Electro-mechanical load transfer from a fiber in a 1–3 piezocomposite with an imperfect interface

Composites: Part B 39 (2008) 1114–1124

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Electro-mechanical load transfer from a fiber in a 1–3 piezocomposite with an imperfect interface Y. Sapsathiarn a, T. Senjuntichai a,*, R.K.N.D. Rajapakse b a b

Department of Civil Engineering, Chulalongkorn University, Bangkok 10330, Thailand Institute for Computing, Information and Cognitive Systems, The University of British Columbia, Vancouver, B.C., Canada V6T 1Z4

a r t i c l e

i n f o

Article history: Received 19 March 2008 Accepted 15 April 2008 Available online 17 June 2008 Keywords: A. Smart materials B. Stress transfer Fiber/matrix bond C. Analytical modeling

a b s t r a c t This paper considers the electro-mechanical interaction between a fiber and a matrix material in a 1–3 piezocomposite due to an axial load and electric charge applied to the fiber. The fiber–matrix interface is assumed to be mechanically imperfect and is represented by a spring-factor model. The interface is either electrically open- or short-circuited. The analytical general solutions corresponding to an infinite piezoelectric fiber with a vertical body force and an electric body charge are derived by using Fourier integral transforms. These solutions together with the analytical general solutions for a transversely isotropic elastic medium are used to formulate the fiber–matrix interaction problem. Selected numerical results for the fiber axial force and vertical electric field, and interfacial stresses are presented for representative 1–3 piezocomposites. The influence of the interface stiffness on the electro-mechanical load diffusion is also examined. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The development of smart structures and composite materials with sensing and actuation capabilities have received increasing attention over the past two decades. Incorporation of piezoelectric or shape memory alloy fibers/rods in polymer matrix materials leads to composite materials with sensing and actuation properties and such composites are commonly known as smart composites. Many recent studies have examined the fabrication of smart composites and their properties (e.g., [1,2]). This paper is concerned with a class of smart composites commonly known as piezocomposites that is produced by embedding piezoelectric ceramic fibers/rods in polymer matrix materials. Piezocomposites were originally developed for underwater hydrophone applications in the low-frequency range, but have also been extended to other applications such as ultrasonic transducers for acoustic imaging, sensors and actuators for biomedical engineering and aerospace engineering. Piezocomposites are classified according to their connectivity [3]. Type 1–3 piezocomposites contain piezoelectric rods embedded in a polymer matrix and aligned through the thickness of the composite (Fig. 1). Performance and reliability of piezocomposites are governed by the interaction between the piezoelectric phase and surrounding matrix material, which is controlled by the material proper* Corresponding author. Tel.: +66 2218 6460; fax: +66 2251 7304. E-mail address: [email protected] (T. Senjuntichai). 1359-8368/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2008.04.001

ties of the two phases, interfacial properties and volume fraction of the piezoelectric phase. The study of the mechanics of fiber–matrix interaction in traditional composite materials has a rich history. Muki and Sternberg [4] examined the threedimensional interaction of a single infinitely long cylindrical fiber in an isotropic matrix by studying the diffusion of an axial load from the fiber into the matrix. They gave an exact analytical solution based on the theory of elasticity for a circular fiber and proposed an approximate solution scheme for a fiber with an arbitrary cross section. Later, they [5] proposed a model based on the 1-D behavior of a cylindrical bar to examine the axial load diffusion from a partially embedded finite elastic bar into an isotropic elastic half-space. The shear lag model, proposed by Cox [6], has also found wide use in the load transfer analysis of unidirectional fiber reinforced composites. The model assumes 1-D behavior for the fiber and the matrix is subjected only to shear deformations. The accuracy of the solutions based on the shear lag model depends largely on the assumed variation of interfacial shear stress. The shear lag model can provide good results for the fiber axial stress and displacement, however, the solution for interfacial shear stress is generally less accurate. There is a need to study the fiber–matrix interaction in piezocomposites to understand the electro-mechanical interaction due to diffusion of an electric charge (actuation) and axial load (sensing) from a piezoelectric fiber into the surrounding matrix. In reality, the fibers in a composite may be imperfectly bonded to the surrounding matrix due to several reasons such as a thin

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Y. Sapsathiarn et al. / Composites: Part B 39 (2008) 1114–1124

Polymer matrix Piezoelectric rod

Fig. 1. Schematic of a 1–3 piezocomposite.

interphase of adhesion coating, chemical action during manufacturing process, or interfacial damage [7,8]. However, the interphasial layer between the fiber and matrix has a very complicated microstructure and different material properties from the fiber and matrix. Therefore, assumptions are employed in the modeling, such as the widely used spring-factor model which assumes the displacement jump across the interface are proportional to the corresponding interfacial stresses [9]. According to authors’ knowledge, the studies on fiber–matrix interaction in piezocomposites are very limited. Rajapakse [10] examined the electroelastic field of a long elastic cylinder with a piezoelectric core subjected to an external radial pressure band. Gu et al. [11] studied a single piezoelectric fiber pullout from an elastic matrix under mechanical and electric loads. Their analysis was simplified by ignoring the radial variations of the stresses and strains in both the fiber and matrix and the electric field in the fiber. According to our knowledge, a rigorous study of the fiber–matrix interaction and electro-mechanical load transfer in 1– 3 piezocomposites is currently not available. This paper examines the fundamental problems involving the diffusion of an electric charge and axial load from a single piezoelectric fiber into a surrounding transversely isotropic elastic matrix. The fiber–matrix interface is assumed to be imperfect and modeled by using a spring-factor model. The interface is either electrically open- or short-circuited. As the optimum properties of 1–3 piezocomposites are achieved for volume fractions around 20% [12,13], it is reasonable to neglect the fiber-to-fiber interaction as a first approximation to study the fundamental mechanics of electro-mechanical load diffusion and the role of imperfect interfaces. Analytical solutions for three-dimensional piezoelectricity based on Fourier integral transforms are employed in the analysis. Selected numerical solutions for fiber axial force, interfacial stresses and electric field are presented for various fiber–matrix systems and interface conditions.

