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Frequency effects on fatigue crack growth and crack tip domain-switching behavior in a lead zirconate titanate ceramic
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Acta Materialia 57 (2009) 3932–3940 www.elsevier.com/locate/actamat
Frequency effects on fatigue crack growth and crack tip domain-switching behavior in a lead zirconate titanate ceramic Soodkhet Pojprapai a,*, Jennifer Russell b, Hou Man c, Jacob L. Jones d, John E. Daniels e, Mark Hoffman b a
School of Ceramic Engineering, Institute of Engineering, Suranaree University of Technology, Thailand b School of Materials Science and Engineering, UNSW, Sydney, NSW, Australia c School of Mechanical and Manufacturing Engineering, UNSW, Sydney, NSW, Australia d Department of Materials Science and Engineering, University of Florida, Gainesville, FL, USA e The European Synchrotron Radiation Facility (ESRF), Grenoble, France Received 10 July 2008; received in revised form 3 April 2009; accepted 28 April 2009 Available online 29 May 2009
Abstract The microstructural origins of the effect of frequency on the electrical fatigue behavior of pre-cracked soft ferroelectric Pb(Zr0.48Ti0.52)O3 is investigated by means of a high spatial resolution hard X-ray synchrotron source. It is found that there is a strong link between the frequency of the applied bipolar field, domain-switching behavior in terms of ferroelastic reorientation of the domains around the crack tip and the resultant crack growth. The crack growth is accentuated under increased ferroelastic switching and, in particular, found to be more pronounced under low-frequency loading. The concept of domain wall viscoelasticity is applied to explain why lower frequencies accelerate crack growth under a bipolar electric field. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Synchrotron radiation; Electroceramics; Ferroelectricity; Texture
1. Introduction Ferroelectric ceramics, such as lead zirconate titanate (PZT), have been used extensively in various applications including sensors and transducer devices. In service environments, these ferroelectric ceramics may experience pure cyclic mechanical loading, pure cyclic electrical loading or mixed loading. In the case of cyclic electrical loading, severe problems are caused by electrical fatigue which leads to the degradation of bulk ferroelectric performance [1–3]. Moreover, under extreme cyclic electrical loading conditions, fatigue may cause cracks to propagate, leading to * Corresponding author. Address: School of Ceramic Engineering, Institute of Engineering, Suranaree University of Technology, 30000, Thailand. Tel.: +66-44-224-542. E-mail address: [email protected] (S. Pojprapai).
mechanical fracture and premature failure of bulk materials [4–6]. On a microscopic scale, these failures have been related to domain-switching behavior [5–7]. The fatigue behavior of ferroelectric materials including crack growth under an electric field have been well documented by Lupascu [8] and Yang [9]. However, a thorough analysis of the relationship between domain-switching behavior and crack propagation under bipolar cyclic electrical loading for bulk ceramics has not yet been developed. Domain-switching behavior in bulk ferroelectric ceramics can be investigated via techniques such as neutron diffraction [10–12] and synchrotron X-ray diffraction [13–15]. By employing either diffraction technique, the interchange of specific diffracted peak intensities is used to describe domain-switching behavior. For this experiment, the synchrotron X-ray diffraction technique is preferred because it provides better spatial resolution that reveal local domain texture. In this
1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.04.054
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work, the degradation of macroscopic polarization reflecting bulk ferroelectric performance was investigated using electrical fatigue testing equipment. Moreover, a state-ofthe-art high-energy X-ray diffraction technique is employed to explore the domain-switching behavior of pre-cracked samples, enabling a quantitative analysis of domain texture around the crack tip under a bipolar cyclic electrical load at various frequencies. The concept of viscosity and domain wall movement is employed to explain the micromechanism of domain-switching-induced crack growth under various loading frequencies. 2. Materials and methods Two testing procedures are used in this experiment. The first procedure describes the sample preparation and testing method for investigating the effect of frequency on remnant polarization in bulk ceramics. The second procedure is used for investigating the effect of frequency on crack propagation and domain-switching behavior around the crack tip. The results from both experiments are used to explain the relationship between loading frequency, remnant polarization, crack propagation and domain-switching behavior in PZT ceramics.
Fig. 1. Experimental setup using high-energy X-ray diffraction to investigate the effect of frequency on: (a) polarization of bulk samples and (b) domain orientation distribution around a crack tip.
2.1. Investigation of frequency effect on remnant polarization in bulk PZT samples Soft PZT samples with tetragonal phase (K350, Piezo Technologies, Indianapolis, IN, USA) were cut into a rectangular shape 3.70 3.00 2.50 mm3. These bulk samples were used to represent the characteristic of the bulk PZT under electrical fatigue testing. The samples were separated into three groups for three different frequencies (10, 50 and 100 Hz) of electrical fatigue testing. The experiment was repeated by testing three virgin samples at the same frequencies. The 3.70 3.00 mm2 surfaces were polished using 1200 and 4000 grit SiC grinding paper and then annealed at 600 °C for 5 h to remove residual stress due to the polishing process. A silver electrode was painted on the polished surfaces using colloidal silver paste. The samples were then submerged in a silicon oil bath and mounted into a fixture connected to a high-voltage power supply. This electrical cyclic testing was conducted using a commercial electrical fatigue testing device (TF Analyzer 2000, aixACCT systems, Germany). A bipolar triangular electric field of amplitude ±1.4 times the coercive field, Ec (equivalent to ±3.5 kV), was applied across the electrode surfaces for up to 104 cycles. A current range of 100 mA, which was automatically adjusted by the machine, was applied to the samples. During electrical cyclic loading, the remnant polarization extracted from hysteresis loops was measured, reflecting the bulk properties of the ferroelectric ceramics. The cyclic testing was repeated using three virgin samples with the same dimensions to obtain the average value of remnant polarization. The experimental geometry is shown in Fig. 1a.
