Anomalies associated with energy release parameters for cracks in piezoelectric materials

Theoretical and Applied Fracture Mechanics 51 (2009) 102–110

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Anomalies associated with energy release parameters for cracks in piezoelectric materials T.J.C. Liu * Department of Mechanical Engineering, Ming Chi University of Technology, Taishan, Taipei County 243, Taiwan

a r t i c l e

i n f o

Article history: Available online 11 April 2009 Keywords: Piezoelectric Impermeable crack Finite element Intensity factor Energy release rate Energy density factor

a b s t r a c t For a central crack in a piezoelectric plate, the mode-I stress intensity factor (KI), electric displacement intensity factor (KD), energy release rates (G, GM) and energy density factor (S) are obtained from the finite element results. For the impermeable crack, the numerical results of KI and KD are coupled; this error is contrary to the uncoupled analytical solutions. The error has little effect on the total energy release rate G and energy density factor S, but in some cases, large errors in the mechanical energy release rate GM are observed. G is global while SED is local. Also G is negative which defies physics where energy cannot be created while crack attempts to extend as implied by G. Computations should be made for the J-integral and also show that J becomes negative. What this shows is that the global fracture energy criterion is not suitable to address the local release of energy because it includes the overall energy which are irrelevant to fracture initiation being a local behavior. In addition, the case study shows that the energy density theory is the better fracture criterion for the piezoelectric material. According to the results of S, it retards the crack growth when the external electric field and piezoelectric poling are on opposite directions. This conclusion agrees with analytical and experimental evidence in the past references. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction In modern electronic industries, piezoelectric material plays an important role in the production of sensors, actuators and smart structures. Therefore, the failure or fracture problem of the piezoelectric material becomes important. For the electro-elastic coupled field in the piezoelectric material, failure or crack propagation is induced by both mechanical and electrical loads. In the last decade, linear electro-elastic fracture mechanics and various crack problems of piezoelectric materials have received a lot of attention [1–8]. The crack tip field in the piezoelectric material also has r1/2 singularity [2–4]. Similar to traditional fracture mechanics, the stress intensity factors (KI, KII, KIII) and electric displacement intensity factor (KIV or KD) at the crack tip were defined as crack tip parameters [2–4]. Furthermore, the energy release rate criterion [4,7] or energy density criterion [8] must be adopted for the fracture criteria. These energy parameters can be computed from the intensity factors KI, KII, KIII and KD. When solving complicated crack problems, numerical methods such as finite elements are very important for the analyses. For the piezoelectric crack, a technique for obtaining the stress intensity factors and electric displacement intensity factor from the finite element results has been developed in Ref. [9]. Using the piezoelec* Tel.: +886 2 29089899x4569. E-mail address: [email protected] 0167-8442/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2009.04.007

tric quarter-point element (QPE), the nodal solutions of the displacements and electric potential near the crack tip can be substituted into formulas in Ref. [9] to obtain the intensity factors. In addition, many studies have applied the finite element analyses to piezoelectric cracks [10–12]. Using the technique of Ref. [9], this paper will also obtain intensity factors from the finite element results. For a central crack in a piezoelectric plate, the effects of the crack tip mesh on numerical errors will be discussed. The software ANSYS [13] is employed to establish finite element models and to calculate numerical results. Numerical results of this paper will be compared with those in Ref. [9] and with analytical solutions presented in Refs. [1–4]. Also, the energy release rates (G, GM) and energy density factors (S) will also be calculated to estimate the effects of numerical errors. Finally, a case study will be discussed for comparing fracture theories of G, GM and S. 2. Problem statements Fig. 1 shows a piezoelectric plate with a central crack for this study. The poling is along the +y-direction. It is remotely subjected to a uniform stress r0 and an electric displacement D0. This sample is a typical mode-I fracture problem. Assumptions for an impermeable or permeable crack can be considered on the crack surfaces [14,15]. When considering an impermeable crack, i.e. when the electric displacement cannot pass through the crack, the analytical

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9 8 rxx > > > > > > > > > > > ryy > > > > > > > > > > > > > > r zz > > > > > > > > > >s > > > > yz > > > = <

