SCIENCE CHINA Physics, Mechanics & Astronomy • Research Paper •

May 2011 Vol.54 No.5: 957–965 doi: 10.1007/s11433-011-4308-y

Electromechanical cracking in ferroelectrics driven by large scale domain switching† CUI YuanQing1 & YANG Wei2* 1

School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China; 2 University Office, Zhejiang University, Hangzhou 310058, China Received December 28, 2010; accepted March 4, 2011; published online March 18, 2011

Experimental results indicate three regimes for cracking in a ferroelectric double cantilever beam (DCB) under combined electromechanical loading. In the loading, the maximum amplitude of the applied electric field reaches almost twice the coercive field of ferroelectrics. Thus, the model of small scale domain switching is not applicable any more, which is dictated only by the singular term of the crack tip field. In the DCB test, a large or global scale domain switching takes place instead, which is driven jointly by both the singular and non-singular terms of the crack-tip electric field. Combining a full field solution with an energy based switching criterion, we obtain the switching zone by the large scale model around the tip of a stationary impermeable crack. It is observed that the switching zone by the large scale model is significantly different from that by the small scale model. According to the large scale switching zone, the switch-induced stress intensity factor (SIF) and the transverse stress (T-stress) are evaluated numerically. Via the SIF and T-stress induced by the combined loading and corresponding criteria, we address the crack initiation and crack growth stability simultaneously. The two theoretical predictions roughly coincide with the experimental observations. ferroelectric ceramics, large scale domain switching, electromechanical loading, crack initiation, crack growth stability, stress intensity factor, transverse stress PACS: 77.84.-s, 77.80.Fm, 46.50.+a, 62.20.Mk

1

Introduction

Ferroelectric ceramics has found extensive applications such as structural health monitoring, micro-electro-mechanical systems, capacitor, and random access memory. In these services, ferroelectrics have to bear external electrical and/or mechanical loadings in a cyclic or a static form. Crack growth in ferroelectrics driven by electrical or mechanical, or electromechanical loading has been an important topic in the past two decades [1–14]. For details, one can refer to the monograph by Yang [15] and the review by Schneider [16]. Polarization switching holds the key to *Corresponding author (email: [email protected]) †Contributed by YANG Wei

© Science China Press and Springer-Verlag Berlin Heidelberg 2011

characterizing the phenomena of hysteresis and nonlinearity in ferroelectric materials. According to the current knowledge about polarization reorientation, existing constitutive models fall into two categories: the phenomenological models [17–24] and the models based on the switching mechanism [25–28]. The energy motivated switching criterion by Hwang et al. [25] has been frequently cited for its simple and intuitive characterization on the hysteresis and nonlinearity of ferroelectrics, especially for analytical purpose. Following the spirit of transformation toughening in zirconia [29] and using the energy based switching criterion, a small scale switching (SSS) model was formulated as an effective tool for the analysis of switching related phenomena [26, 30]. In the case of SSS, the switching zone size is taken as considerably smaller than the specimen size. The SSS model can be used to explain switch toughening, phys.scichina.com