2.2. Investigation of frequency effect on domain-switching behavior near a crack tip Three samples obtained from the same production lot of the bulk samples were cut into a rectangular shape with dimensions of about 4.60 1.50 4.00 mm3. One of 4.60 4.00 mm2 surfaces of each sample was polished using 1200 and 4000 grit SiC grinding paper. A short notch 0.5 mm long was cut on the 4.60 4.00 mm2 surface of samples and a pre-crack initiated on the polished surface at the end of the notch using a 1 kg Vickers indent. To eliminate the residual stress around the crack due to indentation, the samples were annealed at 600 °C for 5 h. After annealing, the crack length on the polished surface of each sample was measured. An electrode was coated on the 4.60 1.50 mm2 faces by applying the previous procedure. The samples were mounted into a fixture under a clamped condition (i.e. the samples were fixed with a screw connected between the electrode surfaces and the power supply). The samples were then submerged in a silicon oil bath and then subjected to a bipolar triangular electrical field of amplitude ±1.4Ec (equivalent to ±5.5 kV), which was applied perpendicular to the crack surface for up to 104 cycles. In this experiment, three samples were tested with three different frequencies: 10, 50 and 100 Hz. After electrical loading, the spatial distribution of domain orientations around the crack tip was investigated using synchrotron X-ray diffraction. The experimental setup for X-ray measurement is shown in Fig. 1b. In this investigation, a high spatial resolution and high-energy X-ray beam
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was obtained from a synchrotron source at beamline ID15B at the European Synchrotron Radiation Facility (ESRF). To obtain the maximum penetration of the PZT samples and to avoid background scatter, the average energy of the X-ray was selected at 87.62 keV, which is just below the energy to excite the background due to the Pb Kedge (88 keV) X-ray fluorescence. The beam size was set to 150 150 lm2 by a set of tungsten slits just prior to the sample. An area of 3.50 2.50 mm2 around the crack tip was scanned by translating the sample through the beam. Diffraction patterns in the form of Debye–Scherrer rings were collected using an X-ray image intensifier coupled with a FReLoN CCD camera. The two-dimensional images (Debye–Scherrer rings) were ‘‘caked” (sectioned into 24 parts with the azimuth angle or eta angle, g, of 15° increments) into one-dimensional diffraction patterns using the radial integration software fit2d [16]. The complete distribution of in-plane-oriented domain orientation may be presented by the 002/200 intensity ratio obtained from different scattering vectors parallel to eta angles, g. The integrated intensity was calculated from curve fitting using a split Pearson VII profile shape function with Matlab (v. 7.0) and used to estimate the domain fractions [17]. 3. Results 3.1. Polarization fatigue of bulk samples The fatigue characteristics of bulk PZT samples without a pre-crack were measured from the remnant polarization during electrical cyclic loading. The remnant polarization under the electrical fatigue environment extracted from ferroelectric hysteresis loops was recorded. To compare the effect of electrical fatigue frequency on the bulk samples, the remnant polarization of each sample was normalized with the initial polarization as shown in Fig. 2. It is seen that the polarization of each sample decreases with an increasing number of cycles. The decay of the polarization under a cyclic bipolar electric field with increase in cycle
number reflects polarization fatigue, which is attributed to the decreasing ability of domains to switch [7]. These results are comparable with previous work [1]. The results show the effect of frequency: the remnant polarization of the sample fatigued at 100 and 50 Hz declined slightly with cycle number, while that of the sample fatigued at 10 Hz dropped off significantly after 104 cycles. The polarization at 104 cycles reduces by about 10%, 31% and 60% for 100, 50 and 10 Hz, respectively. The decrease in polarization may be caused by defects within grains hindering the mobility of domain walls [1,8,18–22]. In the present work, the decrease in domain wall mobility may be considered as one of the causes of polarization fatigue. Under different loading frequencies, the exposure time of the lower-frequency sample (10 Hz) is longer than that of higher frequency (50 Hz or 100 Hz); therefore, the probability of defects hindering the domain wall mobility in the lower-frequency sample is greater. This can lead to more severe polarization fatigue in the case of the lower frequency. 3.2. Domain-switching behavior of pre-cracked samples After the pre-cracked samples were subjected to cyclic electrical loading at frequencies of 10, 50 and 100 Hz, the optical micrographs of crack propagation of each sample were recorded and are shown in Fig. 3a–c for 10, 50 and 100 Hz, respectively. The approximate crack length of each sample before and after being subjected to the cyclic loading was measured and is shown in Table 1. These results show that electrically induced crack propagation rate is much more pronounced if a sample is fatigued with a lower frequency. The crack length after 104 fatigue cycles is about 125 lm (increases by about 38%) after cycling at 10 Hz, 50 lm (increases by about 10%) at 50 Hz, and 25 lm (increases by about 4%) at 100 Hz. In addition to the effect of frequency, the different stress intensities due to the changing crack length may have an influence on crack propagation. Under electrical loading, strain induced by domain switching occurs near the crack tip. As the result of domain switching strain, a ‘‘mechanical” stress concentration is generated around the crack tip [23]. In this case, the crack propagates when the electric field intensity of the crack tip is high enough to induce the critical stress intensity factor (KIC). To consider the relationship between the crack length and the electric field intensity factor, KE, the formula provided by Yang and Suo [23] is considered: KE ¼
E pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2h tanðpai =2hÞ; h
ð1Þ
where h is the distance between electrode surfaces, E is the amplitude of the applied electric field, and ai is the initial crack length. The stress intensity factor, KI, is then given by: Fig. 2. The normalized remnant polarization under a bipolar cyclic field with an amplitude of ±1.4 kV mm–1 at frequencies of 10, 50 and 100 Hz.
KI ¼ b
Y cs KE; Ec
ð2Þ
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Fig. 3. The optical micrographs of samples before and after fatigue testing with an amplitude of ±1.4 kV mm–1 at (a) 10 Hz, (b) 50 Hz and (c) 100 Hz of a bipolar electric field. Black dashed lines are used to indicate the crack path in each sample.