2. Analytical general solution The system considered in the present study is shown in Fig. 2 which consists of a cylindrical piezoelectric fiber of radius ‘a’ imperfectly bonded to a transversely isotropic elastic matrix. A cylindrical polar coordinate system (r, h, z) is defined such that the z-axis coincides with the axis of symmetry of the fiber and matrix. The fiber is subjected to a uniform body force or electric charge at z = 0. The loading and the geometry of the problem are axisymmetric. The constitutive equations for piezoelectric materials which are transversely isotropic or poled along the z-axis can be expressed as [14]:

rrr ¼ cf11 err þ cf12 ehh þ cf13 ezz  ef31 Ez rhh ¼ cf12 err þ cf11 ehh þ cf13 ezz  ef31 Ez rzz ¼ cf13 err þ cf13 ehh þ cf33 ezz  ef33 Ez rrz ¼ 2cf44 erz  ef15 Er Dz ¼ Dr ¼

ef31 err þ ef31 ehh þ 2ef15 erz þ ef11 Er

ef33 ezz

þ

ef33 Ez

ð1aÞ ð1bÞ ð1cÞ ð1dÞ ð1eÞ ð1fÞ

where rrr, rhh, rzz and rrz denote the stress components; err, ehh, ezz and erz denote the strain components; Dr and Dz denote the electric displacement vectors in the r– and z-directions, respectively; Er and Ez denote the electric fields in the r- and z-directions, respectively; cf11 , cf12 , cf13 , cf33 and cf44 denote the elastic constants under zero or constant electric field; ef31 , ef33 and ef15 denote the piezoelectric constants; and ef11 and ef33 denote the dielectric constants under zero or constant strain. The constitutive equations for the matrix material can be obtained from Eqs. (1a)–(1d) by setting efij ¼ efij  0 and replacing cfij by cm ij . The governing equations for a piezoelectric material undergoing axisymmetric deformations about the z-axis can be expressed in terms of the displacements ur and uz and the electric potential / [14]:

!  o2 u o2 ur 1 our 1 o2 u r  z þ u  þ cf44 2 þ cf13 þ cf44 r 2 2 r or r or oz oroz   o2 / þ Fr ¼ 0 þ ef31 þ ef15 oroz ! ! 2   o2 u o2 uz 1 ouz 1 our r f o uz f f þ c þ þ c þ c cf44 þ 33 13 44 r or or2 oz2 oroz r oz ! 2 2 o / 1 o/ o / þ ef33 2 þ F z ¼ 0 þ ef15 þ or 2 r or oz ! ! 2 2   o2 u o uz 1 ouz 1 our r f f o uz f f þ e33 2 þ e31 þ e15 e15 þ þ r or or 2 oz oroz r oz ! 2 o2 / 1 o/ f o /  e þ Q ¼0  ef11 33 or 2 r or oz2

cf11

ð2aÞ

ð2bÞ

ð2cÞ

where Fr, Fz and Q denote the radial and vertical body forces and electric body charge, respectively. To determine the analytical general solution of Eq. (2), a generalized displacement potential function w(r, z) is introduced by relating it to the displacements ur and uz and the electric potential / in the following manner [15]:

ur ¼

Fig. 2. Piezoelectric fiber imperfectly bonded to a matrix.

ow ; or

uz ¼ k1

ow ; oz

/ ¼ k2

ow oz

ð3Þ

where k1 and k2 are unknown constants to be determined. The substitution of Eq. (3) into Eqs. (2a)–(2c) results in the following governing equations to determine w, k1 and k2:

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Y. Sapsathiarn et al. / Composites: Part B 39 (2008) 1114–1124

"

o f c or 11

! # n     o o2 w o2 w 1 ow f f f f f k k þ þ c þ c þ e þ e þ c þ Fr ¼ 0 1 2 44 13 44 31 15 or2 r or oz2

01.0

= 1000

ð4aÞ

0.8

" # !  o2 w 1 ow   o2 w o  f f f þ k þ e k c13 þ cf44 þ cf44 k1 þ ef15 k2 þ Fz ¼ 0 þ c 1 2 33 33 oz or2 r or oz2 ð4bÞ

#

!

2P(z)/ P0

"

= 100

 o2 w 1 ow   o2 w o  f þ e31 þ ef15 þ ef15 k1  ef11 k2 Q ¼0 þ ef33 k1  ef33 k2 2 oz or r or oz2

0.6

= 10 Ef ka

0.4

Present Study Reference [19]

0.2

=1

ð4cÞ

The solution for the potential function w(r, z) composes of the homogeneous solution, denoted by wh(r, z), and the particular solution, denoted by wp(r, z) such that h

p

wðr; zÞ ¼ w ðr; zÞ þ w ðr; zÞ:

00.0 0

1

2

3

4

0.04

ð5Þ

Present Study Reference [19]

The homogeneous solution of the potential function, wh, is derived from Eq. (4) by setting Fr = Fz = Q = 0 and applying Fourier integral transforms with respect to z. The Fourier integral transform of a function f(r, z) with respect to z and its inverse are defined by [16]

Z 1 f ðr; nÞ ¼ p1 ffiffiffiffiffiffiffi f ðr; zÞeinz dz 2p 1 Z 1 1 f ðr; nÞeinz dn f ðr; zÞ ¼ pffiffiffiffiffiffiffi 2p 1

σrz a2 / P0

0.03

2.1. Homogeneous solution

Ef ka

0.02

=1

= 10 0.01

= 100 = 1000

00.00 0

ð6aÞ

1

2

5

ð7aÞ

= 100

-0.001

j¼1

-0.002

= 10 Ef ka

-0.003

where

j ¼ 1; 2; 3

ð7bÞ

-0.004 0

and w is the Fourier transform of the homogeneous solution of the potential function; vj (j = 1, 2, 3) are a set of roots of the characteristic equation corresponding to Eq. (4) as defined in the Appendix; Aj(n) and Bj(n) (j = 1, 2, 3) are arbitrary functions to be determined from the boundary and continuity conditions; In and Kn denote

Present Study; urf (a, z) = urm (a, z) Present Study; urf (a, z) = 0 Reference [19]

=1

1

h

2

3

4

5

z /a Fig. 4. Comparison of (a) fiber axial force; (b) interfacial shear stress and (c) interfacial radial stress of elastic composite with imperfect interface.

the modified Bessel functions of the first and the second kinds of order n, respectively [17]. h h Let k1j and k2j denote the constants k1 and k2 of the homogeneous solution corresponding to the root vj. It can be shown that

1.0

h k1j

Present Study Reference [4]

0.8

2P(z)/ P0

4

= 1000

σrr a2 / P0

3 X  h ðr; nÞ ¼ w ½I0 ðnj rÞAj ðnÞ þ K 0 ðnj rÞBj ðnÞ

0.6

Ef =8 Em

0.4

=4

0

2

ð8aÞ

ð8bÞ

The homogenous general solution given by Eq. (7) corresponds to an infinite annular piezoelectric cylinder. The solution for a solid piezoelectric cylinder of finite radius is obtained from Eq. (7) by setting Bj = 0.