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toughness anisotropy and fatigue crack growth in ferroelectrics [26,30–32] and is further developed by other researchers, such as refs. [33–36]. If the applied electric field is lower than one half of the coercive field of ferroelectrics, the SSS model is applicable. In some applications, however, the applied field is near or above the coercive field. In such a case, a large or global scale domain switching occurs. Accordingly, a large scale switching (LSS) model has to be adopted to analyze this kind of switching. In the case of LSS, the switching zone is comparable with the specimen size, while in the global case the switching zone spans the whole area (see ref. [30]). However, knowledge about the fracture of ferroelectrics under large scale switching is still very limited [37]. For a better understanding about the mechatronic reliability of ferroelectrics subjected to a strong electric field, further explorations are needed. Take the case of purely electric loading as an example. In the SSS model, the switching is driven solely by the singular term of an electric field around a crack tip, and non-singular terms are neglected. Under the loading of a high electric field whose amplitude is comparable with the coercive one, however, non-singular terms are no longer negligible and the LSS model has to be adopted. To perform the LSS analysis, one needs to seek a full field solution of the electric field near the crack tip, a solution that contains a singular term and the other non-singular terms. The present work conducts preliminary theoretical (or almost theoretical) research about the crack initiation and crack growth stability of a stationary crack according to the large scale domain switching. This study is motivated by the experimental data observed in a double cantilever beam (DCB) subjected to electromechanical loading, where the maximum amplitude of the cyclic electric field reaches almost twice the coercive one [38]. Concurrent application of a mechanical load with an electrical load in a DCB specimen allows a quantitative experimental study not only for crack propagation, but also for crack path stability. The theoretical work therefore will address computations of not only stress intensity factor (SIF) to judge the crack initiation, but also transverse stress (T-stress) as a criterion for crack path stability. Three different regimes of crack growth were identified depending on the combinations of mechanical SIF with electric field amplitude: no crack growth, straight crack growth (or stable crack path), and curved crack growth (or unstable crack path) which resulted in eventual fracture of the specimen. A framework of large scale domain switching will be developed to elucidate these regimes. Our theoretical predictions roughly agree with the experimental data.

2 Large scale domain switching driven by combined electromechanical loading At first, we introduce the experimental result of a DCB test

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subjected to joint electromechanical loading. It is observed that a crack takes on one of the three various growth behaviors for different combinations of mechanical loadings with electrical loadings. To interpret the experimental observations, we develop a large scale switching model because the applied electric field is so strong that the SSS model is not applicable any more. To quantify the large scale switching, we have derived a full electric field solution in closed form, reported elsewhere. Combining the full field solution with an energy based switching criterion, we obtain the switching zone around the tip of a stationary impermeable crack by the large scale model. 2.1 A DCB specimen subjected to combined electromechanical loading A brief introduction is given to an existing experimental study of the crack growth of ferroelectric ceramics under the joint electromechanical loading. Westram et al. carried out tests of DCB specimens to investigate crack initiation and crack growth stability under the combination of a cyclic sinusoidal electric field, covering a wide range of amplitudes, with a statically mechanical load [12,38]. The latter alone was maintained at subcritical SIFs. Note that the first quarter of the first sinusoidal cycle of the cyclic electric field loading can be viewed as a poling process; the third quarter acts as a negative electric field loading (see ref. [31]); and the first and the third quarters of all following sinusoidal cycles would also act as a negative electric field loading. For simplicity, the ferroelectric ceramic is assumed to be mono-domain, isotropic and with negligible piezoelectric effect. Thus in an actual test, the loading of the cyclic electric field in the DCB sample is equivalent to a negative electric field loading. In this type of loading, both the initial orientation of the mono-domain and the applied electric field are along the vertical axis, but opposite to each other. A Cartesian coordinate is introduced such that the horizontal axis is along the crack extension line, while the vertical axis normal to the crack. The ferroelectric used in the experiment is PIC 151, a soft commercial PZT with coercive electric field about 1.0 kV/mm. It is a PZT composition on the tetragonal side near the morphotropic phase boundary. Specimens were 40×5×1.5 mm3 in dimension. A sketch of the prepared specimen can be seen in Figure 1 with the final parameters of a=5 mm, h=2.5 mm, and W= 37 mm. One may refer to ref. [38] for the details of the DCB test. A statically mechanical load was applied in terms of a constant SIF in Mode I, with the latter determined by [39,40]: K IM  2 3

Pa  h 1  0.64  . 3  a bh 2 

(1)

Here, P is the applied load in Newton, h, the half height of the specimen, b, the thickness, and a, the distance from the loading point to the crack tip. For the DCB tests under

