Tuning of non-uniform switch toughening in ferroelectric composites by an electric field

Xiaodong Xia·Zheng Zhong

?

RESEARCH PAPER

Tuning of non-uniform switch toughening in ferroelectric composites by an electric field

Xiaodong Xia1·Zheng Zhong1

This paper deals with a mode III interfacial crack subject to anti-plane stress and in-plane electric fields.The analysis concentrates on the tuning of fracture toughness from non-uniform ferroelectric-ferroelastic domain switching by an electric field.The electric loading changes the size of the asymmetric switching zone.Employing the weight function method，we obtain the electrically-dependentswitch toughening forstationary and quasi-static growing interfacial cracks，respectively.Multi-domain solutions are derived for non-poled and fully-poled ferroelectric composites.Numerical results are presented on the electric field tuning of the critical applied stress intensity factor.The research provides ways to optimize fracture properties offerroelectric composites by altering the electric field.

Interfacial crack·Electric field tuning·Domain switching·Ferroelectric-ferroelastic·Toughening

1　Introduction

The discovery of ferroelectrics has led to a significant development in smart materials［1］.Plentiful applications have been made in micro-electro-mechanical devices，such as piezoelectric nanotubes for micro-fluidic system，spintronics，actuators and sensors，etc.［2-4］.However，electromechanical coupling effects are too weak in the existing single-phase ferroelectrics.Ferroelectric composites are designed by fabricating dissimilar ferroelectric layers forpractical applications［5］.Unfortunately，the strength of the interfaces in ferroelectric composites is much weaker than that of the composite constituents［6］.Cracks and flaws inevitably emerge on the interface offerroelectric composites during manufacture and service［7］.Therefore，researches on the fracture toughness of ferroelectric interfacial cracks are of great importance.

? Zheng Zhong zhongk@#edu.cn

1School of Aerospace Engineering and Applied Mechanics，Tongji University，Shanghai 200092，China

In retrospect，there is a substantial body of literature on the interfacial crack of linear piezoelectrics.Comprehensive reviews are provided by Chen and Hasebe［8］and Govorukha et al.［9］.The analysis of piezoelectric interfacial cracks was initiated by Suo et al.［10］，who gave the crack tip field as a linear combination of the coupling oscillating singular field and the non-oscillating singular field by employing Stroh’s method.Based on this framework，Ru［11］gave the exact solution forfinite discontinuouselectrode layersembedded at the interface of bonded dissimilar piezoelectric half-planes. Similarly，Gao and Wang［12］derived Green’s function for an interfacial crack between two dissimilar piezoelectric media under the permeable electric boundary condition，and obtained an explicitclosed-formsolution forthe electric field inside the crack.In addition，Boem［13］investigated the interfacial crack with electrically permeable boundary conditions between two dissimilar piezoelectric ceramics under electromechanical loading，and confirmed an inverse square rootsingularity and a pairofoscillatory singularities nearthe interfacial crack tip fields.Also，Li and Chen［14］performed a similar research for a permeable interface crack in an elastic dielectric/piezoelectric bimaterial.Moreover，Ou and Wu［15］proved that there is no coexistence of the oscillating and non-oscillating parameters in the interfacialcrack-tip generalized stress field for all transversely isotropic piezoelectric bimaterials.

Analysis of the electrically-dependent fracture toughness in a mode III piezoelectric interfacial crack drew more attention due to its physical practicality and mathematical simplicity.One powerful method is the integral transformation.So then Narita and Shindo［16］investigated the anti-plane problem of a piezoelectric layer sandwiched by two dissimilar materials with a crack at the interface based on the Fourier integral transformation method，and found that the stress intensity factor is higher for negative electric fields and lower for positive electric fields.Additionally，Soh et al.［17］gave the analysis of bi-piezoelectric ceramic layers subject to anti-plane shear and in-plane electric loading by means of the Fourier integral transformation，and studied the effects of electric loading on the energy release rate.Further research by Li et al.［18］considered a moving interfacial crack between two bonded piezoelectric materials，and discovered that the distribution of a remote stress field is determined by both the remote mechanical and electric loadings.Another powerful method for crack analysis is the complex function analysis.Then Gao and Wang［19］dealt with a mode III interfacial crack subject to piecewise uniform out-of-plane mechanical loading combined with in-plane electric loading under permeable boundary conditions，and showed that the field singularities are independent of the electric loading.In addition，Li and Kardomateas［20］discussed a mode III interfacial crack located at the interface of the dissimilar bimaterial considering the electro-magnetic fields in the crack by means of Stroh’s method，and concluded that the direction of electric loading has an effect on the possible growth of the interface crack.More research by Li et al.［21］solved a mode III crack terminating at and normal to the interface of two bonded dissimilar materials by solving a singular integral equation，and found that the contribution of electric loading vanishes underthe permeable boundary condition.

