phase field simulations of a permeable crack parallel to the original polarization direction in a...

Engineering Fracture Mechanics 75 (2008) 4886–4897

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Engineering Fracture Mechanics

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Phase field simulations of a permeable crack parallel to the originalpolarization direction in a ferroelectric mono-domain

Jie Wang, Tong-Yi Zhang *

Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

a r t i c l e i n f o

Article history:Available online 3 July 2008

0013-7944/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.engfracmech.2008.06.025

* Corresponding author. Tel.: +852 2358 7192; faE-mail address: [email protected] (T.-Y. Zhang).

a b s t r a c t

Two-dimensional phase field simulations of polarization switching-induced tougheningof an electrically permeable crack parallel to the original polarization direction in a ferro-electric mono-domain were conducted based on the time-dependent Ginzburg–Landauequation. The mono-domain represented a small vicinity region of the crick tip and thecrack was loaded by nominal applied mechanical and/or electrical tip fields. The simulationresults show that a wing-shaped switched zone is formed forwards from the crack tipunder applied loading, which is different from the backward wing-shaped switched zonefor a permeable crack perpendicular to the original polarization direction. The mechanicalload to induce a same size of switched zone for a crack parallel to the original polarizationis much lower than that for a crack perpendicular to the original polarization direction.Consequently, the polarization switching-induced toughening for a crack parallel to theoriginal polarization, which was characterized by the local J-integral, behaves differentlyfrom that of a permeable crack perpendicular to the original polarization direction.

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1. Introduction

In a previous publication [10], polarization switching near the tip of an electrically permeable crack perpendicular to theoriginal polarization direction in a mono-domain ferroelectric was simulated by using a phase field model, which consideredthe polarization gradient energy and the long-range mechanical and electrical interactions between polarizations. The majoradvantage of the phase field simulations lies in that the polarization switching is the result of minimizing the total free en-ergy of the simulated system, which does not need any pre-described switching criteria. For the electrical permeable crackperpendicular to the original polarization direction, the switched zone has a wing shape backwards from the crack tip. Thepolarizations in the switched wing above the crack switch 90� clockwise, while the polarizations in the wing below the crackswitch 90� anticlockwise. There are electric domain walls between the switched and un-switched zones. All polarizationsfollow the head-to-tail arrangements due to the polarization gradient energy and the long-range electric interaction energy.The internal stress field induced by polarizations in the simulations was calculated by the thermal-stress-like method [10].Thus, the polarization-switched zone likes a plastic zone in the elastic–plastic fracture mechanics. In this case, J-integral ispurely mechanical and path-independent as long as the integration contour does not pass through the switched zone. LocalJ-integral was numerically calculated and used as a fracture criterion. Base on the fracture criterion, the simulations [10]illustrate that an applied uniform electric field parallel to the original polarization direction reduces the apparent fracturetoughness, while an applied uniform electric field anti-parallel to the original polarization direction enhances it.

As mentioned in the previous work [10], the fracture behavior of ferroelectric ceramics is complex under mechanical and/or electric loading. Fracture toughness of ferroelectric single crystals and poled ferroelectrics has different values along

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J. Wang, T.-Y. Zhang / Engineering Fracture Mechanics 75 (2008) 4886–4897 4887