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various combinations of electromechanical loading, different crack propagation phenomena were observed, as shown in Figure 2 (Rödel and Westram, Private communications, 2007). In the whole set of experiments, four different mechanical loads in terms of the applied SIF were used: 0.2, 0.3, 0.4, and 0.5 MPa m (the unit will be omitted in the sequel for clarity), as well as eleven different electric field amplitudes. All combinations of the applied SIFs with electric field amplitudes are displayed in Figure 2. Three different regimes of crack growth were observed in the experiments: no crack growth, straight crack growth and curved crack growth which resulted in eventual fracture of the specimen. These three regimes are labelled by solid circles, triangles and asteroids in Figure 2. The same legends will be adopted in related figures in the sequel. For a prescribed level of SIF, say 0.4, the crack propagation mode changes from “no crack growth”, to “stable crack path”, to “unstable crack path” with an increase of electric field amplitude. On the other hand, the crack path was observed to be unstable and the crack gradually kinked and the specimen broke before 60 electric field cycles were accomplished for combinations of higher mechanical load with electrical one [38].

Figure 1 Geometry and loading of a double cantilever beam specimen.

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For the level of electrical and mechanical loads, the largest mechanical SIF is 0.5, less than the intrinsic fracture toughness of PIC 151, 0.7; while the maximum electric field amplitude is 1.7 MV/m, almost twice the coercive field of PIC 151. From the comparison between the two types of loadings, it is argued that the electrical load plays a much more important role than the mechanical one in most cases of combined load. As a rough assumption, we only consider electric-field-induced ferroelectric polarization switching, and ignore the contribution from the mechanical load to domain reorientation. But we count on the contribution from the mechanical load to the total SIF and total T-stress under electromechanical loading. 2.2

Full field solution under electric loading

Following Yang and Suo [41], a process of field-induced polarization switching is decoupled into solving field distribution and domain reorientation driven by the field. As the first step to analyze electric-field-induced switching that takes place in the DCB test, one needs to know the electric field distribution. As demonstrated in Figure 2, the applied electric field is so strong that the non-singular terms of the crack-tip asymptotic field can not be neglected any more. Namely, one has to obtain the full asymptotic expansion of the crack-tip electric field, on which this subsection focuses. The next subsection will deal with the polarization switching driven by the full field. In view of the dimensions of the DCB specimen, the realistic configuration was approximated by an infinitely horizontal strip containing a semi-infinite crack. Besides, since the specimen was entirely immersed in silicon oil such that the pre-crack is covered by insulating oil, the slit surface was modeled as an impermeable electrical boundary condition. Solving the electric field distribution is transformed into a mixed boundary value problem, which was solved by analytic function and conformal mapping, and the full electric field reads [42] E1  iE2   Eapp  exp  π z / h   1



1 2

,

(2)

where Ei is the electric field component, i  1, Eapp the electric field intensity applied to the strip, and z, complex variable. The bar over the complex variable means the complex conjugate. The closed form solution (2) can approximate the electric field distribution within the DCB specimen. 2.3 Figure 2 Crack stability map under different combinations of mechanical SIF with electric field used in the tests. The data are used by courtesy of Rödel and Westram. The dashed curves indicate the approximate boundaries dividing three regimes.

Large scale domain switching

After the determination of the field distribution, we now shift our focus to the electric field induced domain reorientation. We adopt an energy based switching criterion proposed by Hwang et al. [25]. For a mono-domain, polariza-

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tion switching is activated if

 ij  ij  Ei Pi  Wc

(3)

for both 90° and 180° types. In (3),  ij denotes the stress tensor,  ij , the switching strain tensor, Pi , the change in the polarization vector that occurs during the domain switching, and Wc , an energy threshold necessary for domain reorientation. For the switching behavior of ferroelectric ceramic that consists of domains of an arbitrary orientation, the switching criterion is modified as:

 ij  s ij  Ei P s Pi  Wc ,

(4)

where  s and P s denote the magnitudes of switching strain and polarization,  and P , for the angular ij