Table 1 The approximate crack length of the fatigued samples under ±1.4 kV mm–1 of bipolar cyclic loading with different frequencies. Sample
Frequency (Hz)
Initial crack length, ai (lm)
Crack length after 104 cycles of fatigue, af (lm)
A B C
10 50 100
325 500 575
450 550 600
where Y is the Young’s modulus, cs is the electrostrictive strain, Ec is the coercive field, and b is a dimensionless parameter depending on the crack condition (e.g. b 0.25 for an insulating crack). By substituting Eq. (1) into Eq. (2), the relationship between stress intensity factor and the electric field can be expressed as: KI ¼ b
Y cs E pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2h tanðpai =2hÞ: Ec h
ð3Þ
Considering Eq. (3), the stress intensity, KI, is higher if the initial crack length, ai, is longer. Consequently, the crack length effect alone suggests that the greatest crack propagation, Da, in Table 1 should be observed in the sample possessing the longest initial crack length, ai (i.e. sample C). However, the results from this experiment show that the sample with the longest initial crack length (sample C) exhibits the shortest crack propagation, while the sample with the shortest initial crack length (sample A) exhibits the longest or most severe crack propagation. This implies that the effect of cyclic frequency dominates crack propagation under cyclic electric loading. Furthermore, these results show that although the samples were subjected to the same amplitude of cyclic field, the most severe crack propagation occurs under the lowest cyclic frequency of the field. Considering the results from bulk samples and fracture samples, the effect of frequency on polarization and crack propagation can be seen. At the lower frequency, the polarization decays faster and crack propagation per cycle is greater compared with the case of higher frequency. These
Crack growth length, Da mm
%
125 50 25
38 10 4
relationships may be attributed to domain-switching behavior which is the driving force of polarization degradation and crack propagation under cyclic loading. To investigate the micromechanism of domain switching around the crack tip under cyclic electric loading, the fatigued pre-cracked samples were investigated by using a high spatial resolution and high-energy X-ray beam. The domain-switching behavior of the fatigued samples is characterized by the change in the intensities of 002 and 200 peaks representing the c- and a-domains, respectively. The preferred orientations of in-plane domains can be indicated by the integrated intensity ratio of 002 and 200 peaks of which the scattering vector is approximately parallel to the sample surface (see Fig. 1). An example of diffracted intensity of the 002 and 200 peaks as well as fitting curves for the tetragonal phase are shown in Fig. 4. The preferred orientation of c-domains parallel to the electric field direction are indicated by the multiple of a random distribution (MRD), which is expressed by: R I 002 I 002 ; ð4Þ MRD002 ¼ 3 R I 002 I 002 þ 2 I 200 I R200 where I002 and I200 are the integrated intensities of the 002 and 200 peaks of a fatigued sample which possesses preferred domain orientation [24]. The superscript R denotes the integrated intensity of the same peaks for a sample with random domain orientation. From Eq. (4), MRD002 < 1 if the preferred orientation of c-domains is away from the measured direction (i.e. the scattering vector direction); MRD002 = 1 if the orientation of the domains is random; and MRD002 > 1 if there is preferred orientation along
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Fig. 4. 002 and 200 peaks and fitting curves using a split Pearson VII profile shape function.
the measured direction. MRD002 values obtained from the scanning area around a crack tip are plotted as a function of position relative to the crack tip and the direction in the
plane of the sample and are represented as a contour plot. Domain distribution contours of samples fatigued at 10, 50 and 100 Hz are presented in Figs. 5–7, respectively. For each frequency, a set of four contours is shown representing different directions with respect to the crack face and the applied field (this angle is defined as g, see Fig. 1b). The distributions of domain orientations around the crack tip can be interpreted by the combination of two indicators. The first indicator is the direction of a preferred orientation which is represented by an arrow at the left top corner of each contour plot. The second one is the MRD002 values calculated using Eq. (4). For example, in Fig. 5, the preferred c-domain zone (i.e. MRD002 > 1) which orients parallel to the applied field, g = 0° (or perpendicular to the crack face), can be revealed by contour a1 and the orientations g = 45°, 90° and 135° to the applied field are represented by contours a2, a3 and a4, respectively. Moreover, the complete texture of in-plane domains can be obtained by the combination of contours of all g from 0° to 180°. However, for simplicity, the contours presented here are taken from just four selected angles (g = 0°, 45°, 90° and 135°).
Fig. 5. Domain contours around the crack tip in a fatigued sample (sample A) after 104 cycles under a frequency of 10 Hz at different preferred orientations of c-domain. The direction with respect to g of the c-domains is represented by the arrow at the top left corner of each contour. The MRD002 values are indicated by a color bar next to the contour.
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Fig. 6. Domain contours of a fatigue sample (sample B) after 104 cycles at a frequency of 50 Hz at different g angles with respect to the applied field: 0°, 45°, 90°, 135°.
The right-hand side of each plot shows an abnormally high level of MRD002 in each sample. This is related to the interface between the ceramics and electrode, which leads to the very strong preference of the in-plane cdomains in that area. The data can be ignored for this analysis as they are well removed from the crack tip. For Figs. 5–7, the contours of the g = 0° and 90° show symmetry of domain orientation distributions about the crack face, while those of the g = 45° and 135° are asymmetric. The preferred orientations about the crack tip of each fatigued sample can be observed when considering the four g angles together. By comparing the area of preferred c-domain orientations of samples A–C in Fig. 5–7, respectively, it can be seen that they differ with varying frequency. After being fatigued under low frequency (10 Hz), the zone of preferred c-domain orientations, which is represented by MRD002 > 1, is considerably larger than that of the 50 and 100 Hz samples. It is noted that the size of the process zone related to a sample size can affect the level of mechanical constraint. Low mechanical constraint (plane stress) exists if the process zone is large compared to the size of a sample, while high mechanical constraint (plane strain) occurs if the zone
is small. In this case, plane stress exists (particularly for the low frequency) because the size of the process zone (qualified as the region where MRD002 > 1) is approximately 3 3, 2 1 and 0.5 1 mm2 for the samples loaded at 10, 50 and 100 Hz, respectively. The lower-frequency sample exhibits a larger process zone because the total exposure time of the sample to mechanical load during cyclic loading is longer. Reconciliation of the domain contours (Figs. 5–7) and the polarization fatigue measurements (Fig. 2) should be undertaken carefully. The domain contours represent the degree of c-axis alignment around a crack tip subject to cyclic electric field loading at various frequencies. These measurements were conducted ex situ, i.e. after the removal of the electric field, and therefore represent the remnant domain orientations. The polarization fatigue data, however, was measured during the electrical cycling and therefore reflects the degree of domain switching that occurs during the electric field application. The measurements from the two loading scenarios suggest that, while domains can orient a greater amount at lower frequencies (i.e. the degree of domain switching in Fig. 5 is greater than in Fig. 6, which is greater than in Fig. 7), the domain
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Fig. 7. Domain contours of a fatigue sample (sample C) after 104 cycles at a frequency of 100 Hz at different g angles with respect to the applied field: 0°, 45°, 90°, 135°.
switchability declines (i.e. the number of switchable domains decrease, as shown in Fig. 2). In addition to polarization fatigue, the results obtained from domain contours can be used to investigate the relationship between the domain-switching behavior and crack propagation. Non-uniform domain orientation distributions are observed for each fatigued sample. The preferred c-domains oriented 0°, 45° and 135° to the field (or 90°, 135° and 45° to the crack face) are found in front of the crack tip, while those oriented 90° to the field occur at the crack flanks and behind the crack tip. These results agree with the model presented by Zhu et al. [25] and Beom and jeong [6]. Moreover, it can be observed that the switching zone of preferred c-domains oriented at 45° and 135° to the crack face is smaller and exhibits a weaker degree of preferences than those oriented 90° to the crack face. These orientation distributions of preferred c-domains are comparable with the previous measurements using a similar technique [13]. To simplify this explanation, the distributions of preferred c-domain orientations are shown schematically in Fig. 8. In this experiment, the non-uniform distributions of preferred c-domain orientations around the crack tip
may be attributed to the inhomogeneous local field around the crack (Fig. 8b). The inhomogeneous field is caused by the crack conditions (i.e. crack geometry, impermeable or permeable boundary conditions of the crack faces) within the samples. Since the samples were submerged in a silicon bath while being electrically fatigued, it can be expected
Fig. 8. A schematic diagram showing the relationship between the domain orientations and the local field around the crack: (a) represents the random orientations before being fatigued; (b) the local field around the crack tip where the non-uniform distributions of preferred c-domain orientations occur; preferred c-domains align nearly parallel to the electrical field direction.
