Electroelastic analysis of piezoelectric ceramics with surface electrodes

PERGAMON

International Journal of Engineering Science 36 (1998) 1001±1009

Electroelastic analysis of piezoelectric ceramics with surface electrodes Y. Shindo a, *, F. Narita a, H. Sosa b a

Department of Materials Processing, Graduate School of Engineering, Tohoku University, Aramaki Aza-Aoba, AobaKu, Sendai 980-8579, Japan b Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, USA Received 6 August 1997; accepted 24 November 1997

Abstract Following the theory of liner piezoelectricity, we consider the static behavior of the elastic and electric variables in the vicinity of a surface electrode attached to a piezoelectric ceramic. Fourier transforms are used to reduce the mixed boundary value problem to the solution of a pair of dual integral equations. The integral equations are solved exactly and the displacement and electric potential are expressed in closed form. # 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction Piezoelectric ceramics have recently been used as large displacement actuators. Such actuators rely on multilayer concepts, wherein electrodes alternate with ceramic layers. In these new ®elds of application, severe mechanical stressing occurs during operation. For example, in the case of multilayer stacks, the electrodes that terminate an edge inside the ceramic are a source of electric ®eld concentration which can result in stress concentrations high enough to fracture the parts [1, 2]. Failure prevention could be made possible by a knowledge of the service life of piezoelectric components [3, 4]. The electroelastic interaction of piezoelectric ceramics with surface electrodes is also of importance in piezoelectric transducers and various surface wave devices. Accordingly, there is a need to investigate the e€ects produced by surface electrodes in order to understand mechanical and electric failure phenomena and to improve the device's design and performance. Recently, Castro and Sosa [5] conducted the studies from a two* Corresponding author. Tel.: +81-22-217-7341; Fax: +81-22-217-7341; E-mail: [email protected]. 0020-7225/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 9 8 ) 0 0 0 0 7 - X

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dimensional point of view by considering the simplest possible con®guration, namely a strip electrode of ®nite length attached to the surface of a semi-in®nite dielectric solid. Two di€erent approximate boundary conditions were used to model the electrode. Sosa and Castro [6] also studied two fundamental problems of the linear theory of piezoelectricity from a theoretical point of view, namely, the cases of a point force and a point charge acting on the piezoelectric half-plane. Hom and Shankar [7] demonstrated the ®nite element technique by solving the problem of a multilayered actuator and computed both the electric ®eld and stress state near the tip of an internal electrode. In this paper, the static behavior of the elastic and electric variables in the vicinity of a surface electrode attached to a piezoelectric ceramic is studied based on the linear theory of piezoelectricity. Fourier transforms are employed to reduce the electroelastic problem to the solution of a pair of dual integral equations. The integral equations are solved exactly and the displacement and electric potential are expressed in closed form. Numerical calculations are carried out and the stresses and electric displacements in the neighborhood of the electrode's tip are shown graphically for a piezoelectric ceramic. 2. Statement of problem and fundamental equations We consider a two dimensional piezoelectric medium described by Cartesian coordinates vxv < 1 and z>0 and assume plane strain perpendicular to the y-axis. The z-axis is assumed to coincide with the six fold axis of symmetry in the class of a 6 mm crystal class, or with the poling axis in the case of poled piezoelectric ceramics. Consider a piezoelectric body with a perfectly conducting electrode of length 2a attached to a portion of its surface as shown in Fig. 1. The constitutive relationships can be written as sxx ˆ c11 ux,x ‡ c13 uz,z ÿ e31 Ez

szx ˆ c44 …ux,z ‡ uz,x † ÿ e15 Ex

Dx ˆ e15 …ux,z ‡ uz,x † ‡ E11 Ex

Dz ˆ e31 ux,x ‡ e33 uz,z ‡ E33 Ez

szz ˆ c13 ux,x ‡ c33 uz,z ÿ e33 Ez …1† …2†

where sxx, szx=sxz, szz are the stresses, Dx and Dz are the electric displacements, ux and uz are the displacements, Ex and Ez are the electric ®eld intensities, c11, c33, c44, c13 are the elastic sti€ness constants measured in a constant electric ®eld, E11, E33 are the dielectric constants measured at constant strain, e31, e33, e15 are the piezoelectric constants and a comma denotes partial di€erentiation with respect to the coordinates. The electric ®eld components may be written in terms of an electric potential f(x, z) as Ex ˆ ÿf,x

Ez ˆ ÿf,z

…3†

The ®eld equations are obtained as c11 ux,xx ‡ c44 ux,zz ‡ …c13 ‡ c44 †uz,xz ‡ …e31 ‡ e15 †f,xz ˆ 0 …c13 ‡ c44 †ux,xz ‡ c44 uz,xx ‡ c33 uz,zz ‡ e15 f,xx ‡ e33 f,zz ˆ 0

…4†

Y. Shindo et al. / International Journal of Engineering Science 36 (1998) 1001±1009

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Fig. 1. A piezoelectric ceramic with a surface electrode.