different orientations [4], showing an anisotropic nature. In addition, the effect of applied electric fields on the fracturebehavior is also anisotropic. For example, Tobin and Pak [9] reported that an applied electrical field changed the crack lengthof an indentation crack if the crack was perpendicular to the poling direction, while the applied field did not change the cracklength if the crack was parallel to the poling direction. However, Wang and Singh [11] reported that an applied electrical fieldchanged the crack length of an indentation crack no matter the crack was perpendicular or parallel to the poling direction.The complex failure behavior of ferroelectric ceramics is attributed to the coherent nonlinear coupling between the mechan-ical and electric fields. Under mechanical and/or electrical loading, the intensified stress and electric fields in the vicinity of acrack may cause polarization switching, thereby changing the local electrical domain structure. The change in the local do-main structure, in turn, changes the internal electric field near the crack tip and the internal stress field because spontaneousstrains are accompanied with polarizations. The switching-induced internal stress field may, depending on the nature of theinternal stress, shield or anti-shield the crack tip from the applied mechanical load, resulting in switching-toughening orswitching-weakening. Switching-toughening may be regarded as a kind of phase-transformation-toughening. Followingthe classical phase-transformation-toughening theory [3], theoretical studies have been conducted to understand and pre-dict the polarization switching-induced toughening under mechanical and/or electrical loading, which has been summarizedin the review articles [13,12]. Recently, Cui and Yang [1] studied the effect of non-uniform ferroelastic domain switching inthe vicinity of a crack by using an evolution law for the volume fraction of the switched portion under applied electrome-chanical loading. Su and Landis [7] proposed a continuum thermodynamics framework to model the evolution of ferroelec-tric domain structures. They considered the general elastic, piezoelectric and dielectric energies of a ferroelectric materialnear its spontaneously polarized state and their approach was able to yield the Ginzburg–Landau equation. Based on thethermodynamics framework, they derived a principle of virtual work and a finite element formulation. With the developedtheory and numerical methods, Su and Landis [7] investigated the fields near straight 180� and 90� domain walls and deter-mined the electromechanical pinning strength of an array of line charges on the domain walls.

In the theoretical research on fracture of ferroelectric ceramics, two approximate electric boundary conditions are com-monly adopted for an electric insulating crack, namely the electrically permeable and impermeable boundary conditions[13]. Song et al. [8] conducted phase field simulations of polarization switching near an electrically impermeable cracktip under purely mechanical or electrical loading. However, the experimental measurements of electric field distributionsin a pre-cracked ferroelectric ceramic sample [5] indicate that the electrically permeable boundary condition may be moreappropriate. That is why we are focusing on electrically permeable cracks. Following our previous work [10], we simulate thepolarization switching in the vicinity of an electrically permeable crack parallel to the original polarization direction undermechanical and electrical loading.

The present work, studying an electrically permeable crack parallel to the original polarization direction, is a naturalextension of the previous work [10]. Therefore, the theoretical framework and the methodology are the same as those de-scribed in the previous work [10]. For readers’ convenience, we briefly repeat some illustrations and equations here.

2. Simulation methodology

In phase-field simulations of ferroelectrics [2,14,10], it is usually assumed that the mechanical and electric equilibrium beestablished instantaneously once a spontaneous polarization distribution is set down. The spontaneous polarization vector,P = (P1, P2, P3), is used as the order parameter to calculate thermodynamic energies of the ferroelectric phase in the Landauphase transformation theory. The temporal evolution of the spontaneous polarization field is described by the time-depen-dent Ginzburg–Landau equation,

oPiðr; tÞot

¼ �LdF

dPiðr; tÞði ¼ 1;2;3Þ; ð1Þ

where L is the kinetic coefficient, F is the total free energy, dF/dPi(r, t) represents the thermodynamic driving force of the spa-tial and temporal evolution of the simulated system, r = (x1, x2, x3) is the spatial vector, and t denotes time. The total freeenergy includes the standard Landau–Devonshire energy, the polarization gradient energy, the depolarization energy, andthe electrical energy density due to an applied electric field. The standard Landau–Devonshire energy density is given by

fLDðPi;rijÞ ¼ a1ðP21 þ P2

2 þ P23Þ þ a11ðP4

1 þ P42 þ P4

3Þ þ a12ðP21P2

2 þ P22P2

3 þ P21P2

3Þþ a111ðP6

1 þ P62 þ P6

3Þ þ a112½ðP41ðP

22 þ P2

3Þ þ P42ðP

21 þ P2

3Þ þ P43ðP

21 þ P2

2Þ�

þ a123P21P2

2P23 �

12

s11ðr211 þ r2

22 þ r233Þ � s12ðr11r22 þ r22r33 þ r11r33Þ ð2Þ

� 12

s44ðr212 þ r2

23 þ r213Þ � Q 11ðr11P2

1 þ r22P22 þ r33P2

3Þ � Q 12½r11ðP22 þ P2

3Þ

þ r22ðP21 þ P2

3Þ þ r33ðP21 þ P2

2Þ� � Q 44ðr12P1P2 þ r13P1P3 þ r23P2P3Þ;

where a1 = (T � T0)/2e0C0 is the dielectric stiffness, a11, a12, a111, a112, a113, are higher order dielectric stiffnesses, T and T0

denote temperature and the Curie–Weiss temperature, respectively, C0 is the Curie constant; sij are the elastic compliancecoefficients, Qij are electrostrictive constants and rij denote mechanical stresses. Note that mechanical stresses include ap-plied and internal stresses induced by spontaneous strains.