i

distributions associated with an initial polarization angle of the switched portion, whose expressions may be found from refs. [15,30]. The modified criterion incorporates a realistic switching degree of ceramic under different levels of the applied load. Namely, the concept of volume fraction is absorbed into  s , P s , and Wc . Accordingly, these quantities are no longer constants and become load dependent. For the case of domain switching driven by the crack tip field, one may regard these three quantities as taking their respective saturated values. Combining eq. (2) with the equality in eq. (4), one may determine the switching boundary in terms of the LSS model. A theoretical but implicit expression is then obtained for the boundary of LSS. To have a visual representation, one solves the implicit equation of the switching boundary by the method of bisection. With the help of this numerical technique, the boundary of LSS is obtained. The switching zones around a stationary crack tip via the LSS model are demonstrated in Figures 3(a) and 3(b) for electric field intensities of 1.0 and 1.5 kV/mm. The switching zones in terms of the SSS model are also presented for comparison.

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The latter are plotted according to the analytical equation given in ref. [31] under an intensity factor of electric field for a semi-infinite strip. The switching zone refers to the region within which eq. (4) is satisfied. The symmetry allows the illustration of only the lower half of the switching zone. In Figures 3(a) and 3(b), the maximum half-heights of 180° switching zone are 1.26 and 2.5 mm. They are comparable with and equal to the half height of the strip respectively. In these two figures, the zones of 180° switching are extensive and global in scale whereas the zones of 90° switching are approximately small scaled for the former, and large scaled for the latter under these large electric fields. Consequently, the surrounding un-switched portion would provide enough constraint for the switched core in the former, while not enough constraint in the latter. For simplicity, we calculate the switch induced toughness variation of the latter case by neglecting the effect of the insufficient constraint. To facilitate subsequent analysis, we display schematically the switching boundary around the tip of a stationary crack under a negative electric field loading in Figure 4, where E denotes the applied electric field vector, P0, the initial polarization vector, P90, the polarization vector which experiences 90° domain reorientation, and P180, the polarization vector which experiences 180° domain switching. Only the 90° domain reorientation induces a switching strain. Thus the 90° switching zone is the only focus of the present work. As the figure shows, the whole boundary of 90° domain switching is divided into three segments: the crack face 1, the curved boundary 2 for 90° switching induced by the negative electric field, and a radial line 3 dividing the regions of 90° and 180° domain switchings.

3 Stress intensity factor induced by electromechanical loading Up to now, we have obtained the switching zone according

Figure 3 Comparison of the 90° and 180° domain switching boundaries around a crack tip as predicted by the small scale switching (SSS) and the large scale switching (LSS) models. The electric fields in various graphs have the amplitudes of (a) 1.0 kV/mm and (b) 1.5 kV/mm [42].

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tion. Taking the assumption of elastic isotropy, one may arrive at the following expression of Ti around s under a purely deviatoric deformation caused by domain switching [30]: Ti 

Y s  Ti , Ti  ij n j , 1 

(7)

where n j denotes the outward normal of s, Y, the

Figure 4 Borders of the 90° domain switching zone around the tip of a stationary crack.

to the LSS model. Based on it, we will present the solving procedures for the SIF and T-stress induced by the combined electromechanical loading. After polarization switching, constraining traction would appear along the switching boundary. Its presence causes stress redistribution near the flaw, and produces an increment of stress intensity factor, K, and a variation of T-stress at the crack tip. According to the load-induced SIF and the crack-tip SIF criterion, we can judge when the crack would initiate for different combinations of mechanical and electrical loadings. According to the load-induced T-stress and the corresponding T-stress criterion, we can determine when the crack would propagate stably and when the crack would propagate unstably. This section focuses on calculating the load-induced SIF, while the next section focuses on determining the load-induced T-stress. The crack tip SIF is related to the applied one by K tip  K app  K .

s

(6)

where s is the whole boundary of the switching zone that is composed of three segments 1, 2 and 3 shown in Figure 4 along the anti-clockwise direction, Ti , the traction distribution acting along the boundary, and hi , the weight func-

1

hi (r ,  ) 

hi ( )