2

6 6 6 6 6 6 6 6 6 6 sxz ¼ 6 6 > > 6 > > > > 6 > > > > 6 s xy > > > > 6 > > > > 6 > > > 6 > > Dx > > > 6 > > > > 6 > > > > 6 D > > y > > 4 > > > > ; : Dz



Fig. 1. A piezoelectric plate with a central crack.

solutions of intensity factors for an infinite piezoelectric plane are as follows [1–4]:

K I ¼ r0

pffiffiffiffiffiffi pa

ð1Þ

pffiffiffiffiffiffi pa

ð2Þ

K D ¼ D0

It should be noted that KI and KD are uncoupled and independent of material properties. For limited permeable and fully permeable cracks, KI is unchanged and KD becomes [10,11]:

K D ¼ ðD0  DCy Þ

pffiffiffiffiffiffi pa

ð3Þ

where DCy represents the intrinsic crack surface charges. DCy can be calculated from Refs. [10–12]:

DCy ¼ jc

Y 42 r0 þ Y 44 ðD0  DCy Þ Y 22 r0 þ Y 24 ðD0  DCy Þ

ð4Þ

Above equation can be modified as follows 2

Y 24 DCy þ ðY 22 r0 þ Y 24 D0  jc Y 44 ÞDCy þ ðjc r0 Y 42 þ D0 jc Y 44 Þ ¼ 0 ð5Þ where jc is the permittivity of the matter inside the crack. The impermeable boundary conditions can be obtained for the limiting case jc = 0, i.e. DCy = 0. The limiting value of KD for the fully permeable crack can be calculated from Eqs. (3) and (4) by choosing the dielectric constant jc ? 1 [11]. In addition, the matrix YMN (M, N = 1, 2, 3, 4) for this study is [11]:

2

Y MN

1=cL 6 0 6 ¼6 4 0 0

0 1=cT

0 0

0

1=cA

1=d

0

3 0 1=d 7 7 7 0 5

c11

c13

c12

0

0

0

0

e31

0

c33

c13

0

0

0

0

e33

0

c11

0

0

0

0

e31

0

0

0

0

0

e15

c66

0

0

0

0

c44

e15

0

0

j11

0

0

j33

0

c44

sym:

3

j11 8 exx > > > > > > eyy > > > > > > > ezz > > > > > > c > > < yz

c

9 > > > > > > > > > > > > > > > > > > > > > =

xz > > > > > > > > > > c > > xy > > > > > > > > > E > > x> > > > > > > > > > Ey > > > > > > > > > ; : Ez

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

ð7Þ

where c66 ¼ 0:5ðc11  c12 Þ. It should be noted that Eq. (7) is transversely isotropic. The electro-elastic field quantities include the normal stress r, shear stress s, normal strain e, shear strain c, electric displacement D, and electric field E. The material constants c, j, and e are the elastic stiffness constant, permittivity, and piezoelectric constant, respectively. Table 2 lists the material constants of PZT-5H and BaTiO3 [9,11] used in Eq. (7). All material constants must be transformed into ANSYS input definitions. In addition, each permittivity in Table 2 must be converted into the relative permittivity for the input of ANSYS. Fig. 2 shows a typical finite element model created by ANSYS for the mode-I problem. Due to the symmetry, only half the plate is analyzed and symmetrical conditions are applied on the symmetrical plane. To simulate the infinite plane, the ratio a/W of the model is 1/50. The dimensions are: a = 0.01 m, W = 0.5 m and L = 0.5 m. The plane strain condition is adopted. In ANSYS, the element type for the piezoelectric problem is PLANE223, which is a plane element with eight nodes. PLANE223 elements can be modified to QPE’s around the crack tip to simulate the crack tip singularity. For the fully permeable crack in the finite element model, the crack surface condition is

/ðx; 0þ Þ ¼ /ðx; 0 Þ

ð8Þ

where / is the electric potential. The symbols ‘‘0+” and ‘‘0” are ycoordinates which denote the upper and lower crack surface, respectively. For the impermeable crack, no condition needs to be prescribed on the crack nodes, i.e. the crack is charge-free. 3. Numerical computations of intensity factors