…e31 ‡ e15 †ux,xz ‡ e15 uz,xx ‡ e33 uz,zz ÿ E11 f,xx ÿ E33 f,zz ˆ 0

…5†

We assume that the mixed boundary conditions are given by szz …x, 0† ˆ 0

…6†

szx …x, 0† ˆ 0

…7†



…0Rjxj
f…x, 0† ˆ V0 Dz …x, 0† ˆ 0

…8†

where V0 is a constant voltage. Because of the assumed symmetry, it is sucient to consider the problem for 0 Rx < 1, 0 R z < 1 only.

3. Solution procedure The Fourier transform is applied on Eqs. (4) and (5) and the result is …1 … 3 3 2X 2X 1 1 ÿgj az aj Aj …a†e sin…ax† da uz ˆ Aj …a†eÿgj az cos…ax† da ux ˆ p jˆ1 0 p jˆ1 gj 0 3 2X bj fˆÿ p jˆ1 gj

…1 0

Aj …a†eÿgj az cos…ax† da

…9†

…10†

where aj ˆ

…e31 ‡ e15 †…c33 g2j ÿ c44 † ÿ …c13 ‡ c44 †…e33 g2j ÿ e15 †

…c44 g2j ÿ c11 †…e33 g2j ÿ e15 † ‡ …c13 ‡ c44 †…e31 ‡ e15 †g2j

bj ˆ

…c44 g2j ÿ c11 †aj ‡ …c13 ‡ c44 † e31 ‡ e15

The functions Aj(a) (j = 1, 2, 3) are as yet unknown and g2j (j = 1, 2, 3) are the roots of following characteristic equation:

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Y. Shindo et al. / International Journal of Engineering Science 36 (1998) 1001±1009

c44 …c33 E33 ‡ e233 †g6 ‡ fÿ2c44 e15 e33 ÿ c11 e233 ÿ c33 …c44 E11 ‡ c11 E33 † ‡ E33 …c13 ‡ c44 †2 ‡ 2e33 …c13 ‡ c44 †…e31 ‡ e15 † ÿ c244 E33 ÿ c33 …e31 ‡ e15 †2 gg4 ‡ f2c11 e15 e33 ‡ c44 e215 ‡ c11 …c33 E11 ‡ c44 E33 † ÿ E11 …c13 ‡ c44 †2 ÿ 2e15 …c13 ‡ c44 †…e31 ‡ e15 † ‡ c244 E11 ‡ c44 …e31 ‡ e15 †2 gg2 ÿ c11 …c44 E11 ‡ e215 † ˆ 0

…12†

Through Eqs. (6) and (7), the unknowns A2(a), A3(a) are related to the unknown A1(a) by the following equations: 3 X

pj Aj …a† ˆ 0

…13†

qj Aj …a† ˆ 0

…14†

jˆ1 3 X jˆ1

where pj ˆ c13 aj ÿ c33 ‡ e33 bj

  1 e15 bj qj ˆ c44 aj gj ‡ ÿ gj gj

…15†

From the mixed boundary conditions, Eq. (8), we obtain the following pair of dual integral equations for the determination of the unknown function A1(a): …1 …1 V0 A1 …a†cos…ax† da ˆ …0Rx
rj ˆ

bj gj

s1 ˆ 1,

3 2X rj sj p jˆ1

…17†

… j ˆ 1, 2, 3† p3 q1 ÿ p1 q3 s2 ˆ , p2 q3 ÿ p3 q2

…18† 

q1 q2 p3 q1 ÿ p1 q3 s3 ˆ ÿ ‡ q3 q3 p2 q3 ÿ p3 q2



The dual integral equations in Eq. (16) lead to the following expression:

…19†

Y. Shindo et al. / International Journal of Engineering Science 36 (1998) 1001±1009

p 2

…1 0

V0 A1 …a†J0 …xa† da ˆ H…a ÿ x† k1

1005

…x

1 p dx 2 0 x ÿ x2

…20†

where J0() is the zero-order Bessel function of the ®rst kind and H() is the Heaviside step function. By taking the inverse Hankel transform of Eq. (20) and solving the resulting equation for A1(a) in terms of the constant V0, the function A1(a) is found to be 2 V0 A1 …a† ˆ p k1

…a 0

xJ0 …xa† dx

…x

1 aV0 p dx ˆ J1 …aa† 2 k1 a 0 x ÿ x2

…21†

where J1() is the one-order Bessel function of the ®rst kind. The unknowns Aj(a)(j = 1, 2, 3) are related to the constant V0 by the following equation: Aj …a† ˆ sj

aV0 J1 …aa† k1 a

…22†

By substituting Eq. (22) into Eqs. (9) and (10), the displacements and electric potential may be obtained as follows: 3 p 2V0 X aj sj xf1 ÿ 2gj zf ÿ1=2 g ux …x, z† ˆ j pk1 jˆ1   3 2V0 X 1 1=2 rj sj p f j ÿ gj z f…x, z† ˆ ÿ pk1 jˆ1 2