4888 J. Wang, T.-Y. Zhang / Engineering Fracture Mechanics 75 (2008) 4886–4897

In the Ginzburg–Landau theory, the free energy function also depends on the gradient of the order parameter. For ferro-electric materials, the polarization gradient energy represents also the domain wall energy. For simplicity, the lowest orderof the gradient energy density is used here, which takes the form:

fGðPi;jÞ ¼12

G11ðP21;1 þ P2

2;2 þ P23;3Þ þ G12ðP1;1P2;2 þ P2;2P3;3 þ P1;1P3;3Þ

þ 12

G44½ðP1;2 þ P2;1Þ2 þ ðP2;3 þ P3;2Þ2 þ ðP1;3 þ P3;1Þ2� ð3Þ

þ 12

G044½ðP1;2 � P2;1Þ2 þ ðP2;3 � P3;2Þ2 þ ðP1;3 � P3;1Þ2�;

where G11, G12, G44, and G044 are gradient energy coefficients, and Pi,j denotes the derivative of the ith component of the polar-ization vector, Pi, with respect to the jth coordinate and i, j = 1, 2, 3.

To describe the long-range electrical interactions between different domains during the switching, the total free energyshould include depolarization energy induced by spatially inhomogeneous spontaneous polarizations. The depolarizationenergy is a self-electrostatic energy corresponding to the long-range electrostatic interaction of spontaneous polarizationsand is calculated by

fdep ¼ �12ðEd

1P1 þ Ed2P2 þ Ed

3P3Þ; ð4Þ

where Ed1, Ed

2 and Ed3 are the components of depolarization field along the x1, x2 and x3 axes, respectively. The self-electrostatic

field is obtained by solving the electrostatic equilibrium equation by using the finite difference method for a given polari-zation distribution and pre-described boundary conditions. If an externally electric field, Ea

i , is applied to the system, the ap-plied field generates an additional electrical energy density,

felec ¼ �Eai Pi: ð5Þ

Integrating all free energy densities over the entire volume of a simulated ferroelectric material yields the total freeenergy, F, of the simulated ferroelectric material:

F ¼Z

V½fLDðPi;rijÞ þ fGðPi;jÞ þ fdepðPi; E

di Þ þ felecðPi; E

ai Þ�dV ; ð6Þ

where V denotes the volume of the simulated ferroelectric material. Substituting the total free energy of Eq. (6) into the evo-lution equation of Eq. (1) and solving it numerically, the temporal evolution of polarization can be obtained. The finite dif-ference method for spatial derivatives and the Runge–Kutta method of order four for temporal derivatives are employed tosolve Eq. (1) in real space.

Two-dimensional simulations with plane strain condition along the third direction are conducted in the present study.We consider an electrically permeable crack lying inside an infinite ferroelectric single crystal under remote electric and/or mechanical loading, as shown in Fig. 1a. Only a small rectangle area near the crack tip is taken as the simulated system,as shown in Fig. 1b. In the two-dimensional simulations, a uniform electric field, Ea

i and/or a KI tip stress field are applied tothe simulated system. The mode I tip stresses under a given Kapp are expressed by [6]

ra11 ¼

Kappffiffiffiffiffiffiffiffiffi2prp Re

s1s2

s1 � s2

s2

ðcos hþ s2 sin hÞ1=2 �s1

ðcos hþ s1 sin hÞ1=2

!" #;

ra22 ¼


1s1 � s2

s1

ðcos hþ s2 sin hÞ1=2 �s2


!" #; ð7Þ

ra12 ¼


s1s2

s1 � s2

1

ðcos hþ s1 sin hÞ1=2 �1


!" #;