ð6Þ

1=k

The terms cL, cT, cA, k and d are effective material constants [2,11]. The effective constants of the piezoelectric ceramic PZT5H and BaTiO3 with +y-axis poling are listed in Table 1 [9,11]. The constitutive relation of the piezoelectric ceramic with +y-axis (or +x2-axis) poling is [11]:

For finite element analyses, the study in Ref. [9] has developed formulas to obtain the stress intensity factor and electric displacement intensity factor, which are as follows [9]:

9 8 ½4ui ðl=4; pÞ  ui ðl; pÞ  3ui ð0; pÞ > > > > > >   a1i 1 < ½4ui ðl=4; pÞ  ui ðl; pÞ  3ui ð0; pÞ = ¼ > 2> a14 > > ½4/ðl=4; pÞ  /ðl; pÞ  3/ð0; pÞ > > ; : ½4/ðl=4; pÞ  /ðl; pÞ  3/ð0; pÞ

ð9Þ

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Table 1 Effective material constants of PZT-5H and BaTiO3 [9,11]. Properties

PZT-5H

BaTiO3

Elastic constants

cA = 35.4 GPa cL = 56.975 GPa cT = 62.244 GPa

cA = 44.75 GPa cL = 68.42 GPa cT = 72.24 GPa

Piezoelectric constants

d = 78.276 C/m2 8

Permittivity

k = 2.1843  10

d = 112.57 C/m2 k = 1.88  108 C/Vm

C/Vm

Table 2 Material constants of PZT-5H and BaTiO3 [9,11]. Properties

PZT-5H

Elastic constants

c11 = 126 GPa c13 = 53 GPa c44 = 35.3 GPa

c12 = 55 GPa c33 = 117 GPa

BaTiO3 c11 = 166 GPa c13 = 77.5 GPa c44 = 42.9 GPa

c12 = 76.6 GPa c33 = 162 GPa

Piezoelectric constants

e15 = 17 C/m2 e33 = 23.3 C/m2

e31 = 6.5 C/m2

e15 = 11.6 C/m2 e33 = 18.6 C/m2

e31 = 4.4 C/m2

Permittivity

j11 = 1.51  108 C/Vm j33 = 1.3  108 C/Vm

j11 = 1.4343  108 C/Vm j33 = 1.6823  108 C/Vm

Fig. 3. QPE’s around the crack tip (l = AC).

where i = 1, 2, 3. It should be noted that Eqs. (9) and (10) are only applicable under the condition that the poling is perpendicular to the crack [9]. u and / in polar coordinates (r, h) represent the displacement and electric potential at different locations of nodes. h = p and p denote the upper and lower crack faces, respectively. The term l is the length of QPE on the crack. Fig. 3 depicts six important nodes: A, B, C, A0 , B0 , and C0 in Eq. (9) [9]. Points A and A0 represent the same node at the crack tip. KI, KII and KIII are the stress intensity factors of the mode-I, mode-II and mode-III fractures, respectively. KD is the electric displacement intensity factor. The terms cL, cT, cA, k and d are effective material constants. According to finite element results, KI, KII, KIII and KD can be obtained from Eqs. (9) and (10). In the mode-I problem, only KI and KD need to be discussed. 4. Numerical results and discussions 4.1. Comparisons with analytical solutions and numerical results in Ref. [9]

Fig. 2. Finite element model. (a) Global view. (b) Local view.

2 9 8 cL K II > > > > rffiffiffiffiffi6 > > = < 6 KI p6 0 ¼ 6 > > 60 2l K III > > > > 4 ; : KD 0

0

0 2

cT d cT kþd2

0

0

cA 0

cT dk cT kþd2

3

9 8 a11 > > > > 7 cT dk = 7> 12 cT kþd2 7 7 > 0 7 > > a13 > > 5> ; : 2 a14  kd 2 0

cT kþd

ð10Þ

The finite element mesh in Fig. 2 was used for numerical calculations. The crack tip mesh and QPE’s are similar to those in Ref. [9] and the external loads: r0 = 1 MPa and D0 = 0.001 C/m2 [9] were considered. Table 3 lists the analytical and numerical results of the impermeable crack in PZT-5H and BaTiO3. A good agreement is seen with the analytical results and the error is within 2.5%. In addition, the fully permeable crack is also studied. The applied loads are r0 = 20 MPa and D0 = 0.02 C/m2 [11]. Table 4 lists the analytical and numerical results; they are again seen to be in good agreement.