3 2V0 X sj uz …x, z† ˆ pk1 jˆ1 gj



1 p f 2

1=2 j

 ÿ gj z …23†

where q fj ˆ gj ‡ g2j ‡ 4g2j z2 x 2

…24†

gj ˆ g2j z2 ÿ x 2 ‡ a2

…25†

The stresses, electric ®eld intensities and electric displacements can be obtained by making use of Eqs. (1)±(3). Table 1 Material properties of PZT±5H ceramic Elastic sti€nesses (1010 N/m2)

Piezoelectric coecients (C/m2)

Dielectric constants (10ÿ10 C/V m) E11

c11

c33

c44

c13

e31

e33

e15

PZT-5H 12.6

11.7

3.53

5.3

ÿ6.5

23.3

17.0

151

E33 130

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Y. Shindo et al. / International Journal of Engineering Science 36 (1998) 1001±1009

Fig. 2. Stress szz in depth direction: (a) x/a = 0.0, 0.5, 1.0 and 1.5; (b) x/a = 0.99, 1.00 and 1.01 and (c) x/a = 1.0.

Y. Shindo et al. / International Journal of Engineering Science 36 (1998) 1001±1009

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Fig. 3. Stress szx in depth direction: (a) x/a = 0.0, 0.5, 1.0 and 1.5; (b) x/a = 0.99, 1.00 and 1.01 and (c) x/a = 1.0.

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Y. Shindo et al. / International Journal of Engineering Science 36 (1998) 1001±1009

Fig. 4. Electric displacement Dx in depth direction: (a) x/a = 0.0, 0.5, 1.0 and 1.5; (b) x/a = 0.99, 1.00 and 1.01 and (c) x/a = 1.0.

Y. Shindo et al. / International Journal of Engineering Science 36 (1998) 1001±1009

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4. Numerical results and discussion To examine the e€ect of electroelastic interactions on the stresses and electric displacement in the neighborhood of the electrode's tip, numerical calculations are carried out. We can take PZT-5H ceramic as an example of which the engineering material constants are listed in Table 1 [8]. In Fig. 2, the distribution of the normal component of stress szza/V0 is shown as a function of the ratio z/a. Fig. 2(a) shows the results for x/a = 0.0, 0.5, 1.0 and 1.5. The results for x/ a = 0.99, 1.00 and 1.01 are also displayed in Fig. 2(b). The normal component of stress for the ratio x/a = 1.0 increases as the ratio z/a decreases. It is interesting to explore the behavior of the stress near the tip of the electrode. Fig. 2(c) shows the distribution of the normal component of stress in the depth direction for x/a = 1.0. The normal component of stress tends to increase with increasing the ratio z/a reaching a peak and then decreases in magnitude. The peak value of szza/V0 is found to be szza/V0=41.0 (N/V m). Fig. 3 and Fig. 4 give the distributions of the shear component of stress szxa/V0 and the x-component of electric displacement Dxa/V0 corresponding to szza/V0 shown in Fig. 2. The shear component of stress for x/a = 1.0 declines at ®rst and then increases with increasing the ratio z/a reaching a peak. The x-component of electric displacement for x/a = 1.0 begins to climb until it reaches a peak. The peak values of the shear component of stress and the x-component of electric displacement are found to be szxa/V0=10.7 (N/V m) and Dxa/V0=35.3  10ÿ8 (C/V m), respectively. In conclusion, the linear electroelastic problem for a piezoelectric ceramic with a surface electrode has been analyzed theoretically. It is found that high values of stresses and electric displacement arise in the neighborhood of the electrode tip. These values are suciently high to produce mechanical and electrical failure. Due to the concentration the stresses can exceed the strength of the piezoelectric material in the neighborhood of the surface electrode edge. References [1] [2] [3] [4] [5]

Sosa HA, Pak YE. Int J Solids Struct 1990;26:1. Hao TH, Gong X, Suo Z. J Mech Phys Solids 1996;44:23. Shindo Y, Narita F, Tanaka K. Theor Appl Fract Mech 1996;25:65. Shindo Y, Tanaka K, Narita F. Acta Mech 1997;120:31. Castro MA, Sosa HA. ASME mechanics and materials for electronic packaging. Vol. 3. Coupled ®eld behavior in materials. AMD 1994;193:13. [6] Sosa HA, Castro MA. J Mech Phys Solids 1994;42:1105. [7] Hom CL, Shankar N. Int J Solids Struct 1996;33:1757. [8] Shindo Y, Ozawa E, Nowacki JP. Int J Appl Electromagn Mater 1990;1:77.