2538–46. [9] Qiang FW, Wei PJ, Li L. The effective propagation constants of SH wave in composites reinforced by dispersive parallel nanofibers. Sci China Phys Mech 2012; 55: 1172–7. [10] Qiang FW, Wei PJ, Liu XQ. Propagation of elastic wave in nanoporous material with distributed cylindrical nanoholes. Sci China Phys Mech 2013;56:1542–50. [11] Yan C, Nikolova L, Dadvand A, Harnagea C, Sarkissian A, Perepichka DF, Xue D, Rosei F. Multiple NaNbO3/Nb2O5 heterostructure nanotubes: a new class of ferroelectric/ semiconductor nanomaterials. Adv Mater 2010;22(15):1741–5. [12] Kim K, Sirbuly DJ. Enhanced output of nanostructured piezoelectric arrays via controlled
matrix/transducer
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2012;101(21):213114. [13] Fakhzan MN, Muthalif AGA. Harvesting vibration energy using piezoelectric material: Modeling, simulation and experimental verifications. Mechatronics 2013;23(1):61–6. [14] Pan XH, Yu SW, Feng XQ. A continuum theory of surface piezoelectricity for nanodielectrics. Sci China Phys Mech 2011;54(4):564–73. [15] Fang XQ, Yang Q, Liu JX, Feng WJ. Surface/interface effect around a piezoelectric nano-particle in a polymer matrix under compressional waves. Appl Phys Lett 2012; 100(15):151602. [16] Fang XQ, Liu JX, Huang MJ. Effect of interface energy on effective dynamic properties of piezoelectric medium with randomly distributed piezoelectric nano-fibers. J Appl Phys 2012;112(9):094311. [17] Ozmusul MS, Picu RC. Elastic moduli of particulate composites with graded filler-matrix interfaces. Polym Compos 2002;23(1):110–9. [18] Boutaleb S, Zairi F, Mesbah A, Nait-Abdelaziz M, Gloaguen JM, Boukharouba T. Micromechanics-based modelling of stiffness and yield stress for silica/polymer nanocomposites. Int J Solids Struct 2009;46(7-8):1716–26. [19] Levin VM, Michelitsch TM, Gao HJ. Propagation of electroacoustic waves in the transversely isotropic piezoelectric medium reinforced by randomly distributed cylindrical inhomogeneities. Int J Solids Struct 2002;39:5013–51. [20] Shari HZ, Chou TW.Transverse elastic moduli of unidirectional fiber composites with fiber/matrix interfacial debonding. Compos Sci Technol 1995;53:383–91.
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Figures and figure captions:
Fig. 1. Schematic drawing of scale transition, the surface/interface model and the
coating layer from atomistic simulation, and the Effective Field Method employed to compute the dynamic effective properties
1.40
Present paper Experiment data
Shear modulus (GPa)
1.35 1.30 1.25 1.20 1.15 1.10 1.05 1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Volume fraction of fibers Fig. 2. Comparison with the experimental results of carbon/epoxy case in Ref. [20]
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Obtained from this paper Obtained from Ref.[16] Obtained from Ref.[19]
1.30 1.25 1.20
c*44
1.15 1.10 1.05 1.00 0.95 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
@k F
Fig. 3. Comparison with the existing literatures (Refs. [16,19]) ( L 1.0 , n0 0.15 , c44 10.0 , C
S 1
S 2
e15F * 5.0 , C11F * 5.0 , c44 4.0 , e15C* 2.0 , C11C* 2.0 c44 0 , c44 0 , e15S 1* 0 , e15S 2* 0 ,
c*44
C11S 1 0 , C11S 2 0 , R * 1.0 )
1.50 cS1
=0.1,cS2
=0.1 1.45 44 44 S1
S2
c44 =0.1,c44 =9.0 1.40 1.35 cS1
=7.0,cS2
=0.1 44 44 S2
1.30 cS1
=7.0,c =9.0 44 44 1.25 1.20 1.15 1.10 1.05 1.00 0.95 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
@k Fig. 4. Dynamic effective shear modulus as a function of dimensionless wave numbers ( L 1.3 , F
C
n0 0.15 , c44 10.0 , e15F * 5.0 , C11F * 5.0 , c44 4.0 ,
e15C* 2.0 , C11C* 2.0 , e15S 1* 4.0 , e15S 2* 8.0 , C11S 1 13.0 , C11S 2 15.0 , R * 1.0 )
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1.30 cS1
=0.1,cS2
=0.1 44 44
1.25
cS1
=0.1,cS2
=9.0 44 44 cS1
=7.0,cS2
=0.1 44 44
1.20
cS1
=7.0,cS2
=9.0 44 44
c*44
1.15 1.10 1.05 1.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
@k Fig. 5. Dynamic effective shear modulus as a function of dimensionless wave numbers F
C
( L 1.3 , n0 0.15 , c44 10.0 , e15F * 5.0 , C11F * 5.0 , c44 0.5 ,
e15C* 2.0 , C11C* 3.0 , e15S 1* 4.0 , e15S 2* 8.0 , C11S 1 13.0 , C11S 2 15.0 , R * 1.0 )
1.8
L L L
1.7 1.6
c*44
1.5 1.4 1.3 1.2 1.1 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
@k Fig. 6. Dynamic effective shear modulus as a function of dimensionless wave numbers F
F* F* C
C* C* ( n0 0.15 , c44 10.0 , e15 5.0 , C11 5.0 , c44 4.0 , e15 2.0 , C11 2.0 , S 1
S 2
c44 7.0 , c44 9.0 , e15S 1* 4.0 , e15S 2* 8.0 , C11S 1 13.0 , C11S 2 15.0 , R * 1.0 )
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1.40 @c =1.15 @c =1.30 @c =1.50
1.35 1.30
c*44
1.25 1.20 1.15 1.10 1.05 1.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
@k Fig. 7. Dynamic effective shear modulus as a function of dimensionless wave numbers F
F* F* C
C* C* ( n0 0.15 , c44 10.0 , e15 5.0 , C11 5.0 , c44 0.5 , e15 2.0 , C11 5.0 ,
c*44
S 1
S 2
c44 7.0 , c44 9.0 , e15S 1* 4.0 , e15S 2* 8.0 , C11S 1 13.0 , C11S 2 15.0 , R * 1.0 )
1.50 eS1
=0.1,eS2
=0.1 1.45 15 15 S2
1.40 eS1
=0.1,e =8.0 15 15 S1
S2
1.35 e15 =4.0,e15 =0.1 1.30 eS1
=4.0,eS2
=8.0 15 15 1.25 1.20 1.15 1.10 1.05 1.00 0.95 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
@k Fig. 8. Dynamic effective shear modulus as a function of dimensionless wave numbers F
F* F* C
( c 1.3 , n0 0.15 , c44 10.0 , e15 5.0 , C11 5.0 , c44 4.0 , S 1
S 2
e15C* 2.0 , C11C* 2.0 , c44 7.0 , c44 9.0 , C11S 1 13.0 , C11S 2 15.0 , R * 1.0 )
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