=1

1

¼

  h cf11 mj  cf44  ef15 þ ef31 k2j

cf13 þ cf44     2 cf11 mj  cf44 cf33  cf44 mj  cf13 þ cf44 mj h      k2j ¼  ef15 þ ef31 cf33  cf44 mj  cf13 þ cf44 ef33  ef15 mj

=2 0.2

0

3

-00.000

ð6bÞ

where n denotes the Fourier transform parameter. Applying Fourier integral transforms to Eq. (4) with Fr = Fz = Q = 0, the following homogeneous solution can be obtained,

pffiffiffiffi nj ¼ jnj vj ;

5

3 z /a

4

5

6

Fig. 3. Comparison of the fiber axial force for a perfect interface and ideal elastic materials.

2.2. Particular solution corresponding to radially uniform body forces and charges In order to consider the diffusion of an electric charge and axial load from a piezoelectric fiber into a matrix, it is necessary to find

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Y. Sapsathiarn et al. / Composites: Part B 39 (2008) 1114–1124 Table 1 Material properties

e11 e33

2

(10 N/m ) (1010 N/m2) (1010 N/m2) (1010 N/m2) (1010 N/m2) (C/m2) (C/m2) (C/m2) (109 F/m) (109 F/m)

PZT-6B

PZT-4

BaTiO3

Matrix A

Matrix B

16.8 16.3 6.0 6.0 2.71 0.9 7.1 4.6 3.6 3.4

13.9 11.5 7.78 7.43 2.56 5.2 15.1 12.7 6.45 5.62

15.0 14.6 6.6 6.6 4.4 4.35 17.5 11.4 9.87 11.15

1.49 4.72 0.66 0.52 0.47 0.0 0.0 0.0 0.0 0.0

0.2827 0.2827 0.1211 0.1200 0.0808 0.0 0.0 0.0 0.0 0.0

PZT-6B fiber PZT-4 fiber BaTiO3 fiber

0.03

*

c11 c33 c12 c13 c44 e31 e33 e15

10

0.04

Matrix A Matrix B

0.02

0.01

00.00 0

1

2

3

4

5

6

z /a 01.0

-00.000

0.8

0.6

*

2P(z)/ P0

-0.001

0.4

PZT-6B fiber PZT-4 fiber BaTiO3 fiber

0.2

-0.002

-0.003

Matrix A Matrix B

00.0 0

1

2

3

4

5

-0.004

6

0

1

2

z /a

3

4

5

6

z /a

-00.00

Fig. 6. (a) Shear and (b) radial stresses along the fiber–matrix interface of piezoelectric composite (open-circuited) under applied axial load.

cf ef pz  44 33  ; k1 ¼  ef31 þ ef15 ef33 þ cf13 þ cf44 ef33

pz

k2 ¼

ef33 ef33

pz

k1

ð9Þ

E*z

-0.02

Applying Fourier transforms to Eq. (4b), the following solution for the Fourier transform of the particular solution of the potential function can be obtained

-0.04

 pz ðnÞ ¼ w

-0.06 0

1

2

3

4

5

6

z /a Fig. 5. (a) Resultant axial force and (b) vertical electric field along the z-axis of piezoelectric fiber (open-circuited) under applied axial load.

the particular solution of the Eq. (4) under a non-zero axial body force and a body charge. This particular solution does not exist in the literature. The present work focuses on the fundamental problems of a vertical body force and a body electric charge that are uniform in the radial direction and vary only with z. First, consider the case of a vertical body force Fz(z) in the absence of a radial body force Fr and a body charge Q. To satisfy the Eq. (4) for any arbitrary r-value, the particular solution wp must be independent of r. Con2 p  p in Eq. (4) should vanish. The consequently, the terms oorw2 þ 1r ow or stants k1 and k2 of the particular solution corresponding to a pz pz vertical body force are denoted by k1 and k2 . These constants can be determined from Eqs. (4a) and (4c) with Fr = Q = 0 as

iF z ðnÞ pz

ð10Þ

pz

n3 ðcf33 k1 þ ef33 k2 Þ

where F z ðnÞ denotes the Fourier transform of the vertical body force. Now consider the case of an electric body charge with Fr = Fz = 0. pq Similar to the case of a vertical body force, the expressions for k1 pq and k2 for the case of an electric body charge can be determined from Eqs. (4a) and (4b) as

cf ef pq  44 33   ; k1 ¼  f f f e31 þ e15 c33  cf13 þ cf44 ef33

pq

k2 ¼ 

cf33 ef33

pq

k1

ð11Þ

 pq corresponding The Fourier transform of the particular solution w to the case of an electric charge Q(z) is obtained from Eq. (4c) as

 pq ðnÞ ¼ w

iQðnÞ   pq pq n3 ef33 k1  ef33 k2

where Q ðnÞ denotes the Fourier transform of the body charge.

ð12Þ

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Y. Sapsathiarn et al. / Composites: Part B 39 (2008) 1114–1124

-0-.01 0

0.45

PZT-6B fiber PZT-4 fiber BaTiO3 fiber

-00.00 -0.01

Matrix A Matrix B

0.30

-0.02

-0.04

PZT-6B fiber PZT-4 fiber BaTiO3 fiber

-0.05

Matrix A Matrix B

-0.03

0.15

-0.06

00.00 0

1

2

3 z /a

4

5

0

6

1

2

3

4

5

6

4

5

6

z /a

-00.00

0.005

0.004

-0.05

*

0.003

-0.10

0.002

0.001

-0.15 00.000

0-.001 -0

-0.20 0

1

2

3

4

5

0

6

1

2

3

z /a

z /a Fig. 7. (a) Vertical stress and (b) vertical electric field along the z-axis of piezoelectric fiber (open-circuited) under applied electric charge.