where Kapp denotes the applied mode I stress intensity factor. In Eq. (7), r and h denote the distance from the crack tip and thepolar angle, respectively, as shown in Fig. 1b, s1 and s2 are two unequal complex roots with positive imaginary parts of thecharacteristic equation

b11s4i þ ðb12 þ b44Þs2

i þ b11 ¼ 0; ð8Þ

in which

b11 ¼ s11 �s2

12

s11; b12 ¼ s12 �

s212

s11; b44 ¼ s44; ð9Þ

and s11, s12 and s44 are material compliance constants. The mono-domain state without any loads is taken as reference state.When there are no mechanical and/or electrical loads, the elastic stresses and electric field are zero in the simulated systemand the mono-domain state is stable because the electrically permeable crack is treated to be electrically perfect andmechanical defect. If no polarization switching takes place under applied mechanical and/or electrical loads, mechanical

Crack

)(

22

∞aσ

a

rθ

2x1x

P0

a

11σ a

22σ a

12σ

aiE

)(2

∞aE

)(1

∞aE

b

Crack

Fig. 1. Schematic illustration of an electrically permeable crack in a mono-domain ferroelectric: (a) crack in infinite medium under combined remoteloadings and (b) simulated crack tip area subjected to applied K-field stresses of mode I and a uniform electric field of Ea

i , where the poling direction P0 isparallel to the crack.


stresses and electric field in the system are only the raij and Ea

i . However, induced stresses, rinij ; and induced electric field,

Edi , are generated by inhomogeneous distributions of spontaneous strains and spontaneous polarizations if polarization

switching takes place. Therefore, stresses rij in Eq. (2) include two parts: the applied K-field, raij, and the induced stresses,

rinij , i.e.,

rij ¼ raij þ rin

ij : ð10Þ

Following the thermal stress approach, elastic strains are generated when polarization switching occurs in a perfectmono-domain ferroelectric material, which are calculated by

eij ¼ einij � De0

ij; ð11Þ

where De0ij ¼ Q ijklðPkPl � P0

kP0l Þ is the change of spontaneous strain after switching, Qijkl are the electrostrictive coefficients,

P0i is the initial polarization and ein

ij are strains excluding the applied elastic strain, which must be compatible and are definedby

einij ¼

12ðuin

i;j þ uinj;iÞ; ð12Þ

in which ui are displacements. In linear elasticity, stresses are related to elastic strains through Hooke’s law:

rinij ¼ cijklekl ¼ cijklðein

ij � De0ijÞ: ð13Þ


Without any body forces, the mechanical equilibrium equations are expressed by rij,j = 0. Since the applied stress fieldmeets the mechanical equilibrium condition, the internal stress field rin

ij should also satisfy the following static mechanicalequilibrium equation

rinij;j ¼ 0; ð14Þ

which is solved by the finite element method under fixed dimensions.

Fig. 2. Polarization distribution under (a) K�app ¼ 180, (b) K�app ¼ 270 and (c) K�app ¼ 360; without any applied electric field.


The same dimensionless variables and materials parameters as those used in the previous paper [10] are employed in thepresent simulations. The phase field simulations are conducted with fixed dimensions of the simulated rectangular regionbecause it represented an interior part of a large ferroelectric single crystal. This means that the size of the simulated regionis fixed after accommodating the displacements caused by applied mechanical loads. For the same reason, the boundary con-dition, dP/dn = 0, is used along the four edges of the simulated region in solving the spontaneous polarization field of Eq. (1).The boundary conditions along the crack surfaces are electrically permeable and mechanically traction-free.

3. Simulation results and discussion

In the simulations, we use 241 � 180 discrete grids for the rectangle region shown in Fig. 1b with a cell size ofDx�1 ¼ Dx�2 ¼ 1. The superscript * denotes dimensionless variables. Before applying any electric and mechanical loads, thesimulated system is assumed to be a single electric domain with its polarization direction parallel to the crack direction.In the finite element analysis for Eq. (14), the whole rectangle including the crack in Fig. 1b is meshed by 241 � 180 rectan-gular elements with each element of Dx�1 ¼ Dx�2 ¼ 1; which are consistent with the discrete grids in the simulations. The elec-trically permeable crack is electrically perfect but mechanically defect. We set the elastic constants of the elements of thecrack area to be zero in the finite element analysis to model the mechanical defect.