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T.J.C. Liu / Theoretical and Applied Fracture Mechanics 51 (2009) 102–110 Table 3 Comparisons between analytical and numerical results of the impermeable crack (r0 = 1 MPa, D0 = 0.001 C/m2). Intensity factors pffiffiffiffiffi KI (MPa m) KD (C m3/2)

Analytical results from Eqs.(1) and (2)

Numerical results from Ref. [9] (PZT-5H)

Numerical results in this paper (PZT-5H)

Numerical results in this paper (BaTiO3)

1.7725  101 1.7725  104

1.7720  101 (0.03% error) 1.7720  104 (0.03% error)

1.7817  101 (0.5% error) 1.7776  104 (0.3% error)

1.8174  101 (2.5% error) 1.7762  104 (0.2% error)

Table 4 Comparisons between analytical and numerical results of the fully permeable crack (r0 = 20 MPa, D0 = 0.02 C/m2). Intensity factors pffiffiffiffiffi KI (MPa m) KD (C m3/2) (BaTiO3)a KD (C m3/2) (PZT-5H)b DCy b DCy a

Analytical results from Eqs.(1) and (3)

Numerical results from Ref. [11] (BaTiO3)

Numerical results in this paper (PZT-5H)

Numerical results in this paper (BaTiO3)

3.5449 5.9202  104 9.8921  104

3.6610 (3.3% error) 6.1150  104 (3.3% error)

3.5628 (0.5% error)

3.6142 (2.0% error) 6.0359  104 (2.0% error)

9.9421  104 (0.5% error) 2

= 0.0166599 for BaTiO3 under r0 = 20 MPa, D0 = 0.02 C/m . = 0.014419 for PZT-5H under r0 = 20 MPa, D0 = 0.02 C/m2.

Fig. 5. Variation of KD versus r0 for the impermeable crack under D0 = 0.

4.2. Numerically coupled results of KI and KD for the impermeable crack

Fig. 4. Electric displacement vectors near the impermeable crack (a), and the fully permeable crack (b) (unit: C/m2).

Fig. 4 shows the vector field of electric displacement near the crack under r0 = 1 MPa and D0 = 0.001 C/m2. The electric displacement cannot pass through the impermeable crack, but it can pass through the fully permeable crack. Both cases show the electric displacement concentration or singularity at the crack tip.

Equations (1)-(4) express the analytical solutions of the intensity factors. For the impermeable crack, the analytical solutions of KI and KD are uncoupled and independent of the material properties. Considering only a pure mechanical load r0 applied on the finite element model in Fig. 2, the impermeable crack and PZT-5H are adopted. The results in Fig. 5 show the variation of KD versus different values of r0. The numerical values of KD are not zero and depend on the mechanical load r0. This result violates the uncoupled analytical solutions given in Eqs. (1) and (2). On the other hand, by considering only the pure electrical load D0 for the impermeable crack in PZT-5H, Fig. 6 shows the variation of KI versus different values of D0. This result also violates the uncoupled analytical solutions in Eqs. (1) and (2). In summary, incorrect numerical results show coupled KI and KD for the impermeable crack. When the crack conditions for the fully permeable case are considered, from Table 5 which shows the results of KI versus different D0 under r0 = 20 MPa, good agreement between analytical and numerical solutions can be seen.

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T.J.C. Liu / Theoretical and Applied Fracture Mechanics 51 (2009) 102–110

factors. In these cases, KD should be zero according to the analytical solution in Eq. (2). From the numerical results in Table 6, KD still has non-zero values which decrease when the mesh is refined and the value of KI approaches its analytical solution gradually. KD may approach zero when the crack tip mesh is refined enough, however, a too fine mesh of triangular QPE’s may cause element distortion problems and other errors. In Table 6, the values of KD seem to be small under pure mechanical loading. It is not confirmed that the small KD or numerically coupled errors can be ignored. It may affect the scales of energy parameters such as G, GM and S. This topic will be discussed in next two sections. 4.4. Energy release rate

