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Domain switching effect on fracture of piezoelectric solids
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 30 (2003) 267–275 www.elsevier.com/locate/mechrescom
Domain switching effect on fracture of piezoelectric solids Wenjun Zeng a, Majid T. Manzari
b,*
, James D. Lee a, Yin-Lin Shen
a
a
Department of Mechanical and Aerospace Engineering, The George Washington University, 801 22nd Street N.W., Washington, DC 20052, USA b Department of Civil and Environmental Engineering, The George Washington University, Academic Center, T635, 801 22nd Street N.W., Washington, DC 20052, USA Received 26 April 2002
Abstract Rational design of smart sensors and actuators that consist of piezoelectric solids requires a thorough understanding of the constitutive behavior of this material under mechanical and electrical loading. Domain switching is the cause of significant nonlinearity in the constitutive behavior of piezoelectric solids, which may be enhanced in the presence of cracks. In this paper, the response of piezoelectric solids is formulated by coupling thermal, electrical, and mechanical effects. The corresponding finite element equations are derived and applied in the solution of the piezoelectric center crack problems. The effects of domain switching are evaluated on the near tip stress intensity factors. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Domain switching; Piezoelectricity; Electro-mechanical coupling; Finite element; Fracture behavior
1. Introduction Piezoelectric materials are widely used in various electrical and mechanical equipment and devices such as ultrasonic generators, filters, sensors, and actuators. Because of their importance, the fracture behavior of piezoelectric materials has attracted a surge of interest in recent years. Tobin and Pak (1993) conducted several indentation experiments on piezoelectric solids. The results of these experiments indicate that the fracture toughness of piezoelectric materials vary with the applied electric fields. Park and Sun (1995a) conducted compact tension and three-point bending tests on piezoelectric specimens. They observed that the electric fields significantly affect the fracture loads. For cracks perpendicular to the poling directions, a positive electric field parallel to the poling direction tends to open the crack, while a negative electric field tends to close it. These experimentally observed phenomena contradict the results of the simulations using linear electro-elastic fracture mechanics presented by Pak (1992) and Park and Sun (1995b). The linear fracture mechanics for piezoelectric materials imply that the applied electrical field has no influence on the crack propagation normal to the poling direction if the stress intensity
*
Corresponding author. Fax: +1-202-994-0127. E-mail address: [email protected] (M.T. Manzari).
0093-6413/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0093-6413(03)00003-X
268
W. Zeng et al. / Mechanics Research Communications 30 (2003) 267–275
factor is used to characterize the piezoelectric crack. Park and Sun (1995a) proposed the mechanical strain energy release rate instead of the total energy release rate as a fracture criterion for piezoelectric ceramics. Their results show that the mechanical strain energy release rate may be increased or decreased, depending on the direction of electric loading. Park and Sun (1995a) also indicated that all interpretations of these results are based on linear piezoelectricity and there may exist factors beyond the scope of linear piezoelectricity that would affect fracture. Domain switching, which refers to the change of the polarization direction in piezoelectric materials under certain electric or mechanical effects, is observed in piezoelectric materials. Hwang et al. (1995) observed the nonlinear behavior of PLZT (lead lanthanum zirconate titanate) under electrical and mechanical loading. When the polarized ceramic is subjected to compressive stress parallel to the polarization direction, the high magnitude compressive stress changes the polarization of the ceramic by 90° switching. Lynch (1996) conducted compression tests on PLZT specimens and demonstrated the importance of considering nonlinear response when assessing a piezoelectric material for actuator applications. Jiang (1998) performed experimental tests for 180° domain switching in PZT-4 (lead zirconate titanate) under combined electric and compressive loads. These experimental studies show a significant nonlinearity in the response of the material due to domain switching. It is believed that the nonlinear effect caused by domain switching around the crack tip may affect the fracture behavior of piezoelectric materials. Hence it is necessary to develop a suitable computational model and to include the domain switching effect when analyzing the fracture behavior of piezoelectric materials. Numerical studies including the effect of domain switching have been performed by several researchers. Hwang et al. (1995) presented a numerical simulation of the nonlinear behavior of PLZT by using Preisach hysteresis model for each grain. Steinkoff (1999) proposed a finite-element model to study the typical behavior of ferroelastic, ferroelectric and piezoelectric materials. Jiang (1998) and Sun and Chang (2000) presented a number of nonlinear finite element solutions that included the effect of domain switching in the analysis of the piezoelectric materials with cracks perpendicular to the poling direction. They found that a negative electric field increases the fracture toughness while a positive electric field leads to a reduction of fracture toughness. These results were consistent with the experimental observations. In this paper, a general finite element formulation for the analysis of problems involving the coupling of thermal, electrical and mechanical effects is presented. The underlying constitutive equations include the nonlinear effects caused by domain switching. A center crack problem is then analyzed to investigate the effect of domain switching on fracture behavior of piezoelectric solids. The finite element solutions are interpreted in the light of the previously obtained experimental results by other investigators.