However，discrepancies between experimentalresults and theoretical predictions within the linear constitutive models imply that nonlinear phenomena of ferroelectric composites are of great significance［22-24］.Furthermore，Park and Sun［25］tested the fracture of ferroelectric specimens under electric loading and found there might exist factors beyond the scope of linear piezoelectricity that would affect fracture behavior near the crack tip.It is to be noted that Fu and Zhang［26］obtained a nonlinear relationship between the fracture toughness and the applied electric field in fracture experiments of poled commercial lead zirconate titanate（PZT）.In addition，dos Santos e Lucato et al.［27］observed the electrically driven crack propagation in nonpoled and poled PZT，and attributed the reinforcement of poled specimens to domain switching.From the aspect of theoretical researches，Yang and Zhu［28］investigated the influence of the electric loading on the fracture toughness in homogenous ferroelectrics considering the domain switching process.Also，Landis［29］gave complete asymptotic solutions for a crack embedded in a ferroelectric switching material subject to anti-plane shear the in-plane electric fields.Moreover，Wang and Landis［30］explored the effect of in-plane electric field on the toughening behavior of mode I steady-state crack.In addition，Liu and Hsia［31］suggested the likelihood ofdomain switching nearthe interfacial crack tip when exploring the in-plane electric loading in ferroelectric composites.Beyond this，Liu and Fang［32］investigated the domain switching behavior in ferroelectrics by using nonlinear finite element method.Also，Qiao et al.［33］gave phase-field simulation of domain switching around electrically permeable and impermeable cracks in ferroelectric crystals.More research by Li et al.［34］studied the mode I crack propagation of ferroelectric ceramics in open circuits using nonlinear fracture mechanics.Then Xia et al.［35］dealt with a mode III interfacial crack and gave analytic expressions for the switch toughening effects within the elastic range.To the best of the authors’knowledge，theoreticalstudieson the electrically-dependentswitch toughening for the ferroelectric interfacial cracks have not been addressed.

The present paper deals with the tuning of switch toughening in ferroelectric composites by an electric field.The tuning effect is characterized by the non-uniform domain switching induced by the electric field.The plans of the rest of this paper are organized as follows.Section 2 gives the switch-induced strain and polarization driven by the crack tip stress and electric fields.Section 3 introduces the realistic switch-induced strain in ferroelectric solids with the electric field tuning.Section 4 gives the variation of the asymmetric switching zone with respect to the electric loading.Then in Sect.5，the electrically-dependent switch toughening effects for stationary and quasi-static growing interfacial cracks are derived analytically by means ofthe weightfunction method. Section 6 presents the corresponding numerical calculation on the critical applied stress intensity factor with respect to the magnitude and direction ofthe electric field.Finally，conclusions are summarized in Sect.7.

2　Domain switching driven by the crack tip stress and electric fields

Consider a semi-infinite interfacial crack subject to antiplane mechanical loading combined with in-plane electric loading at infinity，as schematically illustrated in Fig.1. Superscripts of“I”and“II”represent the dissimilar ferroelectric constituents in the upper and lower half-planes，respectively.Let（x1，x2，x3）denote a global Cartesian coordinate system attached to the cracked solid，with the origin centered at the current crack tip.（r，?）are the polar coordi-nates corresponding to the in-plane coordinates x1-x2.In addition，a local Cartesian coordinate systemis attached to the tetragonal ferroelectric mono-crystal，whereis along the initial poling axis andalong other optical axes，as depicted in Fig.2.

Fig.1 A semi-infinite interfacial crack between dissimilar ferroelectric constituents subject to anti-plane mechanical loading and in-plane electric loading at infinity

Fig.2 Schematic of 90?and 180?domain switching of tetragonal ferroelectric mono-crystal in the local Cartesian coordinate

In this work，we assume that the ferroelectric constituents are modelled as elastically and dielectrically isotropic［36］. This assumption is also adopted to establish the domain switching criterion［37，38］.The interaction between stress and electric fields is attributed to the domain switching near the crack tip，which is assumed to be weak and negligible outside the switching zone［29，36］.Attention here is focused on the case of small scale switching［28，39-41］，in the sense that the switching zone size is considerably smaller than the specimen size.The influence of residual stress induced by polarization switching is ignored under this assumption.All switching strains serve to toughen the material，and cannot be accommodated by the globaldeformation.The researches of large scale switching can be referred to the work by Cui and Yang［42］，Cui and Zhong［43］.

The crack tip stress field of a mode III interfacial crack is given as［44，45］

where Kappis the applied stress intensity factor at infinity，and the in-plane components ofstress field are zero underthe smallscale switching assumption.In addition，the permeable electric boundary condition is employed for the interfacial crack surface［46］，which overlooks the perturbation on the electric field by the presence of defects［28］.The electric displacement is continuous at the interface.Results for other electric boundary conditions［47］can be derived in a similar way.The electric field of ferroelectric composites is given as a uniform field under the in-plane electric loading and the permeable electric boundary condition，

whereω represents the direction ofelectric field with respect to the x1axis，and E is the magnitude of the electric field. The out-of-plane component of the electric field is zero.