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that the silicon oil film may fill a gap between crack surfaces. Therefore, it can be assumed that a crack becomes insulating (if the oil fully fills the gap and there is no free space between the crack faces) or semi-insulating (if the oil partially fills the gap and there is free space between the crack faces). Due to being shielded by the crack fully or partially filled with the oil, the electric flux lines deflect around the crack tip and concentrate at the crack tip [26–30]. Consequently, this inhomogeneous field causes the non-uniform distribution of preferred domain orientations (i.e. the preferred c-domain oriented 0° to the crack face occurs at the crack face behind the crack tip while that oriented 90°, 45° or 135° to the crack face takes place at the front of the crack tip). This leads to large ferroelastic strain mismatch, which is a driving force for crack propagation [30], between the area in front of the crack tip and behind the crack tip. Under cyclic electrical loading, if the field increases above the coercive field, 90° domain switching occurs and non-uniform domain orientation distribution is induced by the inhomogeneous local field around the crack. Under this condition, the switching process due to electrical energy induces ferroelastic strain along the crack flank leading to stress, which can be considered as a wedging force. When strain mismatch around the crack tip increases, the stress intensity increases. Consequently, if the stress intensity reaches the critical value (KIc) of the ceramics, the crack starts propagating. 3.3. Model of the frequency effect on crack propagation It has been demonstrated that both an inhomogeneous electric field and non-uniform domain orientation distributions around a crack tip influence crack propagation. In addition, the frequency of an applied field also affects the severity of crack propagation. In a previous work, the authors studied the effect of mechanical loading frequency on ferroelastic strain induced by irreversible switched domains [11] and created a model to explain this relationship. In the present experiment, the previous model is applied to explain the micromechanism of crack growth due to domain switching under different frequencies of an electric field. The model describes the frequency-dependent domain behavior and ferroelastic strain using a viscoelasticity-based concept. This viscoelasticity-based model states that the ferroelastic strain induced by 90° domain wall movement of bulk ceramics can be expressed by: eðf Þ ¼ AeN =f h þ C 0 ;
ð5Þ
where e(f) is the ferroelastic strain as a function of frequency, h is the relaxation time, N is the number of cycles, f is the loading frequency, and A and C0 are constants. An example of experimental data of the ferroelastic strain as a function of the number of cycles under different frequencies of a cyclic loading and fitting curves generated by Eq. (5) is shown in Fig. 9. This model and the experimental results of the previous work show that the ferroelastic strain at lower frequency (1 Hz) is larger than that at higher frequency
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Fig. 9. Frequency effect on ferroelastic strain as a function of the number of cycle. The fitting curve is calculated by using a viscoelasticity-based model (Eq. (5)). (Figure from Ref. [11].)
(20 Hz). This implies that domain walls under a lower frequency are able to travel further than that under a higher frequency. In the case of crack propagation under different frequencies of electric loading, the same viscoelasticity-based concept may be used to explain the link between the frequency of the applied field, domain-switching behavior and crack propagation. Under cyclic loading, when the electrical load increases, domain walls move over a certain distance due to a local electric field. When the load decreases during a cycle, some domain walls move back while some domain walls do not revert to the original position. This may be attributed to microscopic defects within grains which reduce the mobility of domain walls [1,8,18–21,31,32]. At the switching zone just behind the crack tip, this movement of domain walls may therefore cause high local strain and initiate crack opening. In the case of low frequency, domains at the switching zone behind the crack tip experience a longer time under the local field for a single cycle; as a result, domain walls are able to travel over a longer distance due to viscoelastic effects. Therefore, a greater local strain is induced at the switching zone and wedges the crack open, resulting in more severe crack propagation. On the other hand, if the higher frequency of an electric field is applied, domain walls move over a shorter distance compared with the first case. Consequently, the lesser local deformation, which takes place at the switching zone just behind the crack tip, may cause less crack propagation per cycle compared with the case of a low-frequency loading. 4. Conclusions Experimental results demonstrate that the degree of polarization fatigue in PZT cycled at an electric field above the coercive field increases with a reduction in cycling frequency. Electrical cycling of a fracture sample, furthermore, exhibited a greater rate of crack propagation with
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decreasing frequency. Spatial analysis of c-domain orientations around a crack tip show a clear zone of preferred domain orientations. The orientation of the c-domains can be predicted using the path of the electric field around an insulating crack. Additionally, the size of the zone of preferred domain orientations increased with decreasing frequency. It is therefore concluded that: (1) The bipolar electrical cyclic loading causes a decrease in domain switchability, reflecting decline in the remnant polarization. (2) The viscoelastic nature of domain wall movement under numbers of loading cycles leads to greater degrees of polarization fatigue at lower frequencies. (3) Cyclic fatigue crack growth in piezoelectric materials is caused by ferroelastic strain mismatch at the crack tip caused by domain reorientations induced by the path of the electric field. (4) A large ferroelastic strain mismatch causes a high stress intensity factor which is a driving force for crack propagation. Eventually, the crack starts propagating when the stress intensity factor reaches the critical value (KIc) of the material. (5) The magnitude of this mismatch decreases with increasing frequency due to reduced amount of domain reorientation per cycle owing to the viscoelastic nature of the domain movement. (6) Hence, fatigue crack growth rates per cycle decrease with increasing cycling frequency. Acknowledgments The authors gratefully acknowledge financial support from an ARC Discovery Grant No. DP0558596. Moreover, the authors would like to thank Dr. Nagarajan Valanoor of the School of Materials Science & Engineering, The University of New South Wales, for his fruitful discussions. Travel support to access these facilities was provided by the Access to Major Research Facilities (AMRF) program, administered by the Australian Nuclear Science and Technology Organisation (ANSTO). J.L.J. acknowledges support for this work from the US National Science Foundation through award number OISE-0402066. S.P.(I.)