2. Formulation The coupling of thermal, mechanical, and electrical effects in a piezoelectric solid is formulated in a quasi-static regime. A finite element approach is proposed. 2.1. Balance laws The fundamental laws of continuum mechanics, without considering magnetic field, can be written as (Eringen and Maugin, 1990) Conservation of mass
q_ þ qti;i ¼ 0
Balance of linear momentum
tkl;k þ qðfl t_ l Þ þ FlE ¼ 0
ð1Þ ð2Þ
W. Zeng et al. / Mechanics Research Communications 30 (2003) 267–275
Balance of moment of momentum Conservation of energy
E tij
¼ Ejit
q_e tkl tl;k þ qk;k qh þ Pk E_ k Jk Ek ¼ 0
Clausius–Duhem inequality
1 qðw_ þ gT_ Þ þ tkl tl;k qk T ;k Pk E_ k Jk Ek P 0 T
269
ð3Þ ð4Þ ð5Þ
where FlE ¼ qe El þ Pk El;k
ð6Þ
Pk ¼ Dk e0 Ek
ð7Þ
e ¼ w þ gT
ð8Þ
E
t, E t, q, f, t, F , e, q, h, P, E, J, w, g, T , qe , D, e0 are stress tensor, symmetric part of stress tensor, mass density per unit volume, body force density per unit mass (excluding electromagnetic force), velocity, electromagnetic body force density per unit volume, internal energy density, heat flux, heat source per unit mass, polarization, electric field intensity, electric current density, free-energy density, entropy density, absolute temperature, electric charge density, dielectric displacement and dielectric constant, respectively. In static case without considering magnetic field, the following equations can be derived from MaxwellÕs equations and Eqs. (2), (4) and inequality (5) tkl;k þ qfl þ FlE ¼ 0
ð9Þ
Ek ¼ /;k
ð10Þ
qk;k ¼ qh
ð11Þ
1 qk T ; k P 0 T where / is electric potential.
ð12Þ
2.2. Constitutive equations If temperature T , strain e~mn and electric field Em are chosen as the independent constitutive variables, the constitutive equations can be written as (Eringen and Maugin, 1990) qk ¼ jkl T;l E tkl
¼ akl bkl T þ rklmn e~mn emkl Em
Pk ¼ Pk0 þ -k T þ ekim e~im þ vEki Ei
ð13Þ ð14Þ ð15Þ
where jkl , akl , bkl , rklmn , emkl , Pk0 , -k , vEki are heat conduction coefficients, the stresses at the reference state, thermal stress moduli, elastic moduli, piezoelectric moduli, the polarization at the reference state, pyroelectric polarizability and dielectric susceptibility, respectively. E tkl is the symmetric part of the stress tensor. The stress tensor and dielectric displacement can be expressed as (Dixon and Eringen, 1965) ~ k T þ ekim e~im ðeki e0 dki Þ/;i Þ/;l tkl ¼ E tkl Pk El ¼ akl bkl T þ rklmn e~mn þ emkl /;m þ ðPk0 þ -
ð16Þ
~ k T þ ekim e~im eki /;i Dk ¼ Pk0 þ -
ð17Þ
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W. Zeng et al. / Mechanics Research Communications 30 (2003) 267–275
where eki ¼ vEki þ e0 dki
ð18Þ
eki and dki are dielectric permittivity and Kronecker delta, respectively (Eringen and Maugin, 1990). 2.3. Constitutive equations after domain switching For piezoelectric materials, an applied stress will deform the crystal structure thereby inducing a relative displacement of the positive and negative ions and thus giving rise to electric charges. Similarly, an applied electric field will change the relative position of the positive and negative ions, which in turn causes deformation of the crystal structure and thus induces strain (Hwang et al., 1995). When an applied stress or electric field exceeds the coercive level that is determined by the change of the internal energy, the poling direction of the piezoelectric materials will change. This phenomenon is called domain switching. Hwang et al. (1995) proposed a Ôdomain switchingÕ criterion based on energy considerations. The criterion requires that the combined electrical and mechanical work exceed the critical value of Gc , i.e. ejk P Gc G ¼ Ei DDi þ tjk D~
ð19Þ c
where the symbols G, E, D, t, ~e, G refer to the change of internal energy, electric field, dielectric displacement, stresses, strains and the critical value of G, respectively. The terms DDi and D~ ejk in Eq. (19) indicate the change in dielectric displacement and the change in strains, respectively. After domain switching, the material remains linear but possesses new constants following the coordinate transformation law corresponding to the direction of polarization switching. In addition, after domain switching the spontaneous stress and spontaneous polarization are included in the constitutive equations to account for the change of the crystal structure (Jiang, 1998; Sun and Chang, 2000). Therefore, the constitutive equations after domain switching can be written as qk ¼ jkl T;l