Because of the orthogonality of tetragonal crystals，each ferroelectric mono-crystal possesses six types of probable orientations，as depicted in Fig.2.The initial poling orientation of ferroelectric mono-crystals is along the x3′axis in the local coordinate.Mode 1 to mode 4 represent four types of 90?domain switching，and mode 5 represents the type of 180?domain switching.Only 90?switching needs to be considered，because 180?switching can be divided into two proceduresof90?switching［48，49］.90?switch-induced strain and polarization in the mono-crystal can be described in the local Cartesian coordinates，as follows:

where εspis the spontaneous strain associated with 90?switching，and Psis the corresponding spontaneous polarization.The switch-induced strain and polarization in the global coordinate are given as，

Here standard index notation is utilized with summation implied over repeated indices.The superscripts of“α”indicate differentdomain switching modes.Mijis the coordinate transformation tensor between the global coordinate and the local coordinate， et al.［35］，we assume that the initial poling orientation is in the x1-x2plane for simplicity by setting θ=π/2 and ψ=π/4.

Then the work released by domain switching is given for these four 90?switching modes as，

The actual domain switching process should proceed to release the maximum work［50］.One obtains the actual domain switching work in the mono-crystal as，

Fig.3 Transformation between the local Cartesian coordinate of the mono-crystal and the global Cartesian coordinate of the cracked solid

where angles（θ，φ）represent the initial orientation of the poling axis with respect to the global Cartesian coordinate，and ψ denotes the rotating angle with respect to the poling axis，with the ranges of 0≤θ≤π，0≤φ≤2π，0≤ψ≤2π，as depicted in Fig.3.Similar to the manipulation of Xia where the symbol|·|denotes the absolute value，△εijand △Piare the actual switch-induced strain and polarization in the mono-crystal.Note that the second and third terms of WDSin Eq.（7）are the actual domain switching work driven by the electric field.In addition，the actual domain switching modes are given in Appendix 1 with respect to the initial poling orientation and the direction of the electric field.

3　Non-uniform domain switching criterion with electric field tuning

In the ferroelectric solid，only a part of ferroelectric crystals undergo the domain switching process［51］.The realistic switch-induced strain in the ferroelectric solid is written as，

where V90denotes the volume fraction of the part that experiences 90?domain switching.

Yang et al.［37］investigated the problem of a ferroelectric grain embedded in a homogeneous matrix to determine the value of V90.Following this framework，Cui and Yang［52］，and Cui and Zhong［38］established the energy-based nonuniform domain switching criterion forferroelectrics subjectto combined electromechanical loadings，which has been used successfully to predictthe domain switching in homogeneousferroelectrics［42，53］and bimaterialferroelectrics［35］through a similar procedure，as follows， Note thatσDSisthe unique controlvariable in the ferroelectricferroelastic domain switching，are the threshold values of σDSto trigger and saturate ferroelectric-ferroelastic domain switching［52］.Moreover，

The non-uniform ferroelectric-ferroelastic switching criterion in Eq.（9）is equivalent to

where W denotes the energy density for a certain value of V90［38］.Substituting the uniform electric field in Eq.（2）into Eq.（10），one obtains that

The electric field influences the domain switching process through perturbing the energy density level.Comparing Eq.（10）with Eq.（11），we modify the energy-based nonuniform domain switching criterion in Eq.（9）as，

where

the non-uniform ferroelectric-ferroelastic domain switching criterion in Eq.（12）can degenerate to the ferroelastic formulation［38］when the electric loading vanishes.

4　Domain switching zone with electric field tuning

4.1Electrically-dependent domain switching zone

Based on the non-uniform ferroelectric-ferroelastic domain switching criterion in Eq.（12），domain switching proceeds whenFor a certain initial poling orientation φ， one gets the contour of the electrically-dependent switching zone via setting

In addition，one obtainsthe maximumheightofthe switching zone Hmax（E，ω）by taking the partial derivative of the height H with respect to ?，

Fig.4 The contours of the domain switching zone around a stationary crack tip in the homogenous ferroelectric material versus the magnitude of electric field under different electric directions，when Kapp= 1 MPa·m1/2and φ=0，for a ω=0，b ω=π

4.2Switching zone geometry with electric field tuning

Combining Eqs.（12）and（14），we can visualize the electric field tuning on the domain switching process by comparison of switching zones under different electric loadings.