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Acta Materialia 57 (2009) 3932–3940 www.elsevier.com/locate/actamat
Frequency effects on fatigue crack growth and crack tip domain-switching behavior in a lead zirconate titanate ceramic Soodkhet Pojprapai a,*, Jennifer Russell b, Hou Man c, Jacob L. Jones d, John E. Daniels e, Mark Hoffman b a
School of Ceramic Engineering, Institute of Engineering, Suranaree University of Technology, Thailand b School of Materials Science and Engineering, UNSW, Sydney, NSW, Australia c School of Mechanical and Manufacturing Engineering, UNSW, Sydney, NSW, Australia d Department of Materials Science and Engineering, University of Florida, Gainesville, FL, USA e The European Synchrotron Radiation Facility (ESRF), Grenoble, France Received 10 July 2008; received in revised form 3 April 2009; accepted 28 April 2009 Available online 29 May 2009
Abstract The microstructural origins of the effect of frequency on the electrical fatigue behavior of pre-cracked soft ferroelectric Pb(Zr0.48Ti0.52)O3 is investigated by means of a high spatial resolution hard X-ray synchrotron source. It is found that there is a strong link between the frequency of the applied bipolar field, domain-switching behavior in terms of ferroelastic reorientation of the domains around the crack tip and the resultant crack growth. The crack growth is accentuated under increased ferroelastic switching and, in particular, found to be more pronounced under low-frequency loading. The concept of domain wall viscoelasticity is applied to explain why lower frequencies accelerate crack growth under a bipolar electric field. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Synchrotron radiation; Electroceramics; Ferroelectricity; Texture
1. Introduction Ferroelectric ceramics, such as lead zirconate titanate (PZT), have been used extensively in various applications including sensors and transducer devices. In service environments, these ferroelectric ceramics may experience pure cyclic mechanical loading, pure cyclic electrical loading or mixed loading. In the case of cyclic electrical loading, severe problems are caused by electrical fatigue which leads to the degradation of bulk ferroelectric performance [1–3]. Moreover, under extreme cyclic electrical loading conditions, fatigue may cause cracks to propagate, leading to * Corresponding author. Address: School of Ceramic Engineering, Institute of Engineering, Suranaree University of Technology, 30000, Thailand. Tel.: +66-44-224-542. E-mail address: [email protected] (S. Pojprapai).
mechanical fracture and premature failure of bulk materials [4–6]. On a microscopic scale, these failures have been related to domain-switching behavior [5–7]. The fatigue behavior of ferroelectric materials including crack growth under an electric field have been well documented by Lupascu [8] and Yang [9]. However, a thorough analysis of the relationship between domain-switching behavior and crack propagation under bipolar cyclic electrical loading for bulk ceramics has not yet been developed. Domain-switching behavior in bulk ferroelectric ceramics can be investigated via techniques such as neutron diffraction [10–12] and synchrotron X-ray diffraction [13–15]. By employing either diffraction technique, the interchange of specific diffracted peak intensities is used to describe domain-switching behavior. For this experiment, the synchrotron X-ray diffraction technique is preferred because it provides better spatial resolution that reveal local domain texture. In this
1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.04.054
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work, the degradation of macroscopic polarization reflecting bulk ferroelectric performance was investigated using electrical fatigue testing equipment. Moreover, a state-ofthe-art high-energy X-ray diffraction technique is employed to explore the domain-switching behavior of pre-cracked samples, enabling a quantitative analysis of domain texture around the crack tip under a bipolar cyclic electrical load at various frequencies. The concept of viscosity and domain wall movement is employed to explain the micromechanism of domain-switching-induced crack growth under various loading frequencies. 2. Materials and methods Two testing procedures are used in this experiment. The first procedure describes the sample preparation and testing method for investigating the effect of frequency on remnant polarization in bulk ceramics. The second procedure is used for investigating the effect of frequency on crack propagation and domain-switching behavior around the crack tip. The results from both experiments are used to explain the relationship between loading frequency, remnant polarization, crack propagation and domain-switching behavior in PZT ceramics.
Fig. 1. Experimental setup using high-energy X-ray diffraction to investigate the effect of frequency on: (a) polarization of bulk samples and (b) domain orientation distribution around a crack tip.
2.1. Investigation of frequency effect on remnant polarization in bulk PZT samples Soft PZT samples with tetragonal phase (K350, Piezo Technologies, Indianapolis, IN, USA) were cut into a rectangular shape 3.70 3.00 2.50 mm3. These bulk samples were used to represent the characteristic of the bulk PZT under electrical fatigue testing. The samples were separated into three groups for three different frequencies (10, 50 and 100 Hz) of electrical fatigue testing. The experiment was repeated by testing three virgin samples at the same frequencies. The 3.70 3.00 mm2 surfaces were polished using 1200 and 4000 grit SiC grinding paper and then annealed at 600 °C for 5 h to remove residual stress due to the polishing process. A silver electrode was painted on the polished surfaces using colloidal silver paste. The samples were then submerged in a silicon oil bath and mounted into a fixture connected to a high-voltage power supply. This electrical cyclic testing was conducted using a commercial electrical fatigue testing device (TF Analyzer 2000, aixACCT systems, Germany). A bipolar triangular electric field of amplitude ±1.4 times the coercive field, Ec (equivalent to ±3.5 kV), was applied across the electrode surfaces for up to 104 cycles. A current range of 100 mA, which was automatically adjusted by the machine, was applied to the samples. During electrical cyclic loading, the remnant polarization extracted from hysteresis loops was measured, reflecting the bulk properties of the ferroelectric ceramics. The cyclic testing was repeated using three virgin samples with the same dimensions to obtain the average value of remnant polarization. The experimental geometry is shown in Fig. 1a.