2.3. Complete general solution

Fig. 8. (a) Shear and (b) radial stresses along the fiber–matrix interface of piezoelectric composite (open-circuited) under applied electric charge. 3 h   i X h h nj cf44 1 þ k1j  ef15 k2j I1 ðnj rÞAj ðnÞ

r rz ðr; nÞ ¼ in

ð13gÞ

j¼1

By using the homogeneous and particular solutions presented above, the complete general solution for the Fourier transforms of the electroelastic field of a solid piezoelectric cylinder can be expressed as:  r ðr; nÞ ¼ u

3 X

 z ðr; nÞ ¼ in u

" 3 X

ð13aÞ #

h pz  pz pq  pq k1j I0 ðnj rÞAj ðnÞ þ k1 w ðnÞ þ k1 w ðnÞ

ð13bÞ

j¼1

" # 3 X h pz  pz pq  pq  k2j I0 ðnj rÞAj ðnÞ þ k2 w ðnÞ þ k2 w ðnÞ /ðr;nÞ ¼ in

ð13cÞ

j¼1 3 n h i   o X h h jnj2 cf11 vj  cf13 k1j  ef31 k2j I0 ðnj rÞ þ cf12  cf11 nj r 1 I1 ðnj rÞ Aj ðnÞ j¼1

 n2

h

   i pz pz  pz pq pq  pq cf13 k1 þ ef31 k2 w ðnÞ þ cf13 k1 þ ef31 k2 w ðnÞ ð13dÞ

3 n h i   o X h h r hh ðr;nÞ ¼ jnj2 cf12 vj  cf13 k1j  ef31 k2j I0 ðnj rÞ þ cf11  cf12 nj r 1 I1 ðnj rÞ Aj ðnÞ j¼1

 n2

h

   i pz pz  pz pq pq  pq cf13 k1 þ ef31 k2 w ðnÞ þ cf13 k1 þ ef31 k2 w ðnÞ

3 X

jnj

j¼1 2

n

2

h

h

h cf13 vj  cf33 k1j

h  ef33 k2j

ð13hÞ

j¼1

Dz ðr;nÞ ¼

3 X

h i h h jnj2 ef31 vj  ef33 k1j þ ef33 k2j I0 ðnj rÞAj ðnÞ

 n2

h

   i pz pz  pz pq pq  pq ef33 k1  ef33 k2 w ðnÞ þ ef33 k1  ef33 k2 w ðnÞ

ð13iÞ

The three arbitrary functions Aj (j = 1, 2, 3) appearing in Eq. (13) can be determined from the three boundary conditions (two mechanical and one electric) associated with the outer surface of a solid piezoelectric cylinder. Next consider the analytical general solution of the matrix material shown in Fig. 2. The matrix is transversely isotropic and free from body forces. The general solution can be derived by following a procedure similar to the piezoelectric case and setting, efij ¼ efij  0 and replacing cfij by cm ij . The details are not presented here for brevity. The general solution of the matrix region is given by

 r ðr; nÞ ¼ u

2 X

nk K 1 ðnk rÞC k ðnÞ

ð14aÞ

k¼1

ð13eÞ

r zz ðr; nÞ ¼

3 h   i X h h nj ef15 1 þ k1j  ef11 k2j I1 ðnj rÞAj ðnÞ

j¼1

nj I1 ðnj rÞAj ðnÞ

j¼1

r rr ðr;nÞ ¼

Dr ðr; nÞ ¼ in

i I0 ðnj rÞAj ðnÞ

 z ðr; nÞ ¼ in u

2 X

nk K 0 ðnk rÞC k ðnÞ

ð14bÞ

k¼1

r rr ðr; nÞ ¼

   i pz pz  pz pq pq  pq cf33 k1 þ ef33 k2 w ðnÞ þ cf33 k1 þ ef33 k2 w ðnÞ

2 n X

  m jnj2 cm 11 wk  c 13 nk K 0 ðnk rÞ

k¼1

ð13fÞ

   1 m  cm 12  c 11 nk r K 1 ðnk rÞ C k ðnÞ

ð14cÞ

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Y. Sapsathiarn et al. / Composites: Part B 39 (2008) 1114–1124

-0.02 0-.02

01.0

00.00

0.8

2P(z)/ P0

0.02 0.6

0.04 0.4

PZT-6B fiber PZT-4 fiber BaTiO3 fiber

0.2

PZT-6B fiber PZT-4 fiber BaTiO3 fiber

0.06

Matrix A Matrix B

0.08

Matrix A Matrix B

0.10

00.0 0

1

2

3 z /a

4

5

0

6

1

2

3

4

5

6

4

5

6

z /a

0.01 -0-.01

-0.05 -0-.05

-00.00

PZT-6B

-00.00 -0.01 *

BaTiO3 -0.05

PZT-4

-0.02

-0.10

-0.03

-0.04 0

1

2

3

4

5

6

z /a

2 n X   m r hh ðr; nÞ ¼ jnj2 cm 12 wk  c 13 nk K 0 ðnk rÞ k¼1

r zz ðr; nÞ ¼

2 X

  m jnj2 cm 13 wk  c 33 nk K 0 ðnk rÞC k ðnÞ

ð14dÞ ð14eÞ

k¼1

r rz ðr; nÞ ¼ in

2 X

nk cm 44 ½1 þ nk K 1 ðnk rÞC k ðnÞ

ð14fÞ

k¼1

where cm ij denotes elastic constants of the matrix material; Ck(n)(k = 1, 2) are arbitrary functions to be determined from the boundary and continuity conditions; and

pffiffiffiffiffiffi nk ¼ jnj wk ;

nk ¼

m cm 11 wk  c 44 ; m cm þ c 13 44

k ¼ 1; 2

ð15Þ

and wk (k = 1, 2) are the roots of the equation m 2 cm 11 c 44 wk þ

h

m 2

c13

i

m m m m m þ 2cm 13 c 44  c 11 c 33 wk þ c 33 c 44 ¼ 0:

0

1

2

3

z /a

Fig. 9. (a) Resultant axial force and (b) vertical electric field along the z-axis of piezoelectric fiber (short-circuited) under applied axial load.