Firstly，we explore the domain switching zone around a stationary crack embedded in the homogeneous ferroelectric material，as shown in Fig.4.In plotting the contour，we set Kapp=1 MPa·m1/2and φ=0.Material constants are selected as“Material I”in Table 1.A symmetric switching geometry is achieved under the ferroelectric-ferroelastic domain switching criterion.Specifically，the size of the domain switching zone decreases with respect to the positive electric field（ω=0），while it increases with respect to the negative electric field（ω=π），which corresponds with the experimentalresults of Jiang etal.［54］.Next，the domain switching zone isinvestigated foran interfacialcrack embedded in ferroelectric composites.Material constants of the each constituentare listed in Table 1.An asymmetric domain switching zone is obtained due to the mismatch of bimaterial properties，as depicted in Fig.5.It implies that the electric loading can tune the switching zone geometry in homogeneous and bimaterial ferroelectrics.Note that the crack-tip stress field fulfils the non-oscillation condition for a mode III interfacial crack in an infinite body［44，55］.Therefore，the appearance of material II does not change the stress field in materialI［45］.Domain switching zones ofthe upper halfplane are identical for the homogeneous case（Fig.4a）and the bimaterial case（Fig.5）under the same external loading.

Table 1 Elastic and domain switching parameters of theferroelectric constituents when the electric field vanishes

Fig.5 The contoursofthe domain switching zone around an interfacial stationary crack tip in ferroelectric composites versus the magnitude of electric field，when Kapp=1 MPa·m1/2，ω=0，and φ=0

5　Non-uniform switch toughening with electric field tuning

5.1The weight function method

Utilizing the crack tip field and domain switching zone obtained in previous sections，we will explore the electric field tuning of switch toughening effect by evaluating the crack tip stress intensity factor（SIF），

where a toughness increment △K appears because of the transformation strain induced by the non-uniform ferroelectric-ferroelastic domain switching［56］.A stationary and a quasi-static growing interfacial crack will be analyzed in detail.

The weightfunction method isapplied here to quantify the interaction between the transformation strain and the interfacial crack.The switch toughening effect can be achieved by a surface integral through the domain switching zone［53］，

Here AIand AIIdenote the domain switching zones in the upper and lower half-planes，respectively，are the weightfunctions ofan anti-plane semi-infinite interfacial crack between dissimilar isotropic materials［55］，given by

where μIand μIIare the shear moduli of ferroelectric constituents，， and other components ofare zero.Owing to the piecewise property of V90，it is convenient to divide the toughness increment △K into three fractions［35］:the uniform switching fraction（denoted as the subscript“uni”），the transitional switching fraction（denoted as the subscript“tran”），and the saturated fraction（denoted as the subscript“sat”），

5.2Electrically-dependent switch toughening effect of a stationary crack

Our calculation begins with a stationary interfacial crack. Following the scheme above，we will evaluate these three fractions of toughening effect separately.Firstly，we concentrate on the uniform switching partin materialI.Substituting the domain switching zone in Eq.（14）into Eq.（18），we obtain

and the transitional switching fraction，

Similar conclusions can be obtained for material II.Finally，we obtain the toughening effect of a stationary interfacial crack by substituting Eqs.（21）-（23）into Eq.（20），

NI（E，ω），NII（E，ω），and ū are three coupling parameters defined to evaluate the influence of bimaterial properties. Note that NI（E，ω）and NII（E，ω）are functions of the magnitude and direction ofthe electric field，reflecting the electric field tuning offracture toughness in ferroelectric composites. On the contrary，ū is independent of the electric field.

Substituting the domain switching zone in Appendix 1 into Eq.（24），we arrive at，

It is concluded that the variation of stress intensity factor is identical to zero for a stationary interfacial crack in spite of the electric loading.The transformation toughening effect is equal to the transformation weakening effect for a stationary interfacial crack in bimaterial ferroelectrics，which is identical to the results of a stationary crack in homogeneous brittle materials［39，40］and ferroelastic materials［28，41，52］.

5.3Electrically-dependent switch toughening effect of a quasi-static growing crack

Next，we calculate the toughening effect of a quasi-static growing interfacial crack.The switching zone is divided into two parts:the front zone and the wake zone，

where the frontzone can be regarded as one partofthe switching area around a stationary crack，while the wake zone is a strip switching area that emerged when the interfacial crack propagates along the interface.

The switch toughening effect of the front zone is obtained by substituting the corresponding initial and final angles into Eq.（24），

Then the contribution of the wake zone is explored.Similarly，we also divide the wake zone into three fractions:the uniform one，the transitional one，and the saturated one，

where h denotes the height of the wake zone，and △a is the increment length of the interfacial crack，h/△a→0 means that the interfacial crack has reached a quasi-static growing condition.The area of the wake zone is an infinite horizontal strip with a finite height.The area elements of the integral are expressed as

Again，the calculation isperformed formaterialIfirst.Substituting Eq.（30）into Eq.（21）yieldsthe contribution ofuniform switching zone in material I，

In addition，we obtain the transitional switching fraction，

and the saturated switching fraction，

Substituting Eqs.（33）-（35）into Eq.（29）yields the switch toughening effect in the wake zone，

Finally，we obtain the crack tip SIF ofa quasi-static growing interfacial crack by substituting Eqs.（27），（28），and（36）into Eq.（17），

where

Note that Eq.（37）can degenerate to the formulation of ferroelastic domain switching［35］when the electric field vanishes.