Fig. 6. Variation of KI versus D0 for the impermeable crack under r0 = 0. Table 5 Variation of KI versus different values of D0 for the fully permeable crack (PZT-5H, r0 = 20 MPa). pffiffiffiffiffi pffiffiffiffiffi D0 (C/m2) KI (analytical) (MPa m) KI (numerical) (MPa m) Errors (%) 0 0.0001 0.001 0.02 0.1 1 10 20 100

3.5449

3.5628 3.5628 3.5628 3.5628 3.5628 3.5628 3.5628 3.5628 3.5628

if jxj < a

r0 x

: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi if jxj > a x2  a2

ð11Þ

and

Dy ðx; 0Þ ¼

8 <0

if jxj < a D0 x : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi if jxj > a x2  a2

1 K 2I K 2II K 2III K I K D þ þ þ 2 cT cL cA d

!

ð12Þ

It should be noted that ry and Dy are only uncoupled along the line y = 0. This result leads to the uncoupled intensity factors given in Eqs. (1) and (2). The numerical errors of coupled KI and KD in Figs. 5 and 6 may reveal whether the finite element size near the crack plane (y = 0) is not small enough. The pure mechanical load r0 = 1 MPa applied on the PZT-5H cracked plate and the impermeable crack were also considered. The local mesh around the crack tip was modified to investigate the errors. Fig. 7 shows four different types of the crack tip mesh. Table 6 shows the effects of the crack tip mesh on the intensity

ð13Þ

!

1 K 2I K 2II K 2III K I K D K I K D K 2D G ¼ GM þ GE ¼ þ þ þ þ  2 cT cL cA d d k

The results of the impermeable crack in Figs. 5 and 6 violate the uncoupled analytical solutions in Eqs. (1) and (2). In fact, there are the same coupled errors in Table 3 even when it shows good agreement. These unusual results may be a result of the finite element mesh around the crack tip. According to Sosa’s analytical solutions for the impermeable crack [3], the stress ry and electric displacement Dy along the line y = 0 (crack plane) are

ry ðx; 0Þ ¼

GM ¼

1 K I K D K 2D GE ¼  2 d k

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

4.3. Effects of the crack tip mesh for the impermeable crack

8 <0

When a proper crack tip mesh such as that shown in Fig. 2 (or Type C in Fig. 7) must be used due to the computational costs involved, the coupled KI and KD of the impermeable crack cannot be avoided. Numerically coupled effects on the fracture criteria are discussed in this section. To estimate the crack propagation in the piezoelectric material, the energy release rate can be adopted for the fracture criterion [4,7,10]. The total energy release rate G can be computed from the intensity factors KI, KII, KIII and KD as follows [10]:

ð14Þ ! ð15Þ

where GM and GE are the mechanical and electrical energy release rates, respectively. The above equations are only adapted for the condition that the poling is perpendicular to the crack [9,10]. In this study, KII = KIII = 0 for the mode-I problem. The mesh in Fig. 2 is used again for this discussion. There are eight loading cases for the impermeable crack in the PZT-5H plate: (1) r0 = 106 Pa and D0 = 103 C/m2; (2) r0 = 106 Pa and D0 = 106 C/ m2; (3) r0 = 103 Pa and D0 = 103 C/m2; (4) r0 = 103 Pa and D0 = 106 C/m2; (5) r0 = 106 Pa and D0 = 103 C/m2; (6) r0 = 106 Pa and D0 = 106 C/m2; (7) r0 = 103 Pa and D0 = 103 C/ m2; (8) r0 = 103 Pa and D0 = 106 C/m2. Table 7 shows the numerical results for KI, KD and G. There are large errors of intensity factors in loading cases 2, 3, 6 and 7. In these cases, the changes of the scales of r0 and D0 reveal the errors of numerically coupled effects. It should be noted that for the total energy release rates, there is still good agreement between analytical and numerical results and the maximum error is 2.6%. According to the calculation in Eq. (15), numerically coupled errors of KI and KD have little effect on the values of total energy release rates. If the total energy release rate G is used for the fracture criterion, the numerically coupled errors of KI and KD for the impermeable crack can be ignored in the analysis. On the other hand, the mechanical energy release rate GM is proposed for the fracture criterion of the piezoelectric material [4,7]. Table 7 also shows analytical and numerical results for the calculation of GM. The large error in GM induced by the error in KI is 22.5% for loading case 3. Other loading cases are discussed for GM under r0 = 105 Pa. Table 8 shows results of GM with different electrical loads and numerically accurate results are observed when the loading ratio |r0/ D0| P 107 N/C. With small loading ratios, there are large errors caused by numerically coupled effects.

