In order to obtain the required asymptotic field, we introduce a local polar coordinate system (ρ, θ)with the origin at the periphery of the crack, which satisfies
ρ=
(r−a)2+z2, θ= tan−1[z/(x−a)]. (141) In the close vicinity of the crack front, i.e.ρ << a,we have
l1j a+ρ 2
cos(θ)−
cos2(θ) +γj2sin2(θ)
, (142a)
l2j a+ρ 2
cos(θ) +
cos2(θ) +γj2sin2(θ)
. (142b)
Upon substitution of these into (73), by neglecting some higher-order infinitesimal terms, the asymptotic expressions for electroelastic field in the vicinity of the crack front
are derived below:
σρρ(ρ, θ) σzz(ρ, θ) σρz(ρ, θ)
2Q π
a 2ρ
3 j=1
β0jajf2j(θ) β1jajf2j(θ)
−β2jajf1j(θ)
, (143a)
sρρ(ρ, θ) szz(ρ, θ) sρz(ρ, θ)
2Q π
a 2ρ
3 j=1
−ajf2j(θ) η3jγj2ajf2j(θ)
−(1 +η3j)γjajf1j(θ)/2
, (143b)
Dρ(ρ, θ) Dz(ρ, θ) Eρ(ρ, θ) Ez(ρ, θ)
2Q π
a 2ρ
3 j=1
−β3jajf1j(θ) β4jajf2j(θ) η4jγjajf1j(θ)
−η4jγj2ajf2j(θ)
, (143c)
wheref1j(θ) andf2j(θ)denote the functions of angle distribution, defined as before. By comparing the above asymptotic electroelastic field of the cracked piezoelectric space with that of the cracked piezoelectric plane, one can find that all dependence of electroelastic field on material properties and angle distribution is identical except for a constant factor 2/π,coinciding with those for purely two-dimensional and three-dimensional elastic media with a straight crack and penny-shaped crack, respectively (Broberg, 1999).
From the above, the intensity factors of stress, strain, electric-displacement and electric- field near the crack front, according to their definitions
Kq = lim
r→a+
2π(r−a)q(r,0) (144)
whereqstands for one amongσzz, Dz,andEz,respectively, can be evaluated as Kσ = 2σ0
π
√πa, (145a)
Ks= 2Q π
√πa 3 j=1
η3jγj2aj, (145b)
KD = 2
D0−D¯ π
√πa, (145c)
KE =−2Q π
√πa 3 j=1
η4jγj2aj. (145d)
As seen from the above, no matter how applied electric loading varies, stress in- tensity factor maintains unchanged, implying that stress intensity factors near the crack front is inapplicable to predicting crack growth of piezoelectric materials. On the con- trary, the intensity factors of strain, electric displacement and electric field depend on the material properties and applied mechanical loading. In particular, if setting ε¯ = 0, we find KD = 2D0
a/π, independent of applied mechanical stress, which is in agreement with existing results such as Chen and Shioya (1999), Karapetian et al.
(2000), Jiang and Sun (2001), whereas if setting ∆φ(r,0) = 0 or ε¯ = ∞, we find KD = 2σ0det [β4, β2, η2]√
a/(det [β1, β2, η2]√
π),dependent solely on applied mechan- ical stress, in agreement with those obtained in Kogan et al. (1996), Yang and Lee (2001).
Figure 12: Ks/(2/π)√
πa vs E∞ for a three-dimensional cracked PZT-4, a) σ∞ = 8 MPa,b)s∞= 2×10−4.
Figure 13: KD/(2/π)√
πa vsE∞ for a three-dimensional cracked PZT-4, a) σ∞ = 8 MPa,b)s∞= 2×10−4.
Therefore, the above obtained conclusions under impermeable and permeable conditions are completely opposite. However, a real crack is neither electrically impermeable nor electrically permeable (at the boundaries of an undeformed crack). Consideration of a dielectric crack results in an important conclusion. That is, applied mechanical loading strongly affects the singularity of the electric displacement near the crack front, and also its intensity factor varies with the dielectric permittivity of the crack interior. Moreover, the impermeable and permeable cracks can be taken as two limiting cases of a dielectric crack.