Employing the stress intensity factor criterion，Ktip= KIIIc，we obtain the electrically-dependent critical applied SIF for a mode III quasi-static growing interfacial crack，

where KIIIcisthe mode IIIfracture toughness offerroelectric composites.

5.4Electrically-dependent multi-domain solution

Now，we concentrate on the ferroelectric composites consisting of domains in different orientations.An orientation distribution function f（φ）is introduced to describe the distribution of initial poling orientations，which is normalized byTwo categories of multi-domain poling configurations，i.e.，the non-poled and fully-poled ferroelectric composites，are investigated as below.

5.4.1Non-poled configuration

For ferroelectric composites in the non-poled configuration，a random orientation distribution function is involved，

Combining Eqs.（37）and（40），we obtain the switch toughening effect for non-poled ferroelectric composites，

5.4.2Fully-poled configuration

Forferroelectric composites in fully-poled configuration，the orientation distribution function［28］is considered as，

where ωpdenotes the poled direction.Similarly，we achieve the switch toughening effect for fully-poled ferroelectric composites by combining Eqs.（37）and（42），

In addition，the value of ?maxfor fully-poled ferroelectric composites is given in Appendix 2 with respect to the poled direction ωp，which is used in Eq.（43）to evaluate the switch toughening effect.

6　Numerical results and discussions

Numericalcalculationsare given inthissection on the electric field tuning of critical applied SIF by using specific material constants.The constituents of ferroelectric composites are selected as PIC 151 and another ferroelectric material，with material properties listed in Table 1.It is convenient to adopt dimensionless parameters in the plot.The normalized magnitude ofelectric field and the normalized criticalapplied SIF are defined as，

where the super-imposed tilde indicates the normalization，and Ec=1 MV/m is the coercive electric field［57］.The variation ofthe criticalapplied SIF is discussed in detailwith respect to the magnitude and direction of the electric field. The mono-domain and multi-domain solutions of a quasistatic growing crack are discussed as below.

6.1Electric field tuning in the mono-domain solution

Firstly，we explore the electric field tuning ofswitch toughening in the mono-domain solution.Figure 6 depicts the critical applied SIF，plotted against the magnitude and direction of the electric field under different initial poling orientations. The direction of the electric field can be divided into the toughening range，

Fig.6 Normalized critical applied stress intensity factors of monodomain quasi-static growing crack versus the normalized magnitude and direction of electric field under different initial poling orientations: a φ=0，b φ=π，c φ=7π/4

and the weakening range，

with respect to the initial poling orientation.The critical applied SIFincreases with the normalized magnitude ofelectric field when the direction of the electric field is within the toughening range，while it decreases with the normalized magnitude of electric field when the direction of the electric field is within the weakening range，as shown in Fig.6a-c. It is similar to the results in the homogeneous ferroelectrics through the phase-field simulation［58，59］.In addition，the optimal toughening and weakening directions of the electric field are given analytically with respect to the initial poling orientation as，

It implies that the electric field can be utilized to tune the fracture toughness of ferroelectric composites.

6.2Electric field tuning in the multi-domain solution

6.2.1Ferroelectric composites in non-poled configuration Then similar analysis is explored for the electric field tuning of switch toughening in the multi-domain solution.Figure 7 reveals the critical applied SIF in non-poled ferroelectric composites plotted againstthe magnitude ofthe electric field under different electric directions.Figure 7a shows the normalized criticalapplied SIF increasesmonotonously with the normalized magnitude of electric field.However，it varies periodically with respect to the direction of the electric field，as depicted in Fig.7b.The normalized critical applied SIF reaches the maximum in the direction of ω=π/4 or ω=5π/4，while it reaches the minimum in the direction of ω=3π/4 or ω=7π/4.

6.2.2Ferroelectric composites in fully-poled configuration Figure 8 shows thatthe criticalapplied SIF in fully-poled ferroelectric compositesvarieswith the magnitude and direction of the electric field under different poled directions.Similar to the case of the mono-domain solution，the direction of the electric field can also be divided into the toughening range and weakening range for fully-poled ferroelectric composites.The optimal toughening and weakening directions of the electric field vary with respect to the poled direction，as depicted in Fig.8a-c.These figures can provide guidance

to determine the optimal direction of the electric field for fully-poled ferroelectric composites.