ð20Þ
0 tkl ¼ a0kl bkl T þ rklmn e~mn þ e0mkl /;m þ Pk0 þ -k T þ e0kim e~im ðeki e0 dki Þ/;i /;l
ð21Þ
0
Dk ¼ Pk0 þ -k T þ e0kim e~im eki /;i
ð22Þ
where s a0kl ¼ akl þ tkl ¼ akl rklij e~sij 0
Pk0 ¼ Pk0 þ Dsk ¼ Pk0 e0kij e~sij
ð23Þ ð24Þ
For 180° domain switching in the X1 X3 plane with X3 as the original poling direction, e~sij ¼ 0 P1s
¼ 0;
ði; j ¼ 1; 3Þ
ð25Þ
P3s ¼ 2P s
For 90° domain switching in the X1 X3 plane with X3 as the original poling direction, e~s11 ¼ e~s ; P1s ¼ P s ;
e~s33 ¼ ~ es ; P3s ¼ P s
e~s13 ¼ e~s31 ¼ 0
ð26Þ
where the spontaneous strain e~s and the spontaneous polarization P s are assumed to be material constants. The spontaneous polarization P s is used in the criterion (Jiang, 1998; Sun and Chang, 2000).
W. Zeng et al. / Mechanics Research Communications 30 (2003) 267–275
271
The new piezoelectric constants e0ijk are obtained from the coordinate transformation law, i.e. e0ijk ¼ Tim Tjn Tkp emnp
ð27Þ
where Tij is the transformation tensor. It is important to note that the constitutive model outlined above leads to a tri-linear approximation of the stress–strain relationship and has been used by other investigators for modeling domain switching phenomenon (e.g., (Sun and Chang, 2000)). Fig. 1 shows how the constitutive model approximates the actual stress–strain behavior of a typical piezoelectric material. The first and last branches of the stress– strain curve are approximated by the linear piezoelectric responses of the material before and after completion of the domain switching. The transition from the initial linear part to the final linear part is approximated by a spontaneous strain (and spontaneous polarization) which allows the initial linear branch be shifted and rotated to the final linear branch. This is what we refer to as a tri-linear approximation. A number of more advanced constitutive models have also become available recently. Some of these models can closely reproduce the essential features of the constitutive behavior of ferroelectric ceramics in electrical and mechanical loading [e.g., (Kamlah and Tsakmakis, 1999) among others]. However, for the purpose of this study the simple constitutive model outlined above will be sufficient in capturing the salient features of electrical and mechanical nonlinearity in a piezoelectric solid. 2.4. Finite element analysis Here we assume that the domain of interest is divided into ne elements, each element is associated with n nodal points and, correspondingly, n shape functions. Then the displacement, temperature and electrical field within an element can be obtained via the corresponding nodal values of that element as (Zienkiewicz and Taylor, 1989) ui ¼ Nia Ua
ða ¼ 1; 2; . . . ; 3nÞ ði ¼ 1; 2; 3Þ
ð28Þ
250
Normal Stress t 33 (MPa)
200
150
100
50
Model Experiment
0 0
2000
4000
-6
6000
8000
Normal Strain e 33 (x10 )
Fig. 1. The constitutive model approximation of the stress–strain behavior of a typical piezoelectric material.
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Fig. 2. Center crack problem: (a) configuration (unit: mm); (b) poling direction is X1 (tensile force is applied); (c) finite element mesh (number of elements: 3000, number of nodal points: 3101).
T ¼ Nb Tb
ðb ¼ 1; 2; . . . ; nÞ
ð29Þ
/ ¼ Nb Ub
ðb ¼ 1; 2; . . . ; nÞ
ð30Þ
W. Zeng et al. / Mechanics Research Communications 30 (2003) 267–275
273
3/2
Stress intensity factor KI (MN/m )
1.6 1.4 1.2 1 0.8 0.6 0.4
E=-0.5 Mv/m E=0 Mv/m
0.2
E=0.5 Mv/m 0 0
0.1
(a)
0.2
0.3
0.4
0.5
0.6
0.7
Normalized distance x/a (%)
3/2
Stress intensity factor KI (MN/m )
1.6 1.4 1.2 1 0.8 0.6 0.4
E=-0.5 Mv/m E=0 Mv/m
0.2
E=0.5 Mv/m 0 0
0.1
(b)
0.2
0.3
0.4
0.5
0.6
0.7
Normalized distance x/a (%)
Fig. 3. Stress intensity factor KI vs. normalized distance x=a (t ¼ 3:0 MPa): (a) without domain switching; (b) including domain switching.