   1 m  cm 11  c 12 nk r K 1 ðnk rÞ C k ðnÞ

-0.15

ð16Þ

3. Analysis of electro-mechanical load diffusion The general solutions presented in the preceding sections are used in the analysis of the fundamental electro-mechanical fiber– matrix interaction problem shown in Fig. 2. The fiber–matrix interface is assumed to be either electrically open- or short-circuited and mechanically imperfectly bonded. The general solution of the fiber and matrix are given by Eqs. (13) and (14), respectively. These

Fig. 10. (a) Vertical stress and (b) vertical electric field along the z-axis of piezoelectric fiber (short-circuited) under applied electric charge.

equations involve five arbitrary functions, A1, A2, A3, C1 and C2 which can be determined from the continuity conditions along the fiber–matrix interface for a specified body force Fz(z) and/or electric charge Q(z). In this paper, the fundamental solutions involving an axial force and an electric charge applied uniformly over the fiber cross section at z = 0 (Fig. 2) are considered. The total magnitude of the body force and electric charge are P0 and Q0, respectively. The corresponding body force functions Fz(z) and Q(z) can be expressed as

F z ðzÞ ¼ P0 dðzÞ=ðpa2 Þ;

QðzÞ ¼ Q 0 dðzÞ=ðpa2 Þ:

ð17Þ

The stress continuity conditions and electrically impermeable condition (open-circuit) along the fiber–matrix interface (r = a) can be expressed as

rfrr ða; zÞ ¼ rmrr ða; zÞ; rfrz ða; zÞ ¼ rmrz ða; zÞ; Dfr ða; zÞ ¼ 0

ð18Þ

in which the superscript f and m are used to identify the quantities corresponding to the piezoelectric fiber (0 6 r 6 a) and elastic matrix (a 6 r < 1), respectively. Alternately, the electric boundary condition in Eq. (18) can be replaced by /f(a, z) = 0 which implies short-circuited condition. The two conditions represent the extreme electric boundary conditions at the interface. In the present study, the mechanically imperfect fiber–matrix interface is represented by a shear spring-factor model which assumes that sufficient cohesion exists on the interface to prevent separation in the radial direction and relates the vertical displacement jump along the interface and the interface shear stress through a spring-factor parameter as follows:

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Y. Sapsathiarn et al. / Composites: Part B 39 (2008) 1114–1124

-0.000

1.0

0.8

*

2P(z)/ P0

-0.005

0.6

Perfect interface ks* = 10 =1 = 0.1 = 0.01 = 0.001

0.4

0.2

-0.010

-0.015

0.0

*

0

1

2

3 z /a

4

5

0

6

1

2

3

4

5

6

4

5

6

z /a

0.04

--0.001 0.001

0.03

-0.000

0.02

-0.001

0.01

-0.002

-0.003

0.00 0

1

2

3

4

5

0

6

1

2

3

z /a

z /a

Fig. 11. Composite of PZT-6B fiber with Matrix A (a) resultant axial force and (b) vertical electric field along the z-axis of piezoelectric fiber; (c) shear and (d) radial stresses along the fiber–matrix interface (open-circuited) under applied axial load for different interface stiffness values.

ufr ða; zÞ ¼ um r ða; zÞ  m  ks uz ða; zÞ  ufz ða; zÞ ¼ rfrz ða; zÞ

ð19aÞ

Z

rI0 ðnrÞdr ¼

ð19bÞ

where ks denotes the spring-factor parameter. For a perfectly bonded interface, ks ? 1 and ufz ða; zÞ ¼ um z ða; zÞ. Eqs. (17)–(19) can be easily expressed in the Fourier transform domain and the substitution of general solutions given by Eqs. (13) and (14) in Eqs. (18) and (19) results in a linear algebraic simultaneous equation system to determine the arbitrary functions A1, A2, A3, C1 and C2. Once arbitrary functions are determined, the full electroelastic field of the system shown in Fig. 2 can be determined. Application of inverse Fourier integral transforms to Eq. (13f) results in the following solution for axial normal stress rfzz ðr; zÞ in the piezoelectric fiber

Z 1n h i 1 h h rfzz ðr; zÞ ¼ pffiffiffiffiffiffiffi jnj2 cf13 vj  cf33 k1j  ef33 k2j I0 ðnj rÞAj ðnÞn2 2p 1 h io    pz pz  pz pq pq  pq cf33 k1 þ ef33 k2 w ðnÞ þ cf33 k1 þ ef33 k2 w ðnÞ einz dn:

rI1 ðnrÞ n

ð22Þ

yields

( 3  X jnj  f an2 h h f f pffiffiffiffi c13 vj  c33 k1j  e33 k2j I1 ðnj aÞAj ðnÞ  2 vj 1 j¼1    h io pz pz  pz pq pq  pq f f f f  c33 k1 þ e33 k2 w ðnÞ þ c33 k1 þ e33 k2 w ðnÞ einz dn:

pffiffiffiffiffiffiffi Z PðzÞ ¼ 2pa

1

ð23Þ The interfacial shear stress and the electric field along the z-axis are given by

1

rrz ða; zÞ ¼ pffiffiffiffiffiffiffi 2p

Z

1

1

in

3 n   o X h h nj cf44 1 þ k1j  ef15 k2j j¼1

inz

I1 ðnj aÞAj ðnÞe dn ð24Þ ) Z 1 (X 3 1 h pz  pz pq  pq n2 k2j Aj ðnÞ þ k2 w ðnÞ þ k2 w ðnÞ einz dn: Ez ð0; zÞ ¼ pffiffiffiffiffiffiffi 2p 1 j¼1

ð20Þ The resultant axial force P(z) in the piezoelectric fiber at a given cross section can be determined from

PðzÞ ¼ 2p

Z 0

ð25Þ 4. Numerical results and discussion

a f zz ðr; zÞrdr;

r

1 < z < 1:

Substitution of the solution of following identity [17]

f zz ðr; zÞ

r

ð21Þ

into Eq. (21) and use of the

The computation of the electroelastic field corresponding to the problem shown in Fig. 2 requires the numerical evaluation of infinite integrals involving inverse Fourier transforms. These integrals

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Y. Sapsathiarn et al. / Composites: Part B 39 (2008) 1114–1124