7　Concluding remarks

This work deals with the tuning of fracture toughness of ferroelectric composites by an electric field.The tuning process is implemented by the non-uniform ferroelectricferroelastic domain switching near the interfacial crack tip.The electrically-dependent switch toughening effect is obtained for stationary and quasi-static growing interfacial cracks by employing the weight function method.Multidomain solutions are derived for non-poled and fully-poled ferroelectric composites.The conclusionsthatcan be reached from the analysis above are:

（1）The size of the domain switching zone varies with respect to the electric loading.However，no toughening effect exists for a stationary interfacial crack in spite of the electric loading.

Fig.7 Normalized criticalapplied stress intensity factors ofnon-poled quasi-staticgrowing crack versus a thenormalized magnitudeofelectric field，and b the direction of electric field

（2）The electric loading can toughen orweaken the fracture toughness of a quasi-static growing interfacial crack through the domain switching process.The optimal toughening and weakening directions of the electric field are derived analytically with respect to the initial poling orientation for a quasi-static growing interfacial crack in the mono-domain solution.

（3）For a quasi-static growing interfacial crack in nonpoled ferroelectric composites，the critical applied SIF increases monotonously with the magnitude ofthe electric field，and varies periodically with the direction of the electric field.For a quasi-static growing interfacial crack in fully-poled ferroelectric composites，the criticalapplied SIFvarieswith respectto the electric loading and poled direction.The electric field can be utilized to tune the fracture toughness of ferroelectric composites.

This work focuses on the possible application of an electric field in the tuning of fracture toughness of ferroelectric composites based on an assumption of elastic and dielectric isotropy.The influence of elastic and dielectric anisotropy will be further investigated in our future work.

Fig.8 Normalized critical applied stress intensity factors of a fullypoled quasi-static growing crack versus the normalized magnitude and direction ofthe electric field under differentpoled directions:a ωp=0，b ωp=π/2，c ωp=π

Acknowledgments Theprojectwassponsored by the NationalNatural Science Foundation of China（Grants 11090334，11572227）.

Appendix 1:Actualdomain switching modesfor ferroelectric mono-crystal with respect to the initial poling orientation and the direction of electric field

Each tetragonal ferroelectric mono-crystal possesses four types of probable 90?domain switching modes.The condition ofdetermines the actual domainswitching mode and the range ofdomain switching zone with respectto the poling orientationφ and the direction ofelectric field ω.

The actual domain switching mode is given under a positive electric field（0≤ω＜π）as

The actual domain switching mode is given under a negative electric field（π≤ω＜2π）as:

Appendix 2:The values of ?max for fully-poled ferroelectric composites with respect to the poled direction ωp

By taking the partial derivative to the height of domain switching zone H（E，ω）with respect to ?，we obtain the value of ?maxassociated with the maximum height of the domain switching zone Hmax（E，ω）:

1.Scott，J.F.:Prospectsforferroelectrics:2012-2022.ISRNMat.Sci. 2013，187313（2013）

2.Scott，J.F.:Applications of modern ferroelectrics.Science 315，954-959（2007）

3.Kim，S.，Bastani，Y.，Lu，H.D.，etal.:Directfabrication ofarbitraryshaped ferroelectric nanostructures on plastic，glass，and silicon substrates.Adv.Mater.23，3786-3790（2011）

4.Lu，H.，Bark，C.W.，de los Ojos，D.E.，et al.:Mechanical writing of ferroelectricpolarization.Science 336，59-61（2012）

5.Pritchard，J.，Bowen，C.R.，Lowrie，F(xiàn).:Multilayeractuators:review. Br.Ceram.Trans.100，265-273（2001）

6.H?usler，C.，Jelitto，H.，Neumeister，P.，et al.:Interfacial fracture of piezoelectric multilayer actuators under mechanical and electrical loading.Int.J.Fract.160，43-54（2009）

7.Furuta，A.，Uchino，K.:Dynamic observation of crack propagation in piezoelectric multilayeractuators.J.Am.Ceram.Soc.76，1615-1617（1993）

8.Chen，Y.H.，Hasebe，N.:Current understanding on fracture behaviors of ferroelectric/piezoelectric materials.J.Intell.Mater.Syst. Struct.16，673-687（2005）

9.Govorukha，V.，Kamlah，M.，Loboda，V.，et al.:Interface cracks in piezoelectric materials.Smart Mater.Struct.25，023001（2016）

10.Suo，Z.，Kuo，C.M.，Barnett，D.M.，et al.:Fracture mechanics for piezoelectric ceramics.J.Mech.Phys.Solids 40，739-765（1992）

11.Ru，C.Q.:Exactsolution forfinite electrode layersembedded atthe interface oftwo piezoelectrichalf-planes.J.Mech.Phys.Solids48，693-708（2000）