2.2. Investigation of frequency effect on domain-switching behavior near a crack tip Three samples obtained from the same production lot of the bulk samples were cut into a rectangular shape with dimensions of about 4.60 1.50 4.00 mm3. One of 4.60 4.00 mm2 surfaces of each sample was polished using 1200 and 4000 grit SiC grinding paper. A short notch 0.5 mm long was cut on the 4.60 4.00 mm2 surface of samples and a pre-crack initiated on the polished surface at the end of the notch using a 1 kg Vickers indent. To eliminate the residual stress around the crack due to indentation, the samples were annealed at 600 °C for 5 h. After annealing, the crack length on the polished surface of each sample was measured. An electrode was coated on the 4.60 1.50 mm2 faces by applying the previous procedure. The samples were mounted into a fixture under a clamped condition (i.e. the samples were fixed with a screw connected between the electrode surfaces and the power supply). The samples were then submerged in a silicon oil bath and then subjected to a bipolar triangular electrical field of amplitude ±1.4Ec (equivalent to ±5.5 kV), which was applied perpendicular to the crack surface for up to 104 cycles. In this experiment, three samples were tested with three different frequencies: 10, 50 and 100 Hz. After electrical loading, the spatial distribution of domain orientations around the crack tip was investigated using synchrotron X-ray diffraction. The experimental setup for X-ray measurement is shown in Fig. 1b. In this investigation, a high spatial resolution and high-energy X-ray beam
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was obtained from a synchrotron source at beamline ID15B at the European Synchrotron Radiation Facility (ESRF). To obtain the maximum penetration of the PZT samples and to avoid background scatter, the average energy of the X-ray was selected at 87.62 keV, which is just below the energy to excite the background due to the Pb Kedge (88 keV) X-ray fluorescence. The beam size was set to 150 150 lm2 by a set of tungsten slits just prior to the sample. An area of 3.50 2.50 mm2 around the crack tip was scanned by translating the sample through the beam. Diffraction patterns in the form of Debye–Scherrer rings were collected using an X-ray image intensifier coupled with a FReLoN CCD camera. The two-dimensional images (Debye–Scherrer rings) were ‘‘caked” (sectioned into 24 parts with the azimuth angle or eta angle, g, of 15° increments) into one-dimensional diffraction patterns using the radial integration software fit2d [16]. The complete distribution of in-plane-oriented domain orientation may be presented by the 002/200 intensity ratio obtained from different scattering vectors parallel to eta angles, g. The integrated intensity was calculated from curve fitting using a split Pearson VII profile shape function with Matlab (v. 7.0) and used to estimate the domain fractions [17]. 3. Results 3.1. Polarization fatigue of bulk samples The fatigue characteristics of bulk PZT samples without a pre-crack were measured from the remnant polarization during electrical cyclic loading. The remnant polarization under the electrical fatigue environment extracted from ferroelectric hysteresis loops was recorded. To compare the effect of electrical fatigue frequency on the bulk samples, the remnant polarization of each sample was normalized with the initial polarization as shown in Fig. 2. It is seen that the polarization of each sample decreases with an increasing number of cycles. The decay of the polarization under a cyclic bipolar electric field with increase in cycle
number reflects polarization fatigue, which is attributed to the decreasing ability of domains to switch [7]. These results are comparable with previous work [1]. The results show the effect of frequency: the remnant polarization of the sample fatigued at 100 and 50 Hz declined slightly with cycle number, while that of the sample fatigued at 10 Hz dropped off significantly after 104 cycles. The polarization at 104 cycles reduces by about 10%, 31% and 60% for 100, 50 and 10 Hz, respectively. The decrease in polarization may be caused by defects within grains hindering the mobility of domain walls [1,8,18–22]. In the present work, the decrease in domain wall mobility may be considered as one of the causes of polarization fatigue. Under different loading frequencies, the exposure time of the lower-frequency sample (10 Hz) is longer than that of higher frequency (50 Hz or 100 Hz); therefore, the probability of defects hindering the domain wall mobility in the lower-frequency sample is greater. This can lead to more severe polarization fatigue in the case of the lower frequency. 3.2. Domain-switching behavior of pre-cracked samples After the pre-cracked samples were subjected to cyclic electrical loading at frequencies of 10, 50 and 100 Hz, the optical micrographs of crack propagation of each sample were recorded and are shown in Fig. 3a–c for 10, 50 and 100 Hz, respectively. The approximate crack length of each sample before and after being subjected to the cyclic loading was measured and is shown in Table 1. These results show that electrically induced crack propagation rate is much more pronounced if a sample is fatigued with a lower frequency. The crack length after 104 fatigue cycles is about 125 lm (increases by about 38%) after cycling at 10 Hz, 50 lm (increases by about 10%) at 50 Hz, and 25 lm (increases by about 4%) at 100 Hz. In addition to the effect of frequency, the different stress intensities due to the changing crack length may have an influence on crack propagation. Under electrical loading, strain induced by domain switching occurs near the crack tip. As the result of domain switching strain, a ‘‘mechanical” stress concentration is generated around the crack tip [23]. In this case, the crack propagates when the electric field intensity of the crack tip is high enough to induce the critical stress intensity factor (KIC). To consider the relationship between the crack length and the electric field intensity factor, KE, the formula provided by Yang and Suo [23] is considered: KE ¼
E pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2h tanðpai =2hÞ; h
ð1Þ
where h is the distance between electrode surfaces, E is the amplitude of the applied electric field, and ai is the initial crack length. The stress intensity factor, KI, is then given by: Fig. 2. The normalized remnant polarization under a bipolar cyclic field with an amplitude of ±1.4 kV mm–1 at frequencies of 10, 50 and 100 Hz.
KI ¼ b
Y cs KE; Ec
ð2Þ
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Fig. 3. The optical micrographs of samples before and after fatigue testing with an amplitude of ±1.4 kV mm–1 at (a) 10 Hz, (b) 50 Hz and (c) 100 Hz of a bipolar electric field. Black dashed lines are used to indicate the crack path in each sample.