2  (1   )

r

,

 3   (1   ) cos  sin  sin   h1   2 2 ,     h2  (1   ) sin   sin  cos 3   2 2 

(8)

where   3  4 for plane strain and    3   1    for plane stress. Corresponding to the division of s into three segments, one can compute the toughness variation from each segment and then get a summation. To clarify the dimensionality of the presentation, we rewrite the contribution from the i-th segment  i as r2

K     Ti hi ds  i

r1

(5)

From the hypothesis in sect. 2, mechanical and electrical loadings make their contributions to Ktip respectively. The term Kapp comes from the mechanical loading, which could be determined according to eq. (1); the other term K comes from the electric-field-induced domain switching, which could be computed by the Eshelby-McMeeking-Evans method [15,29]. The rest of this section is concentrated on its computation. Specifically, one may integrate the constraining body force layer through the corresponding weight function to give the toughness variation: K    Ti hi d i ,

Young’s modulus, and , the Poisson’s ratio. For simplicity, we use the weight function for a semi-infinite crack in an otherwise infinite plate to approximate that for the DCB sample. For a planar configuration, the mode I weight function for a semi-infinite plane crack under a point force acting at point  r ,   was given by [43]



 2

r2 r1

ds Ti hi , r

(9)

where r1 and r2 are the lower and upper bounds of integration variable and

  The quantity 

Y s . 1  1   

(10)

has a stress dimension and reads

Y  / [4(1  )] in the plane strain case. s

4 4.1

2

T-stress under electromechanical loading T-stress induced by mechanical loading

T-stress is the first non-singular term in the Williams expansion [44] and an important parameter in fracture mechanics. Of particular relevance is the proposition of Cotterell and Rice [45] that the T-stress can serve as a criterion for the crack path stability. Specifically, a straight mode I

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crack path is stable for T < 0 and unstable for T > 0. As for the T-stress induced by the joint electromechanical loading, it is well known that mechanical load would result in T-stress; on the other hand, the constraining effect of domain switching can also induce T-stress. Thus for the DCB test, the contributions to T-stress are TM from the mechanical load and TDS from domain switching: T  TM  TDS .

(11)

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the crack faces, x denotes the location where the component of  xx is computed, and the superscript “(1)” refers to the association with the segment of crack face 1 of the switching boundary. By integration, the T-stress at the crack tip resulting from a constant distribution of normal stress along a segment of crack surface from  a, 0  to the crack tip reads

TM 

K IM

1 . a 0.681 h a  0.0685

(12)

The following part of this section is concerned with determining TDS. Similarly, we utilize a Green’s function to calculate it. 4.2

T-stress induced by domain switching

The switch induced T-stress can be obtained by integrating the constraining traction weighted by the corresponding Green’s function along the whole switching boundary s in an anti-clockwise direction: TDS    Ti ti di . s

(13)

Following Cui [48], the Green’s function of T-stress of a pair of symmetric point forces acting at an arbitrary point (not on the crack face) with polar coordinates ( r0 ,  0 ) for a semi-infinite crack in an otherwise infinite plate can be expressed as: ti  r0 ,  0  

ti  0  2 ,  1    r0

 t1  cos3 0   cos  0      . t2   sin 3 0   sin  0 

(14)

Similar to (9), one can calculate the switch induced T-stress of each segment and then get a summation. The contribution of T-stress from the i-th segment  i can be rewritten as r2

T     Ti ti ds  i

r1





r2 r1

ds Ti ti . r

(15)

An exceptional case is encountered when the loading point lies on the crack surface. The Green’s function of T-stress in this case is [49] t    x0 , x    1

1 x , x  0,   x  x0   x0

(16)

where x0 is the loading position of the normal stress on

0

T      yy  x0  lim t    x0 , x  dx0    yy 1

The mechanical contribution for the DCB geometry is given by the following formula [46–48]

a

1

x 0 

x0  0

,

(17)

where  yy denotes the normal traction component. Using the theory of second order weight function, Sham arrived at the same result as the above equation [50].