12.Gao，C.F.，Wang，M.Z.:Green’s functions of an interfacial crack between two dissimilar piezoelectric media.Int.J.Solids Struct. 38，5323-5334（2001）

13.Beom，H.G.:Permeable cracks between two dissimilar piezoelectric materials.Int.J.Solids Struct.40，6669-6679（2003）

14.Li，Q.，Chen，Y.H.:Analysisofapermeableinterfacecrack in elastic dielectric/piezoelectric bimaterials.Acta Mech.Sin.23，681-687（2007）

15.Ou，Z.C.，Wu，X.J.:On the crack-tip stresssingularity ofinterfacial cracks in transversely isotropic piezoelectric bimaterials.Int.J. Solids Struct.40，7499-7511（2003）

16.Narita，F(xiàn).，Shindo，Y.:Layered piezoelectric medium with interface crack under anti-plane shear.Theor.Appl.Fract.Mech.30，119-126（1998）

17.Soh，A.K.，F(xiàn)ang，D.N.，Lee，K.L.:Analysis of a bi-piezoelectric ceramic layerwith an interfacialcrack subjected to anti-plane shear and in-plane electric loading.Eur.J.Mech.A Solids 19，961-977（2000）

18.Li，X.F.，F(xiàn)an，T.Y.，Wu，X.F.:A moving mode-III crack atthe interface between two dissimilar piezoelectric materials.Int.J.Eng. Sci.38，1219-1234（2000）

19.Gao，C.F.，Wang，M.Z.:General treatment of mode III interfacial crack problems in piezoelectric materials.Arch.Appl.Mech.71，296-306（2001）

20.Li，R.，Kardomateas，G.A.:The mode III interface crack in piezoelectro-magneto-elastic dissimilar bimaterials.J.Appl.Mech.73，220-227（2006）

21.Li，X.F.，Liu，G.L.，Lee，K.Y.:Magnetoelectroelastic field induced by a crack terminating atthe interface ofa bi-magnetoelectric material.Philos.Mag.89，449-463（2009）

22.Calderon-Moreno，J.M.:Stress induced domain switching of PZT in compression tests.Mater.Sci.Eng.A 315，227-230（2001）

23.Fang，F(xiàn).，Yang，W.，Zhang，F(xiàn).C.，et al.:Domain structure evolution and fatigue cracking of001-oriented［Pb（Mg1/3Nb2/3）O3］0.67（PbTiO3）0.33ferroelectric single crystals under cyclic electric loading.Appl.Phys.Lett.91，081903（2007）

24.Evans，D.，Schilling，A.，Kumar，A.，et al.:Magnetic switching of ferroelectric domains at room temperature in multiferroic PZTFT. Nat.Commun.4，1534（2013）

25.Park，S.，Sun，C.T.:Fracture criteria for piezoelectric ceramics.J. Am.Ceram.Soc.78，1475-1480（1995）

26.Fu，R.，Zhang，T.Y.:Effects of an electric field on the fracture toughness of poled lead zirconate titanate ceramics.J.Am.Ceram. Soc.83，1215-1218（2000）

27.dos Santos e Lucato，S.L.，Bahr，H.A.，Pham，V.B.et al.:Crack deflection in piezoelectric ceramics.J.Eur.Ceram.Soc.23，1147-1156（2003）

28.Yang，W.，Zhu，T.:Switch-toughening of ferroelectrics subjected to electric fields.J.Mech.Phys.Solids 46，291-311（1998）

29.Landis，C.M.:Uncoupled，asymptotic mode III and mode E crack tip solutionsin non-linearferroelectricmaterials.Eng.Fract.Mech. 69，13-23（2002）

30.Wang，J.，Landis，C.:Effectsofin-plane electric fieldson the toughening behavior of ferroelectric ceramics.J.Mech.Mater.Struct.1，1075-1095（2006）

31.Liu，M.，Hsia，K.J.:Interfacial cracks between piezoelectric and elastic materials under in-plane electric loading.J.Mech.Phys. Solids 51，921-944（2003）

32.Liu，B.，F(xiàn)ang，D.N.:Domain-switching embedded nonlinear electromechanicalfinite elementmethod forferroelectric ceramics. Sci.China Phys.Mech.Astron.54，606-617（2011）

33.Qiao，H.，Wang，J.，Chen，W.Q.:Phase field simulation of domain switching in ferroelectric single crystalwith electrically permeable and impermeable cracks.Acta Mech.Solida Sin.25，1-8（2012）

34.Li，F(xiàn).X.，Sun，Y.，Rajapakse，R.:Effect of electric boundary conditions on crack propagation in ferroelectric ceramics.Acta Mech. Sin.30，153-160（2014）

35.Xia，X.D.，Cui，Y.Q.，Zhong，Z.:A mode III interfacial crack under nonuniform ferro-elastic domain switching.Theor.Appl.Fract. Mech.69，44-52（2014）