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T.J.C. Liu / Theoretical and Applied Fracture Mechanics 51 (2009) 102–110

Fig. 7. Four types of the crack tip mesh.

Table 6 Effects of the crack tip mesh on numerical results of the impermeable crack (PZT-5H, r0 = 1 MPa, D0 = 0). Mesh type

A B C D

a/l

Numbers of QPE’s

10 20 37 50

8 12 32 64

Analytical results pffiffiffiffiffi KI (MPa m)

KD (C m3/2)

1.7725  101

0

Numerical results pffiffiffiffiffi KI (MPa m)

KD (C m3/2)

KI (%)

Errors KD (%)

1.8131  101 1.7943  101 1.7813  101 1.7785  101

1.8713  107 1.6414  107 6.7606  108 5.0333  108

2.3 1.2 0.5 0.3

– – – –

Table 7 Intensity factors and energy release rates for the impermeable crack in PZT-5H. Loading cases

Analytical results pffiffiffiffiffi KD (C m3/2) KI (Pa m)

G (N/m)

GM (N/m)

Numerical results pffiffiffiffiffi KI (Pa m) KD (C m3/2)

G (N/m)

GM (N/m)

KI (%)

KD (%)

G (%)

GM (%)

1 2 3 4 5 6 7 8

1.7725  105 1.7725  105 1.7725  102 1.7725  102 1.7725  105 1.7725  105 1.7725  102 1.7725  102

6.5425  102 2.5278  101 7.1877  101 6.5425  108 8.6812  101 2.5196  101 7.1953  101 8.6812  107

4.5306  101 2.5258  101 2.0094  104 4.5306  107 5.1687  102 2.5216  101 2.0042  104 5.1687  108

1.7817  105 1.7813  105 2.1655  102 1.7817  102 1.7809  105 1.7813  105 1.3970  102 1.7809  102

6.3732  102 2.5544  101 7.2229  101 6.3732  108 8.7159  101 2.5463  101 7.2310  101 8.7159  107

4.5729  101 2.5516  101 2.4618  104 4.5729  107 5.2706  102 2.5475  101 1.5841  104 5.2706  108

0.5 0.5 22.2 0.5 0.5 0.5 21.2 0.5

0.3 38.4 0.2 0.3 0.2 37.9 0.2 0.2

2.6 1.1 0.5 2.6 0.4 1.1 0.5 0.4

0.9 1.0 22.5 0.9 2.0 1.0 21.0 2.0

1.7725  104 1.7725  107 1.7725  104 1.7725  107 1.7725  104 1.7725  107 1.7725  104 1.7725  107

4.5. Energy density factor The energy density criterion is also usually adopted to be the piezoelectric fracture criterion [8,16–18]. For the in-plane crack problems, the energy density factor S of the piezoelectric crack can be computed as follows [8]:

1.7776  104 2.4530  107 1.7769  104 1.7776  107 1.7763  104 1.1009  107 1.7769  104 1.7763  107

Errors

S ¼ A11 K 2I þ A22 K 2II þ A44 K 2D þ 2A12 K I K II þ 2A14 K I K D þ 2A24 K II K D ð16Þ where the material coefficients Aij are functions of h, i.e. the local polar coordinate at the crack tip. The forms of Aij can be found in Ref. [8]. In this paper, KII = KIII = 0 and S is reduced to

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T.J.C. Liu / Theoretical and Applied Fracture Mechanics 51 (2009) 102–110

Table 8 Mechanical energy release rates for the impermeable crack in PZT-5H (r0 = 105 Pa). Electrical load D0 (C/m2)

Analytical results of GM (N/m) 3

0 108 106 104 102 100 102 104 106 108 106 102 100 102 106

2.5496  10 2.5498  103 2.5698  103 4.5729  103 2.0921  101 6.3859  101 4.3829  105 4.3629  109 4.3627  1013 4.3627  1017 2.5293  103 1.9538  101 2.3400  101 4.3425  105 4.3627  1013