5

Discussion

The above two sections have presented the solving procedures for computing the SIF and T-stress induced by the joint electromechanical loading. By combining the load-induced SIF and T-stress with corresponding criteria, this section aims to quantitatively interpret the experimental results about various crack growth behaviors for different load combinations shown in Figure 2. 5.1

Stress intensity factors and crack initiation

The crack tip stress intensity factor, calculated according to (5), is used to judge the cause of crack initiation. In the calculation of SIFs, contribution from Γ2 has to be obtained by numerics, while the contributions from Γ1 and Γ3 are analytically evaluated from the formulae shown in Appendix A. Material parameters of PIC 151 are listed as follows: the Young’s modulus Y=60 GPa, the Poisson’s ratio =1/3, Wc=3.0 MPa, Ps=0.4 C/m2, and s=0.0032. The last three quantities are from the experimental measurement [51]. The value of Wc is determined from the longitudinal compression versus the longitudinal strain curve, since that test involves only 90° domain switching. That value is caused by the product of the spontaneous polarization strain and the saturated compression stress, under which saturated domain switching is achieved. Based on the combined theoretical derivation of sect. 3 and numerical simulation, the predicted SIFs by the LSS model are shown in Figure 5. The experimental results in Figure 2 are also plotted in the same figure for comparison. We combine the load-induced SIF with the near tip SIF criteria to judge when the crack would initiate. The dashed line indicates the predicted dividing boundary between the no crack growth region and that of crack growth. Most experimental points are separated by the dividing boundary of Ktip=0.7. Moreover, from the meeting point between the dashed line and the abscissa axis, one gets the critical elec-

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Figure 5 Predicted contours of the stress intensity factors in comparison with the experimental data.

Figure 6 Contours of T-stress from the theoretical prediction in comparison with the experimental data.

tric field, 1.0 kV/mm, to initiate the crack. That value is consistent with the experimental measurement. Similarly, from the extrapolation crossing point between the dashed line and the vertical axis, the predicted fracture toughness is about 0.73, which agrees well with the intrinsic one. It is concluded that the present theoretical prediction based on the crack tip SIF is in agreement with the crack initiation data of PIC 151.

combination and underestimate that for higher loading combination.

5.2

T-stress and crack path stability

Carrying out the integration in eq. (13) of sect. 4 one obtains the T-stress as a function of the applied electric field. Superposition of this result with the mechanically induced T-stress (12), as anticipated in (11), would yield the variation of T-stress under coupled electromechanical loading. The final results, in conjunction with the experimental data reported in Figure 2, are plotted in Figure 6. We adopt the criteria of crack path stability by T-stress to judge different crack growth behaviors. The dashed dot curves stand for the contour lines of negative T-stress and the solid curves for positive T-stress. The dashed curve stands for zero T-stress and acts as the dividing boundary of stable and unstable crack paths. It is easy to identify that the region of no crack growth and the major region of stable crack path coincide with that of negative T-stress. The contour plots of T-stress clearly indicate that the crack path stability is an outcome of the positive T-stress, as proposed by ref. [45]. Considering the sign and amplitude of T-stress under the combinations of electromechanical loading located in the right of the dashed boundary in Figure 6, the kinking of crack path and the breaking of sample is apparent. The theoretical predictions agree well with the testing data for the mechanical SIF of 0.3 and 0.4, but not so well for the rest. One possible explanation might lie in the assumption of a constantly saturated switching degree for all combinations of mechanical loadings with electrical loadings. This assumption may overestimate the switching degree for lower loading

5.3 Predicting different cracking regimes by combined SIF and T-stress For the sake of a comprehensive comparison, Figures 5 and 6 are combined into Figure 7, where solid curves stand for the contours of SIF and dashed curves for T-stress. In Figure 2, it is clearly seen that three crack propagation regions: no crack growth, stable crack path and unstable crack path, exist for different combinations of mechanical loadings with electric loadings. But in Figure 7, it is worth noting those theoretical predictions are in rough agreement with experimental measurements concurrently in two cases: one case is the coincidence of the dividing boundary between no crack growth and crack initiation; and the other is the consistence of the division curve between stable and unstable crack growth. In addition, for a certain level of mechanical SIF, the crack propagation mode changes from no crack growth, to first stable crack path and then to unstable crack path

Figure 7 Joint contour plots of SIF and T-stress from the theoretical prediction in comparison with the experimental data.