Polarizations in the mono-domain are set to be originally along the x1 positive direction with the initial values of polar-ization of P�1 ¼ 1 and P�2 ¼ 0. The crack is set to be one-element in width and only 60 elements in the crack length are used torepresent the crack tip. The iteration step is 0.04 in the dimensionless time and the polarization distribution become steadyafter 2000 steps of iterations. Here, we report most simulation results after 2000 steps of iterations except for the polariza-tion switching process.

Fig. 2a–c show the polarization distribution under purely applied stress intensity factors of K�app ¼ 180, K�app ¼ 270 andK�app ¼ 360, respectively, without any applied electric field, where the solid line indicates the crack. Under K�app ¼ 180,

50 nm 60nm 70nm 80nm 90nm90nm

100nm

110nm

120nm

130nm

140nm

150nm

crack

Fig. 4. The detailed polarization distribution near the crack tip for the case shown in Fig. 2b.

50 nm 54 nm 58 nm 62 nm 66 nm 70 nm

110nm

114nm

118nm

122nm

126nm

130nm

crack

Fig. 3. The detailed polarization distribution near the crack tip for the case shown in Fig. 2a.


some spontaneous polarizations near the crack tip rotate by angles less than 45�, as shown in Fig. 3, but no distinctswitched zone is formed. A switched zone is formed under a high applied mechanical load, as shown in Fig. 2b forK�app ¼ 270 and in Fig. 2c for K�app ¼ 360. Obviously, the larger the applied mechanical load is, the larger the switched zonewill be. The results indicate that a threshold of applied stress intensity factor might be needed to cause the polarization

55nm 60nm 65nm 70nm 75nm 80nm

105nm

110nm

115nm

120nm

125nm

130nm

135nm

140nm

50nm 55nm 60nm 65nm 70nm 75nm 80nm 85nm 90nm

90 nm

100nm

110nm

120nm

130nm

140nm

150nm

50 nm 60 nm 70 nm 80 nm 90 nm 100nm80nm

90nm

100nm

110nm

120nm

130nm

140nm

150nm

160nm

50 nm 60 nm 70nm 80nm 90nm 100nm 110nm70nm

80nm

90nm

100nm

110nm

120nm

130nm

140nm

150nm

160nm

50 nm 60 nm 70nm 80nm 90nm 100nm 110nm 120nm60 nm

80 nm

100 nm

120 nm

140 nm

160nm

180nm

50nm 60 nm 70nm 80nm 90nm 100nm 110nm50nm

70nm

90nm

110nm

130nm

150nm

170nm

crack

n=1000 n=2000

n=500 n=700c d

e f

n=100 n=300a b

Fig. 5. Temporal evolution of polarization switching at different time steps, n, under K�app ¼ 300 without any applied electric field.


switching at the crack tip. Of course, the value of the threshold stress intensity factor depends on what criterion is usedto gauge the switching.

The shape of the switched zone with two forward wings is almost symmetric with respect to the crack, but theswitching direction is asymmetric. The polarizations in the wing above the crack switch 90� clockwise, while the polar-izations in the wing below the crack switch 90� anticlockwise. To reduce the polarization gradient energy, the polari-zation orientations change gradually from the un-switched region to the switched region, thereby resulting in domainwalls between the two regions, as illustrated in detail in Fig. 4. The wing-shaped switched zone is forward from thecrack tip, which is different from the backward switched zone for the crack perpendicular to the original polarizationdirection [10]. Under the mechanical load of K�app ¼ 270, the switched zone has a wing length of about 40 grids, asshown in Fig. 2b. To have the same wing length for the crack perpendicular to the original polarization direction,the applied mechanical has to reach the level of K�app ¼ 500 [10], which is almost twice of the applied load ofK�app ¼ 270 for the crack parallel to the original polarization direction. The difference in the polarization switchingbehavior for cracks with different orientations to the original polarization direction exhibits the switchinganisotropy.