Table 1 The approximate crack length of the fatigued samples under ±1.4 kV mm–1 of bipolar cyclic loading with different frequencies. Sample
Frequency (Hz)
Initial crack length, ai (lm)
Crack length after 104 cycles of fatigue, af (lm)
A B C
10 50 100
325 500 575
450 550 600
where Y is the Young’s modulus, cs is the electrostrictive strain, Ec is the coercive field, and b is a dimensionless parameter depending on the crack condition (e.g. b 0.25 for an insulating crack). By substituting Eq. (1) into Eq. (2), the relationship between stress intensity factor and the electric field can be expressed as: KI ¼ b
Y cs E pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2h tanðpai =2hÞ: Ec h
ð3Þ
Considering Eq. (3), the stress intensity, KI, is higher if the initial crack length, ai, is longer. Consequently, the crack length effect alone suggests that the greatest crack propagation, Da, in Table 1 should be observed in the sample possessing the longest initial crack length, ai (i.e. sample C). However, the results from this experiment show that the sample with the longest initial crack length (sample C) exhibits the shortest crack propagation, while the sample with the shortest initial crack length (sample A) exhibits the longest or most severe crack propagation. This implies that the effect of cyclic frequency dominates crack propagation under cyclic electric loading. Furthermore, these results show that although the samples were subjected to the same amplitude of cyclic field, the most severe crack propagation occurs under the lowest cyclic frequency of the field. Considering the results from bulk samples and fracture samples, the effect of frequency on polarization and crack propagation can be seen. At the lower frequency, the polarization decays faster and crack propagation per cycle is greater compared with the case of higher frequency. These
Crack growth length, Da mm
%
125 50 25
38 10 4
relationships may be attributed to domain-switching behavior which is the driving force of polarization degradation and crack propagation under cyclic loading. To investigate the micromechanism of domain switching around the crack tip under cyclic electric loading, the fatigued pre-cracked samples were investigated by using a high spatial resolution and high-energy X-ray beam. The domain-switching behavior of the fatigued samples is characterized by the change in the intensities of 002 and 200 peaks representing the c- and a-domains, respectively. The preferred orientations of in-plane domains can be indicated by the integrated intensity ratio of 002 and 200 peaks of which the scattering vector is approximately parallel to the sample surface (see Fig. 1). An example of diffracted intensity of the 002 and 200 peaks as well as fitting curves for the tetragonal phase are shown in Fig. 4. The preferred orientation of c-domains parallel to the electric field direction are indicated by the multiple of a random distribution (MRD), which is expressed by: R I 002 I 002 ; ð4Þ MRD002 ¼ 3 R I 002 I 002 þ 2 I 200 I R200 where I002 and I200 are the integrated intensities of the 002 and 200 peaks of a fatigued sample which possesses preferred domain orientation [24]. The superscript R denotes the integrated intensity of the same peaks for a sample with random domain orientation. From Eq. (4), MRD002 < 1 if the preferred orientation of c-domains is away from the measured direction (i.e. the scattering vector direction); MRD002 = 1 if the orientation of the domains is random; and MRD002 > 1 if there is preferred orientation along
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Fig. 4. 002 and 200 peaks and fitting curves using a split Pearson VII profile shape function.
the measured direction. MRD002 values obtained from the scanning area around a crack tip are plotted as a function of position relative to the crack tip and the direction in the
plane of the sample and are represented as a contour plot. Domain distribution contours of samples fatigued at 10, 50 and 100 Hz are presented in Figs. 5–7, respectively. For each frequency, a set of four contours is shown representing different directions with respect to the crack face and the applied field (this angle is defined as g, see Fig. 1b). The distributions of domain orientations around the crack tip can be interpreted by the combination of two indicators. The first indicator is the direction of a preferred orientation which is represented by an arrow at the left top corner of each contour plot. The second one is the MRD002 values calculated using Eq. (4). For example, in Fig. 5, the preferred c-domain zone (i.e. MRD002 > 1) which orients parallel to the applied field, g = 0° (or perpendicular to the crack face), can be revealed by contour a1 and the orientations g = 45°, 90° and 135° to the applied field are represented by contours a2, a3 and a4, respectively. Moreover, the complete texture of in-plane domains can be obtained by the combination of contours of all g from 0° to 180°. However, for simplicity, the contours presented here are taken from just four selected angles (g = 0°, 45°, 90° and 135°).
Fig. 5. Domain contours around the crack tip in a fatigued sample (sample A) after 104 cycles under a frequency of 10 Hz at different preferred orientations of c-domain. The direction with respect to g of the c-domains is represented by the arrow at the top left corner of each contour. The MRD002 values are indicated by a color bar next to the contour.
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Fig. 6. Domain contours of a fatigue sample (sample B) after 104 cycles at a frequency of 50 Hz at different g angles with respect to the applied field: 0°, 45°, 90°, 135°.
The right-hand side of each plot shows an abnormally high level of MRD002 in each sample. This is related to the interface between the ceramics and electrode, which leads to the very strong preference of the in-plane cdomains in that area. The data can be ignored for this analysis as they are well removed from the crack tip. For Figs. 5–7, the contours of the g = 0° and 90° show symmetry of domain orientation distributions about the crack face, while those of the g = 45° and 135° are asymmetric. The preferred orientations about the crack tip of each fatigued sample can be observed when considering the four g angles together. By comparing the area of preferred c-domain orientations of samples A–C in Fig. 5–7, respectively, it can be seen that they differ with varying frequency. After being fatigued under low frequency (10 Hz), the zone of preferred c-domain orientations, which is represented by MRD002 > 1, is considerably larger than that of the 50 and 100 Hz samples. It is noted that the size of the process zone related to a sample size can affect the level of mechanical constraint. Low mechanical constraint (plane stress) exists if the process zone is large compared to the size of a sample, while high mechanical constraint (plane strain) occurs if the zone
is small. In this case, plane stress exists (particularly for the low frequency) because the size of the process zone (qualified as the region where MRD002 > 1) is approximately 3 3, 2 1 and 0.5 1 mm2 for the samples loaded at 10, 50 and 100 Hz, respectively. The lower-frequency sample exhibits a larger process zone because the total exposure time of the sample to mechanical load during cyclic loading is longer. Reconciliation of the domain contours (Figs. 5–7) and the polarization fatigue measurements (Fig. 2) should be undertaken carefully. The domain contours represent the degree of c-axis alignment around a crack tip subject to cyclic electric field loading at various frequencies. These measurements were conducted ex situ, i.e. after the removal of the electric field, and therefore represent the remnant domain orientations. The polarization fatigue data, however, was measured during the electrical cycling and therefore reflects the degree of domain switching that occurs during the electric field application. The measurements from the two loading scenarios suggest that, while domains can orient a greater amount at lower frequencies (i.e. the degree of domain switching in Fig. 5 is greater than in Fig. 6, which is greater than in Fig. 7), the domain
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Fig. 7. Domain contours of a fatigue sample (sample C) after 104 cycles at a frequency of 100 Hz at different g angles with respect to the applied field: 0°, 45°, 90°, 135°.
switchability declines (i.e. the number of switchable domains decrease, as shown in Fig. 2). In addition to polarization fatigue, the results obtained from domain contours can be used to investigate the relationship between the domain-switching behavior and crack propagation. Non-uniform domain orientation distributions are observed for each fatigued sample. The preferred c-domains oriented 0°, 45° and 135° to the field (or 90°, 135° and 45° to the crack face) are found in front of the crack tip, while those oriented 90° to the field occur at the crack flanks and behind the crack tip. These results agree with the model presented by Zhu et al. [25] and Beom and jeong [6]. Moreover, it can be observed that the switching zone of preferred c-domains oriented at 45° and 135° to the crack face is smaller and exhibits a weaker degree of preferences than those oriented 90° to the crack face. These orientation distributions of preferred c-domains are comparable with the previous measurements using a similar technique [13]. To simplify this explanation, the distributions of preferred c-domain orientations are shown schematically in Fig. 8. In this experiment, the non-uniform distributions of preferred c-domain orientations around the crack tip
may be attributed to the inhomogeneous local field around the crack (Fig. 8b). The inhomogeneous field is caused by the crack conditions (i.e. crack geometry, impermeable or permeable boundary conditions of the crack faces) within the samples. Since the samples were submerged in a silicon bath while being electrically fatigued, it can be expected
Fig. 8. A schematic diagram showing the relationship between the domain orientations and the local field around the crack: (a) represents the random orientations before being fatigued; (b) the local field around the crack tip where the non-uniform distributions of preferred c-domain orientations occur; preferred c-domains align nearly parallel to the electrical field direction.