Table 9 Bij at h = h0 = 0° of PZT-5H [16]. Coefficient

B11

B14

B44

Value

4.050  1012

67.92  1012

5123.0  1012

g 33 g2 þ A44 33 ; b33 b233

B44 ¼

A44 b233

;

B14 ¼

A14 g þ A44 33 b33 b233

ð18Þ

g 33 ¼

1 ½2c13 e31 ðc11  c12 Þ  ðc211  c212 Þe33  detðBÞ

ð19Þ

b33 ¼

1 ½2c2 ðc11  c12 Þ  ðc211  c212 Þe33  detðBÞ 13

ð20Þ

with

  c11  c  12 detðBÞ ¼   c13  e 31

c12

c13

c11

c13

c13 e31

c33 e33

1.0 1.0 1.0 0.9 3.0 218 Very Very Very Very 1.0 1.4 Very Very Very

large large large large

large large large

Table 10 shows the numerical results for KI, KD and Smin. Similar to Table 7, there is good agreement between analytical and numerical results of the energy density factors Smin. In Table 11, good agreements can also be investigated under different electrical loads or loading ratios. When S is used for the fracture criterion, the numerically coupled errors of KI and KD for the impermeable crack can be ignored.

ð17Þ

In addition, the coefficients Aij are related to Bij as follows [8]:

B11 ¼ A11 þ 2A14

Errors (%)

3

2.5236  10 2.5238  103 2.5437  103 4.5304  103 2.0320  101 2.0070  101 2.0067  103 2.0067  105 2.0067  107 2.0067  109 2.5035  103 1.9815  101 2.0065  101 2.0067  103 2.0067  107

S ¼ A11 K 2I þ A44 K 2D þ 2A14 K I K D

Numerical results of GM (N/m)

 e31   e31   e33  j33 

ð21Þ

For the pure mode-I problem in this paper, the crack propagates along the line of h = h0 = 0°, where the minimum energy density factor Smin locates [16]. For the condition that the poling is perpendicular to the crack, the values of Bij at h0 = 0° of PZT-5H are listed in Table 9 [16]. The impermeable crack assumption, finite element mesh and eight loading cases of Section 4.4 are used again for discussions.

4.6. Case study for fracture criteria To compare three kinds of fracture criteria, G, GM and S, for the impermeable and fully permeable cracks in piezoelectric materials, a case study is considered in this section. A smaller PZT-5H plate with L = W = 0.05 m is studied here. The half crack length is a = 0.001 m. The mesh type in Fig. 7c is also used for the crack tip modeling. The mechanical load is r0 = 105 Pa. The electric potential V0 is applied on the bottom surface (electrode) of the plate in Fig. 1. The upper surface (electrode) is subjected to zero electric potential (0 V). It is more practical to use V0 in the analyses because the voltage difference can be easily realized in the experiments. Figs. 8 and 9 depict the results of the impermeable crack under V0 = 400–400 V. It shows that the energy parameters or crack growth are affected by altering the direction of external electrical load. From the results of Smin in Fig. 8, the value of Smin at V0 = 200 V is larger than that at V0 = 200 V. It retards the crack growth when the external electric field and piezoelectric poling are on opposite directions. This conclusion agrees qualitatively with analytical results in Refs. [16–20]. In Ref. [7], the experimental evidence of the cracked piezoelectric plate with positive poling shows that a positive electric field tends to enhance the crack growth while a negative electric field impedes it. This evidence supports the conclusion from the energy density theory for the impermeable crack in this paper. In Fig. 9, the total energy release rates G show negative values in some conditions. Similar to those in Refs. [7,19], the negative

Table 10 Intensity factors and energy density factors for the impermeable crack in PZT-5H. Loading cases

1 2 3 4 5 6 7 8

Analytical results pffiffiffiffiffi KI (Pa m)

KD (C m3/2)

Smin (N/m)

Numerical results pffiffiffiffiffi KI (Pa m)

KD (C m3/2)

Smin (N/m)

Smin (%)

Errors

1.7725  105 1.7725  105 1.7725  102 1.7725  102 1.7725  105 1.7725  105 1.7725  102 1.7725  102