1.0

-0.00

0.8

2 P(z)/ P0

-0.02

0.6

Perfect interface ks* = 10 =1 = 0.1 = 0.01 = 0.001

0.4

0.2

-0.04

0.0

-0.06

0

1

2

3 z /a

4

5

6

0

1

2

3

4

5

6

4

5

6

z /a --2

0.008

-0 ( x 10-4 )

0.010

-2

*

0.006

0.004

-4

0.002

-6

0.000

-8

0

1

2

3

4

5

6

z /a

0

1

2

3

z /a

Fig. 12. Composite of PZT-4 fiber with Matrix B (a) resultant axial force and (b) vertical electric field along the z-axis of piezoelectric fiber; (c) shear and (d) radial stresses along the fiber–matrix interface (open-circuited) under applied axial load for different interface stiffness values.

can be converted to semi-infinite integrals by using the symmetric and anti-symmetric properties of the integrands. In the present study, a globally adaptive numerical quadrature scheme is employed to evaluate the integrals [18]. This scheme subdivides the interval of the integrand and uses a 21-point Gauss–Kronrod rule to estimate the integral over each interval. 4.1. Comparison with ideal elastic composites The accuracy of the present scheme is first verified by comparing with the existing solutions for an ideal elastic composite system. Elastic composites can be analyzed by using the present scheme by setting the piezoelectric coefficients of the fiber to negligibly small values (efij  0). Fig. 3 presents a comparison of the numerical solutions for the fiber axial force of a perfectly bonded fiber with those presented by Muki and Sternberg [4]. The Poisson’s ratio of the fiber and matrix are equal to 0.25 and Ef and Em denote Young’s modulus of the fiber and matrix, respectively. The solutions shown in Fig. 3 are in excellent agreement for various values of Ef/Em confirming the accuracy of the present solution. Further comparisons are presented by considering the case of elastic composites with imperfect fiber–matrix interfaces [19]. The elastic modulus of the fiber, Ef = 68.954 GPa, matrix Poisson’s ratio m = 0.34 and shear modulus l = 2.59 GPa. Lenci and Menditto [19] simplified the analysis by modeling the fiber as a one-dimensional bar whereas the present solution allows consideration of the fiber as a 3-D continuum or 1-D bar. A comparison of the fiber axial force, and interfacial shear and radial stresses are presented in Fig.

4 for various non-dimensional interface stiffness values, Ef/(ksa), in which ks and a denote the interface stiffness (spring-factor) and radius of the fiber, respectively. The results given by Lenci and Menditto [19] and the present scheme agree closely for the fiber axial force and interfacial shear stress. In the case of interfacial radial stress, Fig. 4c shows two solutions obtained from the present scheme: one corresponding to full radial displacement compatibility between the fiber and matrix and other with the constraint that interfacial radial displacement is zero. In the vicinity of the loading plane, the two solutions and the results of [19] agree closely but the differences are noted between the full radial displacement compatibility solution and Ref. [19] as z/a increases. This is due to the fact that the radial deformation is not accounted by the 1D fiber model of [19]. On the other hand, the present solution for interfacial radial stress with the constraint ufr ða; zÞ ¼ 0 agrees very closely with the corresponding solution of [19] for all values of z/a. 4.2. Force and charge diffusion in piezocomposites In this section, selected numerical solutions are presented to demonstrate the basic features of the electro-mechanical load diffusion process in 1–3 piezocomposites. Two elastic polymer matrix materials, identified as matrix A and matrix B, with three different piezoelectric fibers, namely, PZT-6B, PZT-4 and BaTiO3 are used in the numerical study. The properties of these materials are given in Table 1. The case of piezocomposites with perfect bonding along the fiber–matrix interface is first examined. The corresponding solutions

Y. Sapsathiarn et al. / Composites: Part B 39 (2008) 1114–1124

0.08

-0.00

0.06

-0.01

Perfect interface ks* = 10 =1 = 0.1 = 0.01 = 0.001

0.04

0.02

*

*

1122

-0.02

-0.03

-0.04

0.00 0

1

2

3 z /a

4

5

0

6

1

2

3

4

5

6

4

5

6

z /a

-0.000

10

8

( x 10-4 )

-0.005

6

*

4

-0.010

2

0

-0.015

-2

0

1

2

3

4

5

6

z /a

0

1

2

3

z /a

Fig. 13. Composite of PZT-6B fiber with Matrix A (a) vertical stress and (b) vertical electric field along the z-axis of piezoelectric fiber; (c) shear and (d) radial stresses along the fiber–matrix interface of piezoelectric composite (open-circuited) under applied electric charge for different interface stiffness values.

are presented in Figs. 5–10 for fibers with open- and short-circuited conditions along the interface. Note that solutions are presented only for z P 0 as all quantities are either symmetric or anti-symmetric with respect to the loading plane z = 0. Consider the diffusion of an axial load of magnitude P0 applied uniformly over the cross section at z = 0 of the fiber (Fig. 2). The variation of non-dimensional resultant axial force along the fiber length is shown in Fig. 5a for the open-circuited case. Naturally the nondimensional axial load has a unit magnitude at z = 0 and decreases gradually with z. The decay of axial load in the fiber depends on the type of fiber and matrix material. As the matrix A is stiffer than matrix B (Table 1), the axial load is more rapidly transferred to matrix A when compared to matrix B. For example, at a distance six times the fiber-radius from the loading plane, the fiber axial load is approximately 10% and 30% of the applied load for matrix A and matrix B, respectively. The dependence of fiber axial load on fiber material properties is quite negligible when compared to its dependence on the matrix material properties. In the case of piezocomposites, of particular interest to sensor applications are the electric field generated in the fiber due to the applied mechanical loading. Fig. 5b shows the non-dimensional vertical electric field, Ez ¼ Ez ef15 a2 =P0 , along the axis of the piezoelectric fiber. The peak value of vertical electric field occurs near the loading plane (z/a < 1) and thereafter Ez gradually decreases with z. The decay of Ez along the fiber length is significantly affected by the properties of the fiber and matrix. The PZT-4 fiber has the highest Ez followed by the BaTiO3 and PZT-6B fibers which