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with increasing electric field, as demonstrated by the DCB test. 5.4

Remarks

As a preliminary study for the analysis of LSS and related calculations, several hypotheses are made herein and there are even disagreements among those hypotheses. For example, the piezoelectric effect of ferroelectrics is neglected since the current work is dominantly analytical in nature and attention is essentially focused on polarization switching. Improvement is possible by the finite element method analysis. For the electric field above the coercive one, the insufficient constraint in the LSS model would also lead to some error. This is a challenge that may be eliminated by increasing the specimen height or changing the test type and that demands further research. Moreover, the geometry of DCB specimen was approximated as a semi-infinite crack in an otherwise infinite plate in the analytic expression for Green’s function of T-stress. On the other hand, the crack configuration was supposed to be a semi-infinite crack in an infinite strip in solving the full electric field solution. The removal of these inconsistencies will be the object of future work. Despite this, our theoretical predictions are consistent with the experimental data. Especially the predictions can simultaneously describe crack initiation, as well as stable and unstable crack growth.

6

Conclusions

Experimental data about crack propagation are observed in the ferroelectric double cantilever beam specimen subjected to the combined electromechanical loading, with the electric field almost twice the coercive field. Three crack propagation regimes, namely, no crack growth, stable crack path and unstable crack path, exist for different combinations of mechanical SIFs with electric field intensities. The amplitude of the applied electric field clearly suggests the occurrence of a large or global scale domain switching. An analytically full electric field solution is used to construct a large scale domain switching model. Based on the large scale model, we compute the SIF and T-stress induced by the joint electromechanical loading. Combining the load-induced SIF with the criteria of crack initiation by the near tip SIF we can tell the crack initiation. Combining the load-induced T-stress with the T-stress criterion we can judge the crack path stability. The combination of these two theoretical predictions serves to elucidate the experimental data. As a result, the theoretical predictions by the large scale switching model provide a reliable framework of crack initiation, propagation and stability simultaneously. We would like to thank the sharing of illuminating experimental results from Professor Jürgen Rödel and Dr. Ilona Westram of Darmstadt Univer-

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sity of Technology, Germany. This work was supported by the “SinoGerman Center for Research Promotion” under a project of “Crack Growth in Ferroelectrics Driven by Cyclic Electric Loading”, the National Natural Science Foundation of China (Grant No. 10702071), the China Postdoctoral Science Foundation (Grant No. 201003281) and the Shanghai Postdoctoral Scientific Program (Grant No. 10R21415800).

Appendix A

Evaluation of domain switching effect

Attention is first focused on the contribution from the crack surface Γ1. That contribution to SIF is K    1



r   

 2

0

2r    dr Ti hi  1    .  r

(a1)

The T-stress contribution from the crack surface has the following form T     1    . 1

(a2)

Next consider the segment 3 dividing regions of 90° and 180° domain switchings. The toughness increment from this segment is expressed by K    3





0

 T h 2    r

i i

c

1 r

 dr 

2 r  c 

7  3 3 1     sin c  sin c  sin c 2 2 2 2 2 



  . (a3) 

where  c denotes the angle between 3 and the positive abscissa axis. A logarithmic singularity occurs in the calculation of the T-stress resulting from 3, as shown in the formula below, T   3



1





T t  dr   2 sin 4 r      r

i i

c

c

ln

r  c 



.

(a4)

This problem is dealt with by assuming an appropriate value for . The size of lattice parameter of PIC 151, 0.4 nm [52], is used in the numerical calculation of the present work.

1 2 3 4 5

6

7

8

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