36.Zhu，T.，Yang，W.:Toughness variation of ferroelectrics by polarization switch under non-uniform electric field.Acta Mater.45，4695-4702（1997）

37.Yang，W.，Wang，H.T.，Cui，Y.Q.:Composite Eshelby model and domain band geometries of ferroelectric ceramics.Sci.China Ser. E 44，403-413（2001）

38.Cui，Y.Q.，Zhong，Z.:A novel criterion for nonuniform domain switching of tetragonal ferroelectrics.Mech.Mater.45，61-71（2012）

39.McMeeking，R.M.，Evans，A.G.:Mechanics of transformationtoughening in brittle materials.J.Am.Ceram.Soc.65，242-246（1982）

40.Lambropoulos，J.C.:Shear，shape and orientation effects in transformation toughening.Int.J.Solids Struct.22，1083-1106（1986）

41.Ru，C.Q.，Batra，R.C.:Toughening due to transformations induced by a crack tip stress field in ferroelastic materials.Int.J.Solids Struct.32，3289-3305（1995）

42.Cui，Y.Q.，Yang，W.:Electromechanical cracking in ferroelectrics driven by large scale domain switching.Sci.China Phys.Mech. Astron.54，957-965（2011）

43.Cui，Y.Q.，Zhong，Z.:Large scale domain switching around the tip ofan impermeable stationary crack in ferroelectric ceramics driven by near-coercive electric field.Sci.China Phys.Mech.Astron.54，121-126（2011）

44.Suo，Z.:Singularities，interfaces and cracks in dissimilar anisotropic media.Proc.R.Soc.Lond.A 427，331-358（1990）

45.Gao，H.，Abbudi，M.，Barnett，D.M.:Interfacial crack-tip field in anisotropic elastic solids.J.Mech.Phys.Solids40，393-416（1992）

46.Parton，V.Z.:Fracture mechanics of piezoelectric materials.Acta Astron.3，671-683（1976）

47.Wang，B.L.，Han，J.C.:Discussion on electromagnetic crack face boundary conditions for the fracture mechanics of magnetoelectro-elastic materials.Acta Mech.Sin.22，233-242（2006）

48.Hwang，S.C.，Lynch，C.S.，McMeeking，R.M.:Ferroelectric/ferroelastic interactions and a polarization switching model. Acta Metall.Mater.43，2073-2084（1995）

49.Jiang，B.，Bai，Y.，Chu，W.Y.，et al.:Direct observation of two 90 steps of 180 domain switching in BaTiO3single crystal under an antiparallel electric field.Appl.Phys.Lett.93，152905（2008）

50.Yang，W.，F(xiàn)ang，F(xiàn).，Tao，M.:Critical role of domain switching on the fracture toughness of poled ferroelectrics.Int.J.Solids Struct. 38，2203-2211（2001）

51.Hackemann，S.，Pfeiffer，W.:Domain switching in processzonesof PZT:characterization by microdiffraction and fracture mechanical methods.J.Eur.Ceram.Soc.23，141-151（2003）

52.Cui，Y.Q.，Yang，W.:Toughening under non-uniform ferro-elastic domain switching.Int.J.Solids Struct.43，4452-4464（2006）

53.Cui，Y.Q.，Zhong，Z.:Nonuniform ferroelastic domain switching driven by two-parameter crack tip stress field.Eng.Fract.Mech. 96，226-240（2012）

54.Jiang，Y.J.，F(xiàn)ang，D.N.，Li，F(xiàn).X.:In situ observation of electricfield-induced domain switching near a crack tip in poled 0.62PbMg1/3Nb2/3O3-0.38PbTiO3single crystal.Appl.Phys. Lett.90，222907（2007）

55.Gao，H.:Weightfunction method forinterface cracksin anisotropic bimaterials.Int.J.Fract.56，139-158（1992）

56.Kolleck，A.，Schneider，G.，Meschke，F(xiàn).:R-curve behavior of BaTiO3 and PZT ceramics under the influence of an electric field applied parallel to the crack front.Acta Mater.48，4099-4113（2000）

57.Zhou，D.，Kamlah，M.:Room-temperature creep ofsoft PZT under static electrical and compressive stress loading.Acta Mater.54，1389-1396（2006）

58.Wang，J.，Zhang，T.Y.:Phase field simulations of polarization switching-induced toughening in ferroelectric ceramics.Acta Mater.55，2465-2477（2007）

59.Abdollahi，A.，Arias，I.:Phase-field modeling of the coupled microstructure and fracture evolution in ferroelectric single crystals.Acta Mater.59，4733-4746（2011）

2 May 2016/Revised:26 May 2016/Accepted:12 June 2016/Published online:2 September 2016

?The Chinese Society of Theoretical and Applied Mechanics；Institute of Mechanics，Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016