1.7725  104 1.7725  107 1.7725  104 1.7725  107 1.7725  104 1.7725  107 1.7725  104 1.7725  107

1.4806  101 9.9139  102 4.3305  102 1.4806  107 1.3680  101 9.9128  102 4.3294  102 1.3680  107

1.7817  105 1.7813  105 2.1655  102 1.7817  102 1.7809  105 1.7813  105 1.3970  102 1.7809  102

1.7776  104 2.4530  107 1.7769  104 1.7776  107 1.7763  104 1.1009  107 1.7769  104 1.7763  107

1.4939  101 1.0013  101 4.3521  102 1.4939  107 1.3789  101 1.0012  101 4.3510  102 1.3789  107

0.9 1.0 0.5 0.9 0.8 1.0 0.5 0.8

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T.J.C. Liu / Theoretical and Applied Fracture Mechanics 51 (2009) 102–110 Table 11 Energy density factors for the impermeable crack in PZT-5H (r0 = 105 Pa). Electrical load D0 (C/m2) 0 108 106 104 102 100 102 104 106 108 106 102 100 102 106

Analytical results of Smin (N/m) 4

9.9128  10 9.9129  104 9.9189  104 1.4805  103 4.3363  100 4.3297  104 4.3297  108 4.3297  1012 4.3297  1016 4.3297  1020 9.9076  104 4.3250  100 4.3296  104 4.3297  108 4.3297  1016

Numerical results of Smin (N/m) 3

1.0012  10 1.0012  103 1.0018  103 1.4938  103 4.3585  100 4.3518  104 4.3518  108 4.3518  1012 4.3518  1016 4.3518  1020 1.0007  103 4.3470  100 4.3517  104 4.3518  108 4.3518  1016

Errors (%) 1.0 1.0 1.0 0.9 0.5 0.5 0.5 0.5 0.5 0.5 1.0 0.5 0.5 0.5 0.5

values are unphysical and the results contradict with experimental observation related to crack growth enhancement and retardation. Due to above reasons, the mechanical energy release rate GM is proposed in Ref. [7]. Although the parameter GM in Fig. 9 seems to perform more rational results to predict the crack growth, GM cannot be adopted for pure electrical loading (i.e. r0 = 0, D0 – 0). In Eq. (13), the value of GM becomes zero due to r0 = 0 and KI = KII = KIII = 0. According to the results from Refs. [21,22], it demonstrated that the case of pure electrical loading can cause the dielectric breakdown and fracture in the piezoelectric ceramics. The GM criterion cannot describe the fracture behavior under pure electrical loadings or relatively small mechanical loadings [21,22]. For the fully permeable crack in this case study, Smin and G do not depend on the electrical loads V0. The constant values of Smin and G keep as 1.0513  104 N/m and 3.1149  104 N/m, respectively. Above results agree with those in Refs. [19,23] and would contradict experimental findings in Ref. [7]. If the fully permeable crack assumption is used, the crack growth cannot be predicted under different electrical loads. Detail discussions can be found in Refs. [14,23]. Fig. 8. Variations of Smin versus V0 for the impermeable crack.

5. Conclusions

Fig. 9. Variations of G and GM versus V0 for the impermeable crack.

Finite element analyses and its numerical errors of stress intensity factor (KI), electric displacement intensity factor (KD), total energy release rate (G), mechanical energy release rate (GM) and energy density factor (S) of the piezoelectric crack have been obtained and discussed in this paper. For the impermeable crack, a remarkable error of coupled KI and KD was observed. This error violates the analytical solution that KI and KD are uncoupled. Above error has little effect on G and S, but in some cases, large errors in GM were observed. Some unphysical and contradictory results were obtained when the theories of G and GM were used in the case study. The G and GM criteria cannot describe properly the fracture behavior in the piezoelectric materials. On the other hand, if the fully permeable crack assumption is used, the crack growth cannot be predicted under different electrical loads. According to the results of S for the impermeable crack in the case study, it retards the crack growth when the external electric field and piezoelectric poling are on opposite directions. This conclusion agrees qualitatively with analytical results [16–20] and experimental evidence [7]. In addition, when the energy density factor S is used for the fracture criterion, the numerically coupled errors of KI and KD for the impermeable crack can be ignored.

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