implies that PZT-4 is more suitable for applications involving sensing. A softer matrix material result in a higher vertical electric field in the fiber as lesser load is transferred to the matrix. In addition, the decay of electric field is relatively slow in the case of a softer matrix allowing better sensing properties. The variation of non-dimensional shear stress, rrz ¼ rrz a2 =P0 , and radial stress, rrr ¼ rrr a2 =P 0 , along the fiber–matrix interface (r/a = 1) is shown in Fig. 6a and b, respectively. Similar to the case of the fiber axial force, interfacial stresses are mostly influenced by the matrix material when compared to the properties of the fiber. Shear stress along the interface has its maximum value at the loading plane and decreases rapidly in the vertical (fiber) direction. Stiffer matrix materials result in higher interfacial stresses as the load transfer to the matrix is more rapid in this case. Consequently, such composites are more prone to interfacial failure provided that the shear strength at the interface is the same. Radial stress along the fiber–matrix interface is zero at z = 0 and increases rapidly with z, reaching a peak value near z/a = 1 and thereafter decreases rapidly with z. The peak value of the interface radial stress is less than 10% of the peak interface shear stress. Therefore, interfacial strength is mostly governed by the cohesion at the interface instead of Coulomb friction in the present model. Next, consider the diffusion of a patch electric charge from a fiber perfectly bonded to a matrix. Non-dimensional vertical stress, 2 2 f f  f 2 r zz ¼ rzz e15 a =c 44 Q 0 , and vertical electric field, Ez ¼ Ez ðe15 Þ a = cf44 Q 0 , along the z-axis are presented in Fig. 7a and (b), respectively. A substantial dependence of vertical stress on the fiber type is

1123

0.4

-0.00

0.3

-0.05

Perfect interface k s* = 10 =1 = 0.1 = 0.01 = 0.001

0.2

0.1

*

*

Y. Sapsathiarn et al. / Composites: Part B 39 (2008) 1114–1124

-0.10

-0.15

0.0

-0.20

0

1

2

3 z /a

4

5

6

0

1

2

3

4

5

6

4

5

6

z /a

-0.000

12

( x 10-4 )

10 8 6

*

-0.005

4 -0.010 2 0 -2

-0.015 0

1

2

3

4

5

6

z /a

0

1

2

3

z /a

Fig. 14. Composite of PZT-4 fiber with Matrix B (a) vertical stress and (b) vertical electric field along the z-axis of piezoelectric fiber; (c) shear and (d) radial stresses along the fiber–matrix interface of piezoelectric composite (open-circuited) under applied electric charge for different interface stiffness values.

noted. Vertical stress is zero at the loading plane and initially increases rapidly and then decreases slowly along the fiber axis. It is tensile along the upper fiber length, and PZT-4 has the highest non-dimensional value. Fig. 7b show the variation of E z along the fiber axis. In this case, BaTiO3 and PZT-6B fibers have nearly equal but substantially smaller values of E z when compared to a PZT-4 fiber. The decay of the vertical electric field along the fiber f 2 length is not rapid. Non-dimensional shear stress, r rz ¼ rrz e15 a = f f 2 cf44 Q 0 , and radial stress, r ¼ r e a =c Q , at the fiber–matrix rr 15 rr 44 0 interface due to an electric charge are shown in Fig. 8a and b, respectively. The variation of shear and radial stresses along the interface is somewhat similar to the results shown in Fig. 6. However, the dependence of the interface stresses on the fiber and matrix material type is more pronounced in the case of electric charge loading when compared to the axial loading. Fig. 9 shows the resultant axial force and vertical electric field of a short-circuited fiber bonded to an elastic matrix due to an axial force P0. The axial force profiles are similar to those shown in Fig. 5a but the load transfer is slightly more rapid than the open-circuited case and the fiber properties have more influence on the force profiles. Vertical electric field decays very rapidly along the fiber length in contrast to Fig. 5b and show less dependence on the matrix material. The interfacial stresses for the short-circuited case show behavior similar to Fig. 6 but the magnitudes are higher by 20–50%. Fig. 10 shows the variation of vertical stress and vertical electric field when a short-circuited fiber is subjected to an electric charge Q0. The profiles are very different from those shown in Fig. 7 and show very rapid decay of axial stress and vertical elec-

tric field along the fiber length. The magnitudes show considerable dependence on the fiber properties but negligible influence of matrix material. As in the case of Fig. 7 the maximum non-dimensional axial stress and vertical electric field values correspond to PZT-4. Comparisons with Fig. 7 also show that axial stress and vertical electric field are much smaller in the short-circuited case. The effect of imperfect fiber–matrix bonding on the load diffusion characteristics is presented in Figs. 11–14. Two cases of piezocomposites are considered in the numerical study, i.e. composites of PZT-6B fiber with matrix A and PZT-4 fiber with matrix B. The interface behavior is characterized by the spring-factor model described in Eq. (19b). Five values of non-dimensional spring-factor,  ks ¼ ðks aÞ=cf44 ¼ 10; 1; 0:1; 0:01 and 0:001 are considered in the numerical study. Figs. 11 and 12 show the electroelastic field of composites of PZT-6B fiber with matrix A and PZT-4 fiber with matrix B, respectively, under an applied axial load. Numerical results presented in Figs. 11 and 12 demonstrate a substantial dependence of the electroelastic field of both composites on the interface stiffness. Figs. 11a and 12a shows that fiber axial force significantly depends on the stiffness of the interface and as k* decreases (weaker interface) the load transferred to the matrix is obviously reduced. The magnitudes of interfacial shear stress and radial stress obviously decrease as the interface bonding becomes weaker and the decay along the interface becomes less rapid. On the other hand, vertical electric field of the fiber increases in the case of a weaker interface due to less load transfer to the matrix. The magnitude of fiber vertical electric field can be considered a measure of the interface condition (weaker interface yields a higher fiber electric