Fig. 5a–f illustrate the temporal evolution of polarization switching at iteration steps of n = 100, 300, 500, 700, 1000and 2000 under a purely mechanical load of K�app ¼ 300. To show clearly the detailed polarization structure, the scale isdifferent in the figures, as indicated by the grid numbers on the horizontal and vertical axes in each figure. The polar-ization switching is nucleated at the crack tip, as indicated in Fig. 5a for n = 100. Polarizations at the crack tip switchabout 90� anticlockwise above the crack and about 90� clockwise below the crack and their adjacent polarizations switchabout 90� clockwise above the crack and about 90� anticlockwise below the crack, thereby changing about 180� in polar-ization between them. The switched zone grows up in both the wing length direction and the wing width direction bymore polarizations switching about 90� clockwise above the crack and switching about 90� anticlockwise below the crackas the iteration step increases. Fig. 5b and c show polarization structures after 300 and 500 step evolutions, respectively.Obviously, the switched zone grows up faster in the wing length direction than that in the wing width direction, leadingto the wing shape of the switched zone. Furthermore, the switched degree of the polarization inside the wing is moreclose to 90� after 500 evolutions than that after 300 step evolutions. At the time steps of 700 and 1000, The switchedzone continues growing up in its length direction after evolution steps of 700 and 1000, as shown in Fig. 5d and e. Fi-nally, the switched zone grows to its steady state after 2000 step evolutions. The switched zone looks like a pair of wingsforwards the crack tip, as shown by Fig. 5f.

Polarization switching induces internal stress, which can shield and anti-shield the crack tip from applied loading. Toinvestigate the effect of switching-induced internal stress on the fracture behavior of simulated ferroelectrics, we calcu-lated the local J-integral from J ¼

Rsðrijeij=2Þdx2 � Ti

@ui@x1

ds with Ti = rijnj, in which eij, ui and nj are the total stains, totaldisplacements and cosines of unit outward normal vector, respectively. The local J-integral contours around the cracktip used in the present study are the same as those used in the previous study [10]. The numerical calculations indicatethat the local J-integral is independent of the contour size, as described in detail in the previous paper [10]. Fig. 6 showsthat the local J-integral after switching increases monotonically with the applied J-integral, which can be calculated fromthe applied K value, under purely mechanical loading. The applied J-integral is also called the J-integral before switching.This behavior indicates that a purely mechanical load can eventually fracture a ferroelectric sample with the use of localJ-integral as the failure criterion even polarization switching occurs at the crack tip. The similar result has been found forcracks perpendicular to the original polarization direction. The local J-integral after switching is lower than its corre-sponding applied J-integral under purely mechanical loading. For example, the local J-integral after switching is 53.96,higher that the applied value of 49.83, which corresponds to K�app ¼ 300. The polarization switching-induced internal

0 20 40 60 800

20

40

60

80

J*

before

J* afte

r

Fig. 6. Local J-integral after switching versus local J-integral before switching without any electric field.


stress shields the crack tip from applied mechanical loads and toughens the material. The shielding strength for cracksparallel to the original polarization direction is lower than that for cracks perpendicular to the original polarization direc-tion. For example, under the applied J = 80, the local J-integrals are 51.26 and 71.18, respectively, for cracks perpendicularand parallel to the original polarization direction.

Jointly with a mechanical load, an external uniform electric field has a crucial effect on the polarization switching near thecrack tip. In the simulations, the combined mechanical and electrical loading was simultaneously applied on the original

Fig. 7. Polarization distribution under (a) Ea�

1 ¼ �0:5 and K�app ¼ 300; (b) Ea�

1 ¼ 0 and K�app ¼ 300, and (c) Ea�

1 ¼ 0:5 and K�app ¼ 300.