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that the silicon oil film may fill a gap between crack surfaces. Therefore, it can be assumed that a crack becomes insulating (if the oil fully fills the gap and there is no free space between the crack faces) or semi-insulating (if the oil partially fills the gap and there is free space between the crack faces). Due to being shielded by the crack fully or partially filled with the oil, the electric flux lines deflect around the crack tip and concentrate at the crack tip [26–30]. Consequently, this inhomogeneous field causes the non-uniform distribution of preferred domain orientations (i.e. the preferred c-domain oriented 0° to the crack face occurs at the crack face behind the crack tip while that oriented 90°, 45° or 135° to the crack face takes place at the front of the crack tip). This leads to large ferroelastic strain mismatch, which is a driving force for crack propagation [30], between the area in front of the crack tip and behind the crack tip. Under cyclic electrical loading, if the field increases above the coercive field, 90° domain switching occurs and non-uniform domain orientation distribution is induced by the inhomogeneous local field around the crack. Under this condition, the switching process due to electrical energy induces ferroelastic strain along the crack flank leading to stress, which can be considered as a wedging force. When strain mismatch around the crack tip increases, the stress intensity increases. Consequently, if the stress intensity reaches the critical value (KIc) of the ceramics, the crack starts propagating. 3.3. Model of the frequency effect on crack propagation It has been demonstrated that both an inhomogeneous electric field and non-uniform domain orientation distributions around a crack tip influence crack propagation. In addition, the frequency of an applied field also affects the severity of crack propagation. In a previous work, the authors studied the effect of mechanical loading frequency on ferroelastic strain induced by irreversible switched domains [11] and created a model to explain this relationship. In the present experiment, the previous model is applied to explain the micromechanism of crack growth due to domain switching under different frequencies of an electric field. The model describes the frequency-dependent domain behavior and ferroelastic strain using a viscoelasticity-based concept. This viscoelasticity-based model states that the ferroelastic strain induced by 90° domain wall movement of bulk ceramics can be expressed by: eðf Þ ¼ AeN =f h þ C 0 ;
ð5Þ
where e(f) is the ferroelastic strain as a function of frequency, h is the relaxation time, N is the number of cycles, f is the loading frequency, and A and C0 are constants. An example of experimental data of the ferroelastic strain as a function of the number of cycles under different frequencies of a cyclic loading and fitting curves generated by Eq. (5) is shown in Fig. 9. This model and the experimental results of the previous work show that the ferroelastic strain at lower frequency (1 Hz) is larger than that at higher frequency
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Fig. 9. Frequency effect on ferroelastic strain as a function of the number of cycle. The fitting curve is calculated by using a viscoelasticity-based model (Eq. (5)). (Figure from Ref. [11].)
(20 Hz). This implies that domain walls under a lower frequency are able to travel further than that under a higher frequency. In the case of crack propagation under different frequencies of electric loading, the same viscoelasticity-based concept may be used to explain the link between the frequency of the applied field, domain-switching behavior and crack propagation. Under cyclic loading, when the electrical load increases, domain walls move over a certain distance due to a local electric field. When the load decreases during a cycle, some domain walls move back while some domain walls do not revert to the original position. This may be attributed to microscopic defects within grains which reduce the mobility of domain walls [1,8,18–21,31,32]. At the switching zone just behind the crack tip, this movement of domain walls may therefore cause high local strain and initiate crack opening. In the case of low frequency, domains at the switching zone behind the crack tip experience a longer time under the local field for a single cycle; as a result, domain walls are able to travel over a longer distance due to viscoelastic effects. Therefore, a greater local strain is induced at the switching zone and wedges the crack open, resulting in more severe crack propagation. On the other hand, if the higher frequency of an electric field is applied, domain walls move over a shorter distance compared with the first case. Consequently, the lesser local deformation, which takes place at the switching zone just behind the crack tip, may cause less crack propagation per cycle compared with the case of a low-frequency loading. 4. Conclusions Experimental results demonstrate that the degree of polarization fatigue in PZT cycled at an electric field above the coercive field increases with a reduction in cycling frequency. Electrical cycling of a fracture sample, furthermore, exhibited a greater rate of crack propagation with
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decreasing frequency. Spatial analysis of c-domain orientations around a crack tip show a clear zone of preferred domain orientations. The orientation of the c-domains can be predicted using the path of the electric field around an insulating crack. Additionally, the size of the zone of preferred domain orientations increased with decreasing frequency. It is therefore concluded that: (1) The bipolar electrical cyclic loading causes a decrease in domain switchability, reflecting decline in the remnant polarization. (2) The viscoelastic nature of domain wall movement under numbers of loading cycles leads to greater degrees of polarization fatigue at lower frequencies. (3) Cyclic fatigue crack growth in piezoelectric materials is caused by ferroelastic strain mismatch at the crack tip caused by domain reorientations induced by the path of the electric field. (4) A large ferroelastic strain mismatch causes a high stress intensity factor which is a driving force for crack propagation. Eventually, the crack starts propagating when the stress intensity factor reaches the critical value (KIc) of the material. (5) The magnitude of this mismatch decreases with increasing frequency due to reduced amount of domain reorientation per cycle owing to the viscoelastic nature of the domain movement. (6) Hence, fatigue crack growth rates per cycle decrease with increasing cycling frequency. Acknowledgments The authors gratefully acknowledge financial support from an ARC Discovery Grant No. DP0558596. Moreover, the authors would like to thank Dr. Nagarajan Valanoor of the School of Materials Science & Engineering, The University of New South Wales, for his fruitful discussions. Travel support to access these facilities was provided by the Access to Major Research Facilities (AMRF) program, administered by the Australian Nuclear Science and Technology Organisation (ANSTO). J.L.J. acknowledges support for this work from the US National Science Foundation through award number OISE-0402066. S.P.(I.)
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