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Y. Sapsathiarn et al. / Composites: Part B 39 (2008) 1114–1124

field). Therefore, it is possible to obtain a qualitative non-destructive assessment of the interface bond condition by comparing vertical electric field of fibers. A comparison of Figs. 11 and 12 indicate that an imperfect interface has a relatively lesser effect on the load diffusion in a PZT-4/Matrix B composite and the interfacial stresses are also smaller. The effect of an imperfect interface on electric charge transfer is shown in Fig. 13 for a composite of PZT-6B fibers with matrix A and in Fig. 14 for a composite of PZT-4 fibers with matrix B. The influence of interface bonding condition is relatively small when compared to the effects observed in Figs. 11 and 12 for an axial load. Vertical electric field is negative, its absolute value is increased as the interface becomes weaker. The fiber tensile axial stress is also increased as the interface becomes weaker. However, near the loading plane vertical stress and electric field show negligible influence of the interface stiffness. Non-dimensional shear and radial stresses indicate that interfacial stresses decrease as the interface becomes weaker similar to the behavior observed for the axial load diffusion but the stresses have signs opposite to those corresponding to the axial load transfer case. A comparison of Figs. 13 and 14 shows that the influence of interface stiffness is less in the case of PZT-4/Matrix B composites under electric charge loading similar to the case of axial loading. 5. Conclusions A theoretical study of the electro-mechanical load diffusion from a fiber in a 1–3 piezocomposite with an imperfect interface is presented within the framework of 3-D axisymmetric theory of piezoelectricity. The boundary-value problems of axial load and electric charge transfer are successfully formulated by using Fourier integral transform based analytical general solutions. The accuracy and stability of the present solution scheme for a wide range of composites are confirmed by comparison with the existing solutions for load transfer of elastic composites. Selected numerical solutions show that the fiber axial force and interfacial stresses due to an applied axial load are primarily controlled by the stiffness of the matrix material but show minor dependence on the piezoelectric fiber properties whereas vertical electric field in the fiber depends on both fiber and matrix properties. Similarly fiber axial force and interfacial stresses due to an electric charge depends on both matrix and fiber properties. PZT-4 and PZT-6B fibers show the highest and lowest electro-mechanical coupling, respectively. An imperfect interface results in lower axial load transfer to the matrix and hence lower interfacial stresses but a higher fiber vertical electrical field. In the case of electric charge transfer, interfacial stresses decrease and fiber vertical electric field increase as the interface becomes weaker. The electric boundary condition (short- or open-circuit) of the fiber shows significant influence on the fiber electric field and interfacial stresses particularly in the electric charge loading case. The solutions presented in this study provide an insight into the fundamental understanding of the fiber–matrix interaction in 1–3 piezocomposites. Acknowledgements The work presented in this paper was supported by the Thailand Research Fund and the 90th Anniversary of Chulalongkorn University Fund (Ratchadaphiseksomphot Endowment Fund). This support is gratefully acknowledged.

Appendix The characteristic roots vj appearing in Eq. (7) are determined from the following equation

Av3 þ Bv2 þ Cv þ D ¼ 0

ðA1Þ

where the coefficients A, B, C and D are constants expressed in terms of material properties as

   2 ðA2Þ ef15 þ cf44 ef11 n    o B ¼ cf13 2ef15 cf13 þ 2cf44 þ ef11 cf13 þ 2cf44    cf11 2ef15 ef33 þ cf33 ef11 þ cf44 ef33 ðA3Þ

   n    o 2 C ¼ cf11 ef33 þ cf33 ef33 Þ þ cf13 2ef33 ef15 þ ef31 þ ef33 cf13 þ 2cf44

A ¼ cf11

 2   þ cf33 ef15 þ ef31 þ cf44 2ef31 ef33  cf33 ef11

   2 D ¼ cf44 ef33 þ cf33 ef33

ðA4Þ ðA5Þ

The three roots of Eq. (A1) are denoted by vj (j = 1, 2, 3) with v1 assumed to be a positive real number and v2 and v3 either positive real numbers or a pair of complex conjugates with positive real parts. References [1] Bent AA, Hagood NW, Rodgers JP. Anisotropic actuation with piezoelectric fiber composites. J Intell Mater Syst Struct 1995;6:338–49. [2] Nelson LJ. Smart piezoelectric fibre composites. Mater Sci Technol 2002;18:1245–56. [3] Newnham RE, Skinner DP, Cross LE. Connectivity and piezoelectric– pyroelectric composites. Mater Res Bull 1978;13:525–36. [4] Muki R, Sternberg E. On the diffusion of an axial load from an infinite cylindrical bar embedded in an elastic medium. Int J Solids Struct 1969;5:587–605. [5] Muki R, Sternberg E. Elastostatic load-transfer to half-space from a partially embedded axially loaded rod. Int J Solids Struct 1970;6:69–90. [6] Cox HL. The elasticity and strength of a paper and other fibrous materials. Brit J Appl Phys 1952;3:72–9. [7] Kim JK, Mai YW. Engineered interfaces in fiber reinforced composites. Oxford: Elsevier; 1998. [8] Hammamia H, Arousb M, Lagachea M, Kallel A. Experimental study of relaxations in unidirectional piezoelectric composites. Composites 2006;A37:1–8. [9] Hashin Z. Thermoelastic properties of fiber composites with imperfect interface. Mech Mater 1990;8:333–48. [10] Rajapakse RKND. Electroelastic response of a composite cylinder with a piezoceramic core. In: Proceedings of SPIE, California, vol. 2715; 1996. p. 684– 94. [11] Gu B, Liu HY, Mai YW. A theoretical model on piezoelectric fibre pullout with electric input. Eng Fract Mech 2006;73:2053–66. [12] Li L, Sottos NR. A design for optimizing the hydrostatic performance of 1–3 piezocomposites. Ferroelect Lett 1996;21:41–6. [13] Montgomery RE, Richard C. A model for the hydrostatic pressure response of a 1–3 composite. IEEE Trans Ultrason Ferroelect Freq Contr 1996;43:457–66. [14] Parton VZ, Kudryavtsev BA. Electromagnetoelasticity. New York: Gordon and Breach Science Publishers; 1988. [15] Wang Z, Zheng B. The general solution of three-dimensional problems in piezoelectric media. Int J Solids Struct 1995;32:105–15. [16] Sneddon I. The use of integral transforms. New York: McGraw-Hill; 1970. [17] Watson GN. A treatise on the theory of bessel functions. Cambridge University Press; 1962. [18] Piessens R, deDoncker E, Uberhuber C, Kahaner D. QUADPACK: a subroutine package for automatic integration. Berlin: Springer; 1983. [19] Lenci S, Menditto G. Weak interface in long fiber composites. Int J Solids Struct 2000;37:4239–60.