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60.80

0.85

0.90

0.95

1.00

J *

before=53.96

J* afte

r / J

* befo

re

Electric field, E1

a *

Fig. 8. Normalized local J-integral versus Ea�

1


mono-domain. Then, iterations were carried out until a steady state was reached. Fig. 7a–c show the polarization patternsunder uniform electric fields of Ea�

1 ¼ �0:5, Ea�

1 ¼ 0 and Ea�

1 ¼ 0:5, respectively, where the mechanical load of K�app ¼ 300remained the same for the three cases. The switched zone reduces its size as Ea�

1 changes from �0.5 to 0, to 0.5. The resultsindicate that an electric field anti-parallel to the original polarization direction makes the switching easier and increases thesize of the switched zone, while a parallel electric field makes the switching harder and decreases the size of the switchedzone. The effect of an applied electric field on polarization switching at the parallel crack tip is the same as that at the per-pendicular crack tip [10]. The shape of the switched zone under combined mechanical and electrical loading remains thesame as that under purely mechanical loading, as shown in Fig. 7. The change in the switched zone size changes the shieldingbehavior. Fig. 8 shows the variation of local J-integral with the applied electric field parallel or anti-parallel to the originalpolarization direction when the mechanical load is K�app ¼ 300. A positive or negative electric field weakens or strengthensthe shielding effect caused by the mechanical load. The trend of the influence of a parallel or anti-parallel electric field on thelocal J-integrals is the same as that for the perpendicular cracks [10]. However, the shielding effect for the parallel cracks ismuch weaker in comparison with the shielding effect for the perpendicular cracks. The present and previous results suggestthat the effect of polarization switching on the shielding is dependent on the crack orientation with respect to the originalpolarization direction.

The polarization switching under an applied uniform electric field perpendicular to the original polarization directionis also simulated with a given applied stress field. Fig. 9 shows polarization distributions under different perpendicularelectric fields of Ea�

2 ¼ �0:05, Ea�

2 ¼ 0 and Ea�

2 ¼ 0:05, respectively, when the applied mechanical load is K�app ¼ 300. An ap-plied perpendicular electric field has an asymmetric influence on the switched zone. An electric field anti-parallel to thex2 positive direction makes the switched zone below the crack disappear, whereas an electric field parallel to the x2 po-sitive direction makes the switched zone above the crack disappear, as shown in Fig. 9a and c, respectively. The similarphenomenon was observed for the perpendicular cracks, see Fig. 13 in the previous work [10]. Fig. 10 illustrates the var-iation of local J-integral with an applied electric field perpendicular to the original polarization direction. Although thereare some changes in the local J-integral, no distinctive difference is caused by the electric field perpendicular to the ori-ginal polarization direction. This behavior is also the same as that for cracks perpendicular to the original polarizationdirection.

4. Concluding remarks

As an extension of the previous work [10], the present work simulates polarization switching-induced toughening ofan electrically permeable crack parallel to the original polarization direction in a mono-domain ferroelectric material byusing a phase field model and local J-integral. The phase field simulations show that the location of the switched zone isstrongly dependent on the crack orientation with respect to the original polarization direction. The wing-shaped switchedzone is backwards from the crack tip and located behind the crack tip for a permeable crack perpendicular to the originalpolarization direction, while the wing-shaped switched zone is forwards from the crack tip and located in front of thecrack tip for a permeable crack parallel to the original polarization direction. Although polarization switching occursmuch easier at the tip of a parallel crack, the shielding strength is lower in comparison with the cases for a perpendicularcrack.

To simplify the simulations, the crack width is set to be only one element. In this case, it is difficult to calculate the electricfield inside the crack in the numerical simulations if the electric property of the crack interior is different from the electric

Fig. 9. Polarization distribution under (a) Ea�

2 ¼ �0:05 and K�app ¼ 300, (b) Ea�

2 ¼ 0 and K�app ¼ 300, and (c) Ea�

2 ¼ 0:05 and K�app ¼ 300.


property of the material. That is why the crack is treated to be electrically perfect and mechanically defect in the presentwork. In the near future, we shall simulate a general crack by setting few elements as the crack width, which will allowus to investigate the effect of the crack interior property on the polarization switching-induced toughening of ferroelectricceramics.

-0.2 -0.1 0.0 0.1 0.2

0.6

0.8

1.0

1.2

J*

before=53.96

J* afte

r/J* be

fore

Electric field, E2

a *

Fig. 10. Normalized local J-integral versus Ea�

2 .


Acknowledgement

This work was fully supported by a Direct Allocation Grant, DAG05/06. EG35, from the Research Grants Council of theHong Kong Special Administrative Region, China.

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phase field simulations of a permeable crack parallel to the original polarization direction in a...

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