Here the variations of the field intensity factors with applied electric field when sub- jected to applied far-field stress σ∞ = 8MPa or far-field strain s∞ = 2×10−4 for a cracked PZT-4 are displayed in Figs. 12-14. Some treads similar to the two-dimensional case can be found for the three-dimensional case. An apparent difference lies in that the curves fors∞ = 2×10−4corresponding toεr = 0have a turning point, which manifests that the locus of starting opening of an impermeable crack.
Figure 14: KE/(2/π)√
πavsE∞ for a three-dimensional cracked PZT-4, a) σ∞ = 8 MPa,b)s∞= 2×10−4.
5 Fracture Criterion
In the analysis of the stability of crack, a significant parameter is fracture criterion. In purely elastic media, many fracture criteria have been established based on the asymptotic field of stress, strain, elastic displacement in the vicinity of the crack tip, and the energy, the energy density, and so on. However, for piezoelectric materials where elastic and electric fields are coupled, the existing fracture criteria for purely elastic media cannot simply been extended to piezoelectric solids. For example, due to the fact that stress intensity factor for a cracked piezoelectric material is independent of applied electric loading, it is apparently not suitable for predicting crack growth in a piezoelectric ceramic since there are many experiments il- lustrating that applied electric fields change the fracture toughness of piezoelectric ceramics (Tobin and Pak, 1993; Park and Sun, 1995; Wang and Singh, 1997; Shang and Tan, 2001;
etc.).
The above-mentioned experiments exhibit some conflicting experimental results on the effects of electric fields on crack propagation such as those by Park and Sun (1995), who found that positive (negative) electric fields can aid (hinder) crack growth and by Wang and Singh (1997), who found that negative (positive) electric fields can aid (hinder) crack growth. In addition, Shang and Tan (2001) also observed that purely electric field might induce crack propagation in the absence of mechanical loading. Such experimental ob- servations cannot be successfully explained by using stress intensity factor, energy release rate. Up to date, some fracture criteria applicable to cracked piezoelectric materials have been proposed by researchers such as Park and Sun (1995), who suggested the mechanical strain energy release rate as a fracture criterion, and Gao et al. (1997) and Fulton and Gao (2001), who presented a saturation of electric displacement near the crack tip similar to nonlinear region in the well-known Dugdale mode and suggested a local energy release rate as a fracture criterion. Furthermore, Zuo and Sih (2000) generalized the classical energy density factor to piezoelectric media. In Li and Lee (2004b, c), the strain intensity factor has been formulated as a fracture criterion and compared with other fracture criteria and some experimental observations.
As we know, crack growth is the result of elastic deformation. Therefore, elastic dis- placement and strain are responsible for crack advance. Based on these considerations, here
Figure 15:KCOD/√
πavsE∞for a two-dimensional cracked PZT-4, a)σ∞= 8MPa,b) s∞= 2×10−4.
Figure 16: KCOD/(2/π)√
πavsE∞for a three-dimensional cracked PZT-4, a)σ∞ = 8 MPa,b)s∞= 2×10−4.
we define COD intensity factor by KCOD = lim
r→a−
π
2 (a−r)uz(r,0), (146)
so we obtain
KCOD =Q√ πa
3 j=1
η3jγjaj, (147)
for a two-dimensional cracked piezoelectric ceramic, and KCOD = 2
πQ√ πa
3 j=1
η3jγjaj, (148)
for a three-dimensional cracked piezoelectric ceramic, respectively.
As an example, consider a PZT-4 ceramic with a dielectric crack. Figs. 15 and 16 show the variation of COD intensity factors versus applied electric fieldE∞ for various values
ofεrfor a two- and three-dimensional cases, respectively. From Figs. 15 and 16, one can find that under prescribed far-field stressσ∞ = 8MPa,KCOD increases with the increase ofE∞forεr= 1,inferring that a positive electric field can promote crack growth, whereas a negative one can hinder crack growth. These conclusions are in agreement with the ex- perimental observations of Park and Sun (1995). In contrast,KCODremains unchanged for εr = ∞, implying that applied electric fields have no effects on conducting cracks, while KCODfor the case ofεr= 0rises withE∞increasing whenE∞is larger than one certain critical value at which crack starts to open. On the other hand, for prescribed far-field strain, it is easily found that some treads are reversed compared to prescribed stress. That is, ap- plied positive electric fields impede crack growth, whereas negative ones aid crack growth, which coincide with the experimental observation by Wang and Singh (1997), who found that positive electric fields increase its fracture toughness, and negative ones decrease its fracture toughness. It is interesting to note that all curves intersect atE0 = 0.This sug- gests that values ofKCODunder purely mechanical loading are the same for any dielectric crack in a piezoelectric solid. That is, the dielectric permittivity of the crack interior affects KCOD only in the presence of electric fields, as expected. In fact, these conclusions are easily understood from the basic constitutive equations. The reason is that applied positive field can induce piezoelectric ceramics to expand along the poling direction, and it aids crack growth when the cracked piezoelectric solid does not be constrained. Opposite to the above, if the cracked piezoelectric solid is fixed at certain distance, which can be described by prescribed strain or elastic displacement, this, in turn, causes the cracked piezoelectric solid to expand along the opposite direction, i.e. toward an opening crack, which imposes the crack to close. Hence, positive electric fields impede crack growth. Therefore, it is concluded that far-field boundary conditions of prescribed stress or prescribed strain play a crucial role in determining crack growth of piezoelectric materials.
6 Conclusion
Under the action of applied electromechanical loadings, the electroelastic analysis of a cracked piezoelectric material has been made within the framework of the theory of linear piezoelectricity. The associated mixed boundary-value problems are different from those studied previously. The electric boundary conditions are governed by the CODs. By using the Fourier and Hankel transforms to solve the electroelasticity problems related to a crack of finite length and a penny-shaped crack, respectively, a full electroelastic field is deter- mined explicitly. In particular, the asymptotic electroelastic field near the crack front are derived, and the field intensity factors are given. The dependence of field intensity factors on applied electric field for various dielectric permittivities is displayed graphically. Fur- thermore, similar to the strain intensity factor, the COD intensity factor can be used as a suitable fracture criterion of a cracked piezoelectric material. Based on this criterion, the results indicate that applied positive electric fields decrease fracture toughness and negative ones increase fracture toughness for prescribed remote stress. In contrast, applied positive electric fields increase fracture toughness and negative ones decrease fracture toughness for prescribed remote strain or displacement. Therefore, far-field mechanical boundary con- ditions play a crucial role in studying the stability of a crack embedded in a piezoelectric ceramic, which might account for conflicting experimental observations.
References
Abramowitz, M. and Stegun, I. A. 1972. Handbook of Mathematical Functions. Dover, New York.
Broberg, K.B. 1999. Cracks and Fracture, Academic Press, San Diego.
Cao, H. and Evans, A. G. 1994. Electric-field-induced fatigue crack growth in piezo- electrics. J. Am. Ceram. Soc. 77, 1783-1786.
Dunn, M.L., 1994. The effects of crack face boundary conditions on the fracture mechan- ics of piezoelectric solids. Eng. Fract. Mech.48, 25-39.
Chen, W. Q. and Shioya, T. 1999. Fundamental solution for a penny-shaped crack in a piezoelectric medium. J. Mech. Phys. Solids 47, 1459-1475.
Fabrikant, V. I. 1991. Mixed Boundary Value Problems of Potential Theory and Their Applications in Engineering, Kluwer Academic Publishers, Dordrecht.
Fabrikant, V. I. 2003. Computation of infinite integrals involving three Bessel functions by introduction of new formalism. ZAMM.83, 363-374.
Fulton, C. C. and Gao, H. 2001. Effect of local polarization switching on piezoelectric fracture. J. Mech. Phys. Solids, 49, 927-952.
Gao, C.-F. and Wang, M.-Z. 2001. Green’s functions of an interfacial crack between two dissimilar piezoelectric media. Int. J. Solids Struct.38, 5323–5334.
Gao, C.-F. and Fan, W. X. 1999. Exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack. Int. J. Solids Struct.36, 2527-2540.
Gao, H., Zhang T.-Y. and Tong, P. 1997. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic, J. Mech. Phys. Solids,45, 491-510.
Gradshteyn, I. S. and Ryzhik, I. M. 1980. Table of Integrals, Series and Products. Acad- emic Press, New York.
Haertling, G. H. 1987. PLZT electrooptic materials and applications-a review. Ferrl- electrics, 75, 25-55.
Hao, T. H. and Shen, Z. Y. 1994. A new electric boundary condition of electric fracture mechanics and its applications. Eng. Fract. Mech.47, 793-802.
Huang, J. H. 1997. A fracture criterion of a penny-shaped crack in transversely isotropic piezoelectric media. Int. J. Solids Struct. 34, 2631-2644.
Ikeda, T. 1990. Fundamentals of Piezoelectricity. Oxford University Press.
Jiang, L. Z. and Sun, C. T. 2001. Analysis of indentation cracking in piezoceramics. Int.
J. Solids Struct. 38, 1903-1918.
Karapetian, E., Sevostianov, I. and Kachanov, M. 2000. Penny-shaped and half-plane cracks in a transversely isotropic piezoelectric solid under arbitrary loading. Arch.
Appl. Mech. 70, 201-229.
Kogan, L., Hui, C. Y. and Molcov, V., 1996. Stress and induction field of a spheroidal inclusion of a penny-shaped crack in a transversely isotropic piezoelectric material.
Int. J. Solids Struct.33, 2719–2737.
Kwon, S. M. and Lee, K. Y. 2000. Analysis of stress and electric fields in a rectangular piezoelectric body with a center crack under anti-plane shear loading. Int. J. Solids Struct. 37, 4859-4869.
Li, X.-F. 2002. Electroelastic analysis of an anti-plane shear crack in a piezoelectric ce- ramic strip. Int. J. Solids Struct.39, 1097-1117.
Li X-F and Duan X-Y. 2001. Closed-form solution for a mode-III crack at the mid-plane of a piezoelectric layer. Mech. Res. Comm.28, 703-710.
Li, X.-F. and Lee K. Y. 2004a. Effects of electric field on crack growth for a penny-shaped dielectric crack in a piezoelectric layer J. Mech. Phys. Solids. 52, 2079-2100.
Li, X.-F. and Lee K. Y. 2004b. Crack growth in a piezoelectric material with a Griffith crack perpendicular to the poling axis. Phil. Mag.84, 1789-1820
Li, X.-F. and Lee K. Y. 2004c. Fracture analysis of cracked piezoelectric materials. Int. J.
Solids Struct.41, 4137–4161.
Li, X.-F. and Lee K. Y. 2004d. Three-dimensional electroelastic analysis of a piezoelectric material with a penny-shaped dielectric crack. ASME J. Appl. Mech.71, 866-878.
Liu, B., Fang, D.-N., Soh, A. K. and Hwang, K.-C. 2001. An approach for analysis of poled/depolarized piezoelectric materials with a crack. Int. J. Fract. 111, 395-407.
McMeeking, R. M. 1999. Crack tip energy release rate for a piezoelectric compact tension specimen. Eng. Fract. Mech.64, 217-244.
McMeeking, R. M. 2001. Towards a fracture mechanics for brittle piezoelectric and di- electric materials. Int. J. Fract.108, 25-41.
Pak, Y. E. 1990. Crack extension force in a piezoelectric material. J. Appl. Mech.57, 647-653.
Pak, Y. E. 1992. Linear electro-elastic fracture mechanics of piezoelectric materials. Int.
J. Fract.54, 79-100.
Park, S. and Sun, C. T. 1995. Fracture criteria for piezoelectric ceramics. J. Am. Ceram.
Soc.78, 1475-1480.
Parton, V.Z., 1976. Fracture mechanics of piezoelectric materials. Acta Astronaut. 3, 671–683.
Pisarenko, G. G. Chushko, V. M. and Kovalev, S. P. 1985. Anisotropy of fracture tough- ness in piezoelectric ceramics. J. Am. Ceram. Soc.68, 259-265.
Qin, Q. H. 2001. Fracture Mechanics of Piezoelectric Materials. Boston, WIT Press, Southampton
Qin, Q-H and Yu, S. W. 1997. An arbitrarily-oriented plane crack terminating an interface between dissimilar piezoelectric materials. Int. J. Solids Struct.34, 581-590.
Rao, S. S. and Sunar, M. 1994. Piezoelectricity and its use in disturbance sensing and control of flexible structures: a survey. Appl. Mech. Rev.47, 113-123.
Ru, C. Q. 1999. Electric-field induced crack closure in linear piezoelectric media. Acta Mater.47, 4683-4693.
Schneider, G. A., Felten, F. and McMeeking, R. M. 2003. The electrical potential differ- ence across cracks in PZT measured by Kelvin Probe Microscopy and the implica- tions for fracture. Acta Mater.51, 2235-2241.
Shang, J. K. and Tan, X. 2001. A maximum strain criterion for electric-field-induced fatigue crack propagation in ferroelectric ceramics. Mater. Sci. Eng.A 301, 131-139.
Shindo, Y., Tanaka, K., Narita, F., 1997. Singular stress and electric fields of a piezo- electric ceramic strip with a finite crack under longitudinal shear. Acta Mech. 120, 31-45.
Shindo, Y., Watanabe, K. and Narita, F. 2000. Electroelastic analysis of a piezoelectric ceramic strip with a central crack. Int. J. Engng. Sci.38, 1-19.
Sneddon, I. N. 1951. Fourier Transform. McGraw-Hill, New York.
Sosa, H. and Khutoryansky, N., 1996. New developments concerning piezoelectric mate- rials with defects. Int. J. Solids Struct.33, 3399-3414.
Suo, Z., Kuo, C.-M., Barnett, D. M. and Willis, J. R. 1992. Fracture mechanics for piezo- electric ceramics. J. Mech. Phys. Solids.40, 739-765.
Tobin, A. G. and Pak, Y. E. 1993. Effect of electric fields on the fracture behaviour of PZT ceramics. Proc. SPIE, 1916/79, 78-86.
Uchino, K. 1996. Piezoelectric actuators/ultrasonic motors. Norwell, Nassachusetts:
Kluwer Academic Publishers.
Wang, B. L. and Mai, Y.-W. 2003. On the electrical boundary conditions on the crack surfaces in piezoelectric ceramics. Int. J. Engng. Sci. 41, 633-652.
Wang, T. C. and Han, X. L. 1999. Fracture mechanics of piezoelectric materials. Int. J.
Fract.98, 15-35.
Wang, X. D. and Jiang, L. Y. 2002. Fracture behaviour of cracks in piezoelectric media with electromechanically coupled boundary conditions. Proc. R. Soc. Lond. A 458, 2545-2560.
Wang, Z. and Zheng, B. 1995. The general solution of three-dimensional problems in piezoelectric media. Int. J. Solids Struct.32, 105-115.
Xu, X.-L. and Rajapakse, R. K. N. D. 2001. On a plane crack in piezoelectric solids. Int.
J. Solids Struct.38, 7643-7658.
Yang, F. 2001. Fracture mechanics for a Mode I crack in piezoelectric materials. Int. J.
Solids Struct.38, 3813-3830.
Yang, J. H. and Lee, K.Y. 2001. Penny shaped crack in three-dimensional piezoelectric strip under in-plane normal loadings. Acta Mech.148, 187–197.
Zhang, T.-Y. and Tong, P. 1996. Fracture mechanics for a mode III crack in a piezoelectric material. Int. J. Solids Struct.33, 343–359.
Zhang, T.-Y., Zhao, M., and Tong, P. 2002. Fracture of piezoelectric ceramics. Adv. Appl.
Mech.38, 147-289.
Zhang, T.-Y., Qian, C.-F. and Tong, P. 1998. Linear electro-elastic analysis of a cavity or a crack in a piezoelectric material. Int. J. Solids Struct.35, 2121-2149.
Zhong, Z. and Meguid, S. A. 1997. Analysis of a circular arc-crack in piezoelectric mate- rials. Int. J. Fract.84, 143-158.
Zuo, J. Z. and Sih, G. C. 2000. Energy density theory formulation and interpretation of cracking behavior for piezoelectric ceramics. Theo. Appl. Fract. Mech. 34, 17-33.
Editor: Peter L. Reece, pp. 113-157 © 2006 Nova Science Publishers, Inc.
Chapter 3