Before domain switching, the finite element equations can be obtained as Kab Ua þ Pab Ta þ Cab Ua þ Rabd Ua Ud þ Sabd Ta Ud þ Vabd Ua Ud Fb ¼ 0
ð31Þ
Hab Ta Qb ¼ 0
ð32Þ
Mab Ua þ Wab Ta þ Lab Ua Yb ¼ 0
ð33Þ
Similarly, after domain switching, the finite element equations are in the same format as Eqs. (32), (33) and (34). Matrices and vectors K, P, S, V, H, Q, W and L remain unchanged but matrices and vectors C, R, F, M and Y change in accordance with the transformation of tensors due to the change of coordinate system (see (Zeng et al., 2003) for details).
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W. Zeng et al. / Mechanics Research Communications 30 (2003) 267–275
3. Numerical results and discussion The computer program developed based on the formulation presented in Section 2 is used to analyze the center crack problem shown in Fig. 2(a). Due to the symmetry, only a quarter of the specimen is analyzed (Fig. 2(b)). The crack surfaces are assumed to be electrically insulated surfaces. In this work, four-node plain strain elements are employed. All of the stresses reported are stresses at the center of elements. Fig. 2(c) presents the finite element mesh. The total number of elements is 3000 and the element size near the crack tip is about 0.0004% of the crack length. It is shown in Fig. 2(b) that the tensile force is applied and the original poling direction of PZT is parallel to the crack (parallel to X1 -direction). The Mode I near tip stress intensity factors ahead of crack tip and along X1 axis are used to characterize the fracture behavior of piezoelectric crack (Sih and Liebowitz, 1968). Fig. 3 shows the results of fully coupled electromechanical analysis of the cracked plate under tension with and without domain switching. Fig. 3(a) indicates the fact that the electric fields have influence on the stress intensity factors when domain switching is disabled in the program. The positive electric field leads to smaller stress intensity factor but the case with negative electric field shows larger stress intensity factor. Fig. 3(b) shows the stress intensity factors calculated for three different electric fields. If the peak stress intensity factors are compared for different electric fields, it is clear that a negative E increases the stress intensity factor while a positive E reduces it. Since the tensile force is applied, it means that a negative E tends to increase the tensile stress while a positive E tends to reduce it. Therefore, a negative E tends to open the crack while a positive E tends to close the crack. Hence, it appears that the original poling direction and the direction of applied electric field play a role when domain switching is considered.
4. Conclusions The domain switching effect is included in a general 3-D finite element code to solve the fully coupled thermal-electro-mechanical problem. A center crack problem is analyzed. The distribution of the stress intensity factors ahead of the crack tip is obtained. It is observed that in the near tip region, the stress intensity factors are significantly affected by domain switching. Therefore, it appears that the effect of domain switching must be considered in the evaluation of the fracture toughness of piezoelectric material. In the case that poling direction of PZT is parallel to the crack, if a tensile force is applied to the specimen, it is found that a negative E increases the peak stress intensity factor while a positive E reduces it.
References Dixon, R.C., Eringen, A.C., 1965. A dynamical theory of polar elastic dielectrics––I. Int. J. Eng. Sci. 3, 359–377. Eringen, A.C., Maugin, G.A., 1990. Electrodynamics of Continua I. Springer-Verlag, New York. Hwang, S.C., Lynch, C.S., McMeeking, R.M., 1995. Ferroelectric/ferroelastic interactions and a polarization switching model. Acta Metall. Mater. 43, 2073–2084. Jiang, L., 1998. Fracture and fatigue behavior of piezoelectric materials. Ph.D. Thesis, Purdue University. Kamlah, M., Tsakmakis, C., 1999. Phenomenological modeling of the non-linear electro-mechanical coupling in ferroelectrics. Int. J. Solids Struct. 36 (5), 669–695. Lynch, C.S., 1996. The effect of uniaxial stress on the electro-mechanical response of 8/65/35 PLZT. Acta Mater. 44, 4137–4148. Pak, Y.E., 1992. Linear electro-elastic fracture mechanics of piezoelectric materials. Int. J. Fract. 54, 79–100. Park, S., Sun, C.T., 1995a. Fracture criteria for piezoelectric ceramics. J. Am. Ceram. Soc. 78, 1475–1480. Park, S., Sun, C.T., 1995b. Effect of electric field on fracture of piezoelectric ceramics. Int. J. Fract. 70, 203–216. Sih, G.C., Liebowitz, H., 1968. Mathematical theories of brittle fracture. In: Liebowitz, H. (Ed.), Fracture. Academic Press, New York.
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Steinkoff, T., 1999. Finite-element modeling of ferro-electric domain switching in piezoelectric ceramics. J. Euro. Ceram. Soc. 19, 1247– 1249. Sun, C.T., Chang, I., 2000. Singular stress field near crack tip in piezoceramics under electrical and mechanical loading. In: Proceedings of the International Conference on Mesomechanics, II. pp. 819–826. Tobin, A.G., Pak, Y.E., 1993. Effect of electric fields on fracture behavior of PZT ceramics. In: Proceedings of the SPIE, Smart Structures and Materials, vol. 1916. pp. 78–86. Zeng, W., Manzari, M.T., Lee, J.D., Shen, Y., 2003. Fully-coupled nonlinear analysis of piezoelectric solids involving domain switching. Int. J. Num. Meth. Eng. 56, 13–34. Zienkiewicz, O.Z., Taylor, R.L., 1989. The Finite Element Method. McGraw-Hill, England.
Mechanics Research Communications 30 (2003) 267–275 www.elsevier.com/locate/mechrescom
Domain switching effect on fracture of piezoelectric solids Wenjun Zeng a, Majid T. Manzari
b,*
, James D. Lee a, Yin-Lin Shen
a
a
Department of Mechanical and Aerospace Engineering, The George Washington University, 801 22nd Street N.W., Washington, DC 20052, USA b Department of Civil and Environmental Engineering, The George Washington University, Academic Center, T635, 801 22nd Street N.W., Washington, DC 20052, USA Received 26 April 2002
Abstract Rational design of smart sensors and actuators that consist of piezoelectric solids requires a thorough understanding of the constitutive behavior of this material under mechanical and electrical loading. Domain switching is the cause of significant nonlinearity in the constitutive behavior of piezoelectric solids, which may be enhanced in the presence of cracks. In this paper, the response of piezoelectric solids is formulated by coupling thermal, electrical, and mechanical effects. The corresponding finite element equations are derived and applied in the solution of the piezoelectric center crack problems. The effects of domain switching are evaluated on the near tip stress intensity factors. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Domain switching; Piezoelectricity; Electro-mechanical coupling; Finite element; Fracture behavior
1. Introduction Piezoelectric materials are widely used in various electrical and mechanical equipment and devices such as ultrasonic generators, filters, sensors, and actuators. Because of their importance, the fracture behavior of piezoelectric materials has attracted a surge of interest in recent years. Tobin and Pak (1993) conducted several indentation experiments on piezoelectric solids. The results of these experiments indicate that the fracture toughness of piezoelectric materials vary with the applied electric fields. Park and Sun (1995a) conducted compact tension and three-point bending tests on piezoelectric specimens. They observed that the electric fields significantly affect the fracture loads. For cracks perpendicular to the poling directions, a positive electric field parallel to the poling direction tends to open the crack, while a negative electric field tends to close it. These experimentally observed phenomena contradict the results of the simulations using linear electro-elastic fracture mechanics presented by Pak (1992) and Park and Sun (1995b). The linear fracture mechanics for piezoelectric materials imply that the applied electrical field has no influence on the crack propagation normal to the poling direction if the stress intensity
*
Corresponding author. Fax: +1-202-994-0127. E-mail address: [email protected] (M.T. Manzari).
0093-6413/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0093-6413(03)00003-X
268
W. Zeng et al. / Mechanics Research Communications 30 (2003) 267–275
factor is used to characterize the piezoelectric crack. Park and Sun (1995a) proposed the mechanical strain energy release rate instead of the total energy release rate as a fracture criterion for piezoelectric ceramics. Their results show that the mechanical strain energy release rate may be increased or decreased, depending on the direction of electric loading. Park and Sun (1995a) also indicated that all interpretations of these results are based on linear piezoelectricity and there may exist factors beyond the scope of linear piezoelectricity that would affect fracture. Domain switching, which refers to the change of the polarization direction in piezoelectric materials under certain electric or mechanical effects, is observed in piezoelectric materials. Hwang et al. (1995) observed the nonlinear behavior of PLZT (lead lanthanum zirconate titanate) under electrical and mechanical loading. When the polarized ceramic is subjected to compressive stress parallel to the polarization direction, the high magnitude compressive stress changes the polarization of the ceramic by 90° switching. Lynch (1996) conducted compression tests on PLZT specimens and demonstrated the importance of considering nonlinear response when assessing a piezoelectric material for actuator applications. Jiang (1998) performed experimental tests for 180° domain switching in PZT-4 (lead zirconate titanate) under combined electric and compressive loads. These experimental studies show a significant nonlinearity in the response of the material due to domain switching. It is believed that the nonlinear effect caused by domain switching around the crack tip may affect the fracture behavior of piezoelectric materials. Hence it is necessary to develop a suitable computational model and to include the domain switching effect when analyzing the fracture behavior of piezoelectric materials. Numerical studies including the effect of domain switching have been performed by several researchers. Hwang et al. (1995) presented a numerical simulation of the nonlinear behavior of PLZT by using Preisach hysteresis model for each grain. Steinkoff (1999) proposed a finite-element model to study the typical behavior of ferroelastic, ferroelectric and piezoelectric materials. Jiang (1998) and Sun and Chang (2000) presented a number of nonlinear finite element solutions that included the effect of domain switching in the analysis of the piezoelectric materials with cracks perpendicular to the poling direction. They found that a negative electric field increases the fracture toughness while a positive electric field leads to a reduction of fracture toughness. These results were consistent with the experimental observations. In this paper, a general finite element formulation for the analysis of problems involving the coupling of thermal, electrical and mechanical effects is presented. The underlying constitutive equations include the nonlinear effects caused by domain switching. A center crack problem is then analyzed to investigate the effect of domain switching on fracture behavior of piezoelectric solids. The finite element solutions are interpreted in the light of the previously obtained experimental results by other investigators.
2. Formulation The coupling of thermal, mechanical, and electrical effects in a piezoelectric solid is formulated in a quasi-static regime. A finite element approach is proposed. 2.1. Balance laws The fundamental laws of continuum mechanics, without considering magnetic field, can be written as (Eringen and Maugin, 1990) Conservation of mass
q_ þ qti;i ¼ 0
Balance of linear momentum
tkl;k þ qðfl t_ l Þ þ FlE ¼ 0
ð1Þ ð2Þ
W. Zeng et al. / Mechanics Research Communications 30 (2003) 267–275
Balance of moment of momentum Conservation of energy
E tij
¼ Ejit
q_e tkl tl;k þ qk;k qh þ Pk E_ k Jk Ek ¼ 0
Clausius–Duhem inequality
1 qðw_ þ gT_ Þ þ tkl tl;k qk T ;k Pk E_ k Jk Ek P 0 T
269
ð3Þ ð4Þ ð5Þ
where FlE ¼ qe El þ Pk El;k
ð6Þ
Pk ¼ Dk e0 Ek
ð7Þ
e ¼ w þ gT
ð8Þ
E
t, E t, q, f, t, F , e, q, h, P, E, J, w, g, T , qe , D, e0 are stress tensor, symmetric part of stress tensor, mass density per unit volume, body force density per unit mass (excluding electromagnetic force), velocity, electromagnetic body force density per unit volume, internal energy density, heat flux, heat source per unit mass, polarization, electric field intensity, electric current density, free-energy density, entropy density, absolute temperature, electric charge density, dielectric displacement and dielectric constant, respectively. In static case without considering magnetic field, the following equations can be derived from MaxwellÕs equations and Eqs. (2), (4) and inequality (5) tkl;k þ qfl þ FlE ¼ 0
ð9Þ
Ek ¼ /;k
ð10Þ
qk;k ¼ qh
ð11Þ
1 qk T ; k P 0 T where / is electric potential.
ð12Þ
2.2. Constitutive equations If temperature T , strain e~mn and electric field Em are chosen as the independent constitutive variables, the constitutive equations can be written as (Eringen and Maugin, 1990) qk ¼ jkl T;l E tkl
¼ akl bkl T þ rklmn e~mn emkl Em
Pk ¼ Pk0 þ -k T þ ekim e~im þ vEki Ei
ð13Þ ð14Þ ð15Þ
where jkl , akl , bkl , rklmn , emkl , Pk0 , -k , vEki are heat conduction coefficients, the stresses at the reference state, thermal stress moduli, elastic moduli, piezoelectric moduli, the polarization at the reference state, pyroelectric polarizability and dielectric susceptibility, respectively. E tkl is the symmetric part of the stress tensor. The stress tensor and dielectric displacement can be expressed as (Dixon and Eringen, 1965) ~ k T þ ekim e~im ðeki e0 dki Þ/;i Þ/;l tkl ¼ E tkl Pk El ¼ akl bkl T þ rklmn e~mn þ emkl /;m þ ðPk0 þ -
ð16Þ
~ k T þ ekim e~im eki /;i Dk ¼ Pk0 þ -
ð17Þ
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W. Zeng et al. / Mechanics Research Communications 30 (2003) 267–275
where eki ¼ vEki þ e0 dki
ð18Þ
eki and dki are dielectric permittivity and Kronecker delta, respectively (Eringen and Maugin, 1990). 2.3. Constitutive equations after domain switching For piezoelectric materials, an applied stress will deform the crystal structure thereby inducing a relative displacement of the positive and negative ions and thus giving rise to electric charges. Similarly, an applied electric field will change the relative position of the positive and negative ions, which in turn causes deformation of the crystal structure and thus induces strain (Hwang et al., 1995). When an applied stress or electric field exceeds the coercive level that is determined by the change of the internal energy, the poling direction of the piezoelectric materials will change. This phenomenon is called domain switching. Hwang et al. (1995) proposed a Ôdomain switchingÕ criterion based on energy considerations. The criterion requires that the combined electrical and mechanical work exceed the critical value of Gc , i.e. ejk P Gc G ¼ Ei DDi þ tjk D~
ð19Þ c
where the symbols G, E, D, t, ~e, G refer to the change of internal energy, electric field, dielectric displacement, stresses, strains and the critical value of G, respectively. The terms DDi and D~ ejk in Eq. (19) indicate the change in dielectric displacement and the change in strains, respectively. After domain switching, the material remains linear but possesses new constants following the coordinate transformation law corresponding to the direction of polarization switching. In addition, after domain switching the spontaneous stress and spontaneous polarization are included in the constitutive equations to account for the change of the crystal structure (Jiang, 1998; Sun and Chang, 2000). Therefore, the constitutive equations after domain switching can be written as qk ¼ jkl T;l
ð20Þ
0 tkl ¼ a0kl bkl T þ rklmn e~mn þ e0mkl /;m þ Pk0 þ -k T þ e0kim e~im ðeki e0 dki Þ/;i /;l
ð21Þ
0
Dk ¼ Pk0 þ -k T þ e0kim e~im eki /;i
ð22Þ
where s a0kl ¼ akl þ tkl ¼ akl rklij e~sij 0
Pk0 ¼ Pk0 þ Dsk ¼ Pk0 e0kij e~sij
ð23Þ ð24Þ
For 180° domain switching in the X1 X3 plane with X3 as the original poling direction, e~sij ¼ 0 P1s
¼ 0;
ði; j ¼ 1; 3Þ
ð25Þ
P3s ¼ 2P s
For 90° domain switching in the X1 X3 plane with X3 as the original poling direction, e~s11 ¼ e~s ; P1s ¼ P s ;
e~s33 ¼ ~ es ; P3s ¼ P s
e~s13 ¼ e~s31 ¼ 0
ð26Þ
where the spontaneous strain e~s and the spontaneous polarization P s are assumed to be material constants. The spontaneous polarization P s is used in the criterion (Jiang, 1998; Sun and Chang, 2000).
W. Zeng et al. / Mechanics Research Communications 30 (2003) 267–275
271
The new piezoelectric constants e0ijk are obtained from the coordinate transformation law, i.e. e0ijk ¼ Tim Tjn Tkp emnp
ð27Þ
where Tij is the transformation tensor. It is important to note that the constitutive model outlined above leads to a tri-linear approximation of the stress–strain relationship and has been used by other investigators for modeling domain switching phenomenon (e.g., (Sun and Chang, 2000)). Fig. 1 shows how the constitutive model approximates the actual stress–strain behavior of a typical piezoelectric material. The first and last branches of the stress– strain curve are approximated by the linear piezoelectric responses of the material before and after completion of the domain switching. The transition from the initial linear part to the final linear part is approximated by a spontaneous strain (and spontaneous polarization) which allows the initial linear branch be shifted and rotated to the final linear branch. This is what we refer to as a tri-linear approximation. A number of more advanced constitutive models have also become available recently. Some of these models can closely reproduce the essential features of the constitutive behavior of ferroelectric ceramics in electrical and mechanical loading [e.g., (Kamlah and Tsakmakis, 1999) among others]. However, for the purpose of this study the simple constitutive model outlined above will be sufficient in capturing the salient features of electrical and mechanical nonlinearity in a piezoelectric solid. 2.4. Finite element analysis Here we assume that the domain of interest is divided into ne elements, each element is associated with n nodal points and, correspondingly, n shape functions. Then the displacement, temperature and electrical field within an element can be obtained via the corresponding nodal values of that element as (Zienkiewicz and Taylor, 1989) ui ¼ Nia Ua
ða ¼ 1; 2; . . . ; 3nÞ ði ¼ 1; 2; 3Þ
ð28Þ
250
Normal Stress t 33 (MPa)
200
150
100
50
Model Experiment
0 0
2000
4000
-6
6000
8000
Normal Strain e 33 (x10 )
Fig. 1. The constitutive model approximation of the stress–strain behavior of a typical piezoelectric material.
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Fig. 2. Center crack problem: (a) configuration (unit: mm); (b) poling direction is X1 (tensile force is applied); (c) finite element mesh (number of elements: 3000, number of nodal points: 3101).
T ¼ Nb Tb
ðb ¼ 1; 2; . . . ; nÞ
ð29Þ
/ ¼ Nb Ub
ðb ¼ 1; 2; . . . ; nÞ
ð30Þ
W. Zeng et al. / Mechanics Research Communications 30 (2003) 267–275
273
3/2
Stress intensity factor KI (MN/m )
1.6 1.4 1.2 1 0.8 0.6 0.4
E=-0.5 Mv/m E=0 Mv/m
0.2
E=0.5 Mv/m 0 0
0.1
(a)
0.2
0.3
0.4
0.5
0.6
0.7
Normalized distance x/a (%)
3/2
Stress intensity factor KI (MN/m )
1.6 1.4 1.2 1 0.8 0.6 0.4
E=-0.5 Mv/m E=0 Mv/m
0.2
E=0.5 Mv/m 0 0
0.1
(b)
0.2
0.3
0.4
0.5
0.6
0.7
Normalized distance x/a (%)
Fig. 3. Stress intensity factor KI vs. normalized distance x=a (t ¼ 3:0 MPa): (a) without domain switching; (b) including domain switching.
Before domain switching, the finite element equations can be obtained as Kab Ua þ Pab Ta þ Cab Ua þ Rabd Ua Ud þ Sabd Ta Ud þ Vabd Ua Ud Fb ¼ 0
ð31Þ
Hab Ta Qb ¼ 0
ð32Þ
Mab Ua þ Wab Ta þ Lab Ua Yb ¼ 0
ð33Þ
Similarly, after domain switching, the finite element equations are in the same format as Eqs. (32), (33) and (34). Matrices and vectors K, P, S, V, H, Q, W and L remain unchanged but matrices and vectors C, R, F, M and Y change in accordance with the transformation of tensors due to the change of coordinate system (see (Zeng et al., 2003) for details).
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3. Numerical results and discussion The computer program developed based on the formulation presented in Section 2 is used to analyze the center crack problem shown in Fig. 2(a). Due to the symmetry, only a quarter of the specimen is analyzed (Fig. 2(b)). The crack surfaces are assumed to be electrically insulated surfaces. In this work, four-node plain strain elements are employed. All of the stresses reported are stresses at the center of elements. Fig. 2(c) presents the finite element mesh. The total number of elements is 3000 and the element size near the crack tip is about 0.0004% of the crack length. It is shown in Fig. 2(b) that the tensile force is applied and the original poling direction of PZT is parallel to the crack (parallel to X1 -direction). The Mode I near tip stress intensity factors ahead of crack tip and along X1 axis are used to characterize the fracture behavior of piezoelectric crack (Sih and Liebowitz, 1968). Fig. 3 shows the results of fully coupled electromechanical analysis of the cracked plate under tension with and without domain switching. Fig. 3(a) indicates the fact that the electric fields have influence on the stress intensity factors when domain switching is disabled in the program. The positive electric field leads to smaller stress intensity factor but the case with negative electric field shows larger stress intensity factor. Fig. 3(b) shows the stress intensity factors calculated for three different electric fields. If the peak stress intensity factors are compared for different electric fields, it is clear that a negative E increases the stress intensity factor while a positive E reduces it. Since the tensile force is applied, it means that a negative E tends to increase the tensile stress while a positive E tends to reduce it. Therefore, a negative E tends to open the crack while a positive E tends to close the crack. Hence, it appears that the original poling direction and the direction of applied electric field play a role when domain switching is considered.
4. Conclusions The domain switching effect is included in a general 3-D finite element code to solve the fully coupled thermal-electro-mechanical problem. A center crack problem is analyzed. The distribution of the stress intensity factors ahead of the crack tip is obtained. It is observed that in the near tip region, the stress intensity factors are significantly affected by domain switching. Therefore, it appears that the effect of domain switching must be considered in the evaluation of the fracture toughness of piezoelectric material. In the case that poling direction of PZT is parallel to the crack, if a tensile force is applied to the specimen, it is found that a negative E increases the peak stress intensity factor while a positive